THE ^— " MESSENGER OF MATHEMATICS. EDITED BY J. W. L. GLAISHEE, Sc.D., F.R.S., FELLOW OF TRINITY COLLEGE, CAMBRIDGE. VOL. XLV. [May 1915— April 1916]. D aranttirilrge : BOWES & BOWES. iLontton: MACMILLAN & CO. Ltd. aSlasgotD: JAMES MACLEHOSE & SONS. 1916. CONTENTS OF VOL. XLV. '> On certain solutions of Maxwell's equations. By H. Bateman On certain infinite series. By S. Ramanujan . . . The expansion of the square of a Bessel function in the form of a series of Bessel functions. By A. E. Jolliffe - - - ,- Some properties of the tetrahedron and its spheres. By T. C. Lewis Determinants whose elements are alternating numbers. By T. MuiR An inequality associated with the Gamma function. By G. N. Watson Notes on a differential equation. By G. W. Walker A set of criteria for exact derivatives. By E. B. Elliott ... A transformation of central motion. By R. Hargrf.aves . . . On the steady motion of fluid under gravity. By W. Burnside A transformation of the partial differential equation of the second order. By J. R. Wilton . . . Factorisation of A'=(r-^+l) and (Z^-^ + Y^^ ). By Lt.-Col. Allan Cunningham Note on class relation formulas. By L. J. Mordell .... Some formulse in the analytic theory of numbers. By S. Ramanttjan An interpretation of pentaspherical coordinates. By T. C. Lewis - The Brocard and Lemoine circles. By T. C. Lewis .... Successive transforms of an operator with respect to a given operator. By G. A. Miller Systems of particles equimomental with a uniform tetrahedron. By E. B. Neville Note on an elimination. By E. J. Nanson Time and electromagnetism. By Prof. H. Bateman ... A condition for the validity of Taylor's expansion. By T. \V. Chaundy PAGE 1 11 IS 17 21 23 31 33 39 43 46 - 49 - 7G 81 85 89 92 94 96 97 115 IV CONTENTS. Note on the primary minors of a circulant having a vanishing sum of elements. By SiR Thomas MuiR 120 The twisted cubic of constant torsion. By W. H. Salmon - ... 125 On Briggs's process for the repeated extraction of square roots. By J. W. L. Glai.shhii 129 Determinants of cyclically repeated arrays. By Sirt '1'homas MuiR - - 142 Solution of a problem in linear diophantine approximation. By VV. E. H. Bhuwick 154 On equipotential curves as possible free paths. By S Brodictsky - - 1(31 Criteria for exact derivatives. By T. VV. Chaundv 168 An arithmetical proof of a class relation formula. By L. J. Moudbll - 177 Note on the zeros of Riemann's ^-function. By J. R. Wilton . . - 180 Determinants of cyclically repeated arrays. By Pkof. W. Burnsidu - 183 FaJitorisation of i\'^=(a;;':^2/'^). By Lt. -Col. Allan CnKNiNGHAM - - 185 Erratum— p. 46, last line, omit " = 1." MESSENGER OF MATHEMATICS. ON CERTAIN SOLUTIONS OF MAXWELL'S EQUATIONS. By H. Bateman. Vector-fields with moving singular curves. §1.1 HAVE shown elsewhere* that it is possible to obtain a solution of Maxwell's equations which represents a vector field in which the electric and magnetic intensities are infinite at a moving point Q, whose coordinates at time a are ^, 17, ^, and also along a moving curve attached to this point; the curve being the locus of a series of points projected from the different positions of Q^ and travelling along straight lines with the velocity of light. The direction of projection for any position of Q was chosen so that it made an angle 6 with the tangent to the path of Q such that cco^6 = v, wiiere v is the velocity of Q, and c the velocity of light. This condition is, however, not invariant under the transformations of the theory of relativity, and I now find that it is not necessary to restrict the direction of pi'ojection in the way described ; the introduction of the restriction was due to the mistaken idea that the second of equations (291) f is a consequence of the first. Let a and ^ be defined as before by the equations {oi^-iy'r{y-vy-^{^-^f=c'{t-rr (i), l{x-l) + m{y-'n) + n[z-^) = c'p{t-r) (2), * The Mathematical Analysis of Electrical and Optical Wave Motion on the basis of AfaxivelV s Equations, p. 128. This will be cited later as E. t E, p. 129. The error in the proof occurs when the axis of y is chosen so that n, = 0; this introduces a restriction, since /j.m,.w, are generally complex. The same error occurs in one of my previous papers. Annals of Mathematics (1914). VOL. XLV. B 2 J//\ Bateman, Certain solutions of Maxwell's equations. where ^, 77, ^, t are functions of a only, and ?, m, n, p are functions of a and ^ which depend linearly on /3, so that 1 = 1^1^-1^^, m = /3m,-»i„, n = /3»,-»,, p = ^p^-i), ...[?>), To make the values of a and 13 unique, we write T = a and introduce the inequality t </. The quantities /, w, «, p must, moreover, he chosen so that r -\- vr -\- n' = c' p\ and so we have the relations We now use the symbol / to denote an arbitrary function of a and jS, and write P=^[x-l) + v'[U-'n)+K'{^-K)-<i'[t-a)....{b). Tlie vector field which will be the subject of discussion is that defined by the electro-magnetic potentials J Jl A ='^ A=^ 4> = '€ (6). These have been shown to be wave-functions which satisfy the relation c dt On calculating the components of the electric and magnetic intensities with the aid of the relations rr , , p \ dA d<^ c 6t dx we find, as before, that the component of the electric intensity along the radius from ^, 77, ^, a to a;, y, z, t is To make the electric charge associated with the singularity ^, 7;, ^, a a constant quantity 47r, we choosey so that nt, however, to have a vs !Q shall introduce the con ^Vi~ ''i^'~ '",»?'- ",^'= 0 ; It will be convenient, however, to have a value of ^ indepen- dent of /3, and so we shall introduce the condition Mr. Bateman, Certain solutions of MaxivelVs equations. 3 as before, the value ofy is then given bj the equation* We shall suppose that /„, w^, ?V i^ ^'"^ ''^^'' theny'is real and /,, 7??,, ??,, p^ are generally complex quantities. Let 7,, «?,, /7|, j^^ be the conjugate complex quantities, then we have the relations j^ow let a set of real quantities I^, m^, 7?^, p^ be chosen so that /«'+ »'o'+ "o'= C'A'. Vf + '"o«^. + "n^o - C>oZ^o= ^'^ O) Then if we write x — ^'—X, y — r} — Y, z — ^— Z, t — o. = T, c'p.T- J^X- m, Y-n^Z= S, cyj- l^X-mJ-n^Z= ^ \ ep^T-\X-mJ-n^Z^ U, c%T-lX-mJ-n,Z^ W (10), we find that, if li is suitably chosen, there is an identity of the type _ „ SS-VU=h{c''T-X'-Y'-Z') = 0 (11), where Jc is a function of a whose value may be determined by replacing T by 1, X by t,\ Yhy r}\ and Z by ^' in the identity. We thus find that fk=l^ + m,v^\K'-c% (12). By considering the relations satisfied by /,, m^, /;,, p,, we see that ^' = \l^+fil^, 7]'=\m^ + nm^, ^'=X7i^ + fj,n^, l=Xp^ + fip^...{l3), where \ and /j, are quantities to be determined. We deduce at once from these equations that * It should be noticed that when this condition is satisfied the field specified by potentials of type A^" = p + p is conjugate to the field specified by Lienard s potentials of type /I j;'=^- and the relation A:c''Aj:' + Ay''AJ + A^'A/-^''i>' =0 is satisfied. Compare this with the remark E, p. 135. 4 Mr. Bateman, Certain solutions of MaxwdVs equations. Hence 2\ = ^ -^" -^^ ~" =/ (14). The expression for A^ can now be thrown into a more convenient form. Differentiating equation (2) we obtain also ^S= U, hence we may write A -l^-^ '~ Fdx 'Ud8_dJT S dot. 9a + Now let A^ be the complex quantity conjugate to A^, then m + iLrhoJ^]. A =-Ci" and so if 2a^ = ^^ + ^^, we Lave ^/8a = pax Now differentiate the identity (11) with regard to a, keeping x, y, z, t constant, we obtain Substituting In (16), making use of the relation UU= SS, we find that 2a —- /8a Pdx Ufd, (U 8_ 8a ,,^, [tJLU\ 2A'P1 Ufd. (fiU\ 7i^t^8~.8~a^^''^'^i'a^S'«^''^^^^'i^ac 'Ut^j' J/r. Bateman^ Certain solutions ofMaxwelVs equations. 5 Now it follows at once from (13) that \U+ /mU = —P, and 2X=f, hence we have 2a. fiPUdxdoL Pdxda^ 2kfdoi + W^ ,- //"f^^ V -bx Pd -J^^(b) No d 3^,(lo.P)4 + 3^,f,(Io.^P], consequently our expression for A^ can be written in the form «,= - ^ + — ex P ^^-iHi.u) lu + dx (logP) XU d , ffiU\ J\Iaking- use of the value of fk given by (12) and using djda to denote a differentiation with regard to a when X, Y, Z, and T are regarded as constant^ we obtain the simple formula r da. d J- dF " P dx da. ^ dx ;i7), where i^ is a function Avhose exact form is not needed. The corresponding expressions for a^, a^, and ^ may be written down by analogy. Separating tiie expressions for a^, a , a^, (p into two parts, we obtain terms of the type —^jP representing the potentials of an electro-magnetic field with an electric charge iir associated with the point singularitv (s, Vi ^1 a.), and also terms of type 0 da. d —- dF .. A ° = — - ^ bgf/+,r— (18). dx da ^ dx ^ ^ These will be regarded as the potentials of an elementary cethereal field whose singular curve is obtained by putting U= 0. We easily find that U= 0 when and only when L ^0 Z T ,(19). b2 6 Mr. Bateman, Certain solutions ofMaxtvelVs equations. The singular curve is thus built up of points which travel along straight lines with the velocity of light ; moreover, there is no restriction on the direction ot" projection, for we can make Z^, wj^, and n^ arbitrary functions of a and construct the corresponding expressions of type (18) without having to determine /^, m^, n^, jf?^, for X U= — P— /j, f/, and /j, is determined by the equation 2/x (7„r + m^v' + nX " cj,) = V + v"' + K"- c\ It should be noticed that if the direction of projection does \ dU . , , . . not varv with a, ^= — is either zero, or a function ot a, and U da. then the components of the electric and magnetic intensities in the {ethereal field defined by (18) are null. Hence the ctthereal field exists onhj in regions of space and time corres- pondiiifj to values of a for tvltich the direction of i^rojection of the singularities varies with the time.* It" the direction of projection is originally constant, then varies for a short interval, and finally becomes constant again, the singular curve of the sBthereal field will at any instant be of finite length. With regard to the directions of projection specified by /„, m^, ??^, 2>o and l^, Wi^, «^, p^^ it follows at once from (13) that they lie in a plane containing the tangent to the path of Q. If, moreover, we draw a sphere of radius c having the point Q as centre, and measure a length u in the direction of the motion of Q so as to obtain a point F for which QV=v, the points on the sphere which correspond to the directions /^, m^, «,^, 7„, ¥i^, «^|, will lie on a line through V. This is an immediate consequence of equations (4), (8), and (9), which indicate that the tangent planes at the points f — , — ^ , -^ J , ( I m n \ • 1- ■''^'' 1 -^^^ ^0 ( =? , ~, z^\ of the sphere intersect m a line which meets ^Po Pu VJ (I tn 11 \ (I m. « \ the sphere in two points -L, — , — , [=^, ^, ~], which ' V;^, p, pj \p, p, pj lie on the polar plane of Fon account of the relations of type * Cf. E, p. 130, where equations (292) should read jj ' J p ' p > p The signs need correction in the succeeding equations, also in the equations on pp. 117 and 118. 31r, Bateman, Certain solutiojis ofMaxiceWs equations. 7 Expressions for the electric and magnetic intensities. §2. Since we may evidently write ^^o 1-1 -, dm. _ _ c7^ = ^'. + ^ '. + "^h 5 -^g^ = ^'". + ^ "' . + »?"^« , and two similar equations, hence wliere e and rj are functions of a and the symbol M is used to denote the real part of the quantity which follows. Now, if ^ is the complex quantity conjugate to ^, we have f:iS= U, and we find that the components of the magnetic force II are _ _ a (?/, 2) ' " d {z, X)' ' d {X, y) On account of the characteristic properties* of the functions a and /3, we may now write, for the components of the com- plex vector J/= H. + iE, and two similar equations. The field specified by the poten- tials (18) thus possesses all the characteristics of the fethereal fields described in n)y paper in the structure of the sether ;t in particular, it is conjugate to the electro-magnetic field of Lieuard's type, with ^, 77, ^, a as a moving point charge, and Poynting's vector at x, ?/, z, t is along the line drawn to this point from the associated position'^, 77, ^, a of the point charge. The direction of Voynlinfs vector in a general type of field. § 3. Let us now determine the direction of Poynting's vector in a field specified by a complex vector M given by equations of type (20), in which |3 is replaced by /3, and e is * E, §5, 43. t Bull Amer. Math. Soc, March, 1915. 8 Mr. Bateman, Certain solutions of MaxweWs eqiiations. a function of the two quantities a and y5, which are defined by equations (l) and (2) with tlie modification that ^, t), ^, t, Z, ?/i, ??, p are now supposed to be functions of both a and /3. , As we liave shown elsewhere* these equations may be rephiced by two equations of type z-ct = <^-\e{x\iij), z + ct=-ip--^{x-iy)...{2l), where 0, (p, '<p are functions of a and /3 provided the + sign is taken in equations of type (20). Differentiating equations (21) with regard to x, y, z, t iu turn, and writing 6=0^ + 16.^, \-'=(^^'+e^\ where ^, and 0.^ are real, we find that cz dz 0, cz dz 73 ca p 8/3 (■22\ where the values of F, Q, B, S need not be written down. These equations give cz <>( da. ^ r/3 (23). 8a Combining these with the equations of type J/ ^ 9_(Ml) ^ ^ 3(«. /3) ' ' '9(y, ^)~ c d[x, t)' Amer, Jour, of Math., ApvU, 1915. Ml-. Bateman, Certain solutions of MaxivelV s equations. 9 we find at once that - 2\^,.V + 2\6^ .¥ + (X - 1) il/ = 0 I 2\d^M,-[\-\)M^-i[\+\)M = ^] Since the coefficients of the coniponenta of M iu these equations are real, it follows that Poyntlng's vector is in the direction of the line whose direction cosines are , 2\d^ 2X0.^ \-l . . Now I have shown in a previous paper* that when a possible pair of complex values ot a and /3 have been chosen, there are co * corresponding sets of real values of x, y, z, t, and these are associated with the different positions of a point which travels with the velocity of light along a straight line whose direction cosines are I, on, n. Hence, if we regard Poynting's vector as an indicator of the direction in which energy flows through the field, we may conclude that the energy in the field under consideration flows along a series of lines whose directions are given by equations (25). The direction of the flow of enei-gy at (cc, y, z, t) is, moreover, the same whatever be the form of the function e. Faraday tubes. §4. I have shown elsewheref that the lines of electric force in the field due to a moving point charge (^, 77, ^, a) can be obtained as loci of points travelling along straight lines with the velocity of light by considering directions of projection which satisfy differential equations of type ^|^=X^"+(r-cO(/r + tnV' + ^'r) (26), dot. where \ = c-l^ -mi]' -n^' ., fi = c^ - K" -v" - K'% and I, m, ?i are the direction cosines of the direction of projection at time a. It is clear from these equations that, if ^"=v"=^"=^, ^j ^"j and n do not vary with a. Hence, if we consider an elementary aethereal field whose singular curve is always along a line of electric force, it appears that the aethereal Held only exists In those domains of x^ ?/, z, t which correspond to values of a, for which the velocity of the point ^, ?;, ^, a is not uniform. Hence ♦ See last reference. t Bull. Amev. Math. Soc, March, 1915. 10 Mr. Batetmm, Certain solutions ofMaxivelTs equations. the radiation in this type of cethereal field is due to the accelera- tion of the point ^, 7), ^, a. We shall now show that the differential equations (26) are covariant under a Lorentz transformation. The idea that the Faraday lines of force or Faraday tubes are the ' fibres' of an element of the asther* is thus compatible with the theory of relativity. The simplest way of obtaining the required result is to remark that the two-way generated by a moving line ot electric force satisfies the differential equation f EJ{y,z) + Ed{z,x) + E/l[x,y) - cHJ {x, t] ~ cHd [y, t) - cBd {z, t) = 0, and is built up of the paths of particles which are projected from different positions of the point charge and travel along straight lines with the velocity of light. The differential equation and the property just mentioned are known to be covariant under a transformation which leaves Maxwell's equations unaltered in form, and so the result follows. To obtain a direct proof, we write s (1 4- n) = Z + m, cr(l+n) = ?- i'm, the differential equation satisfied by s Is then ds Now apply the Lorentz transformation | = |:^, 7} = 'r]^, i;= ^Q cosh u-oa„ si nhu^ ca = ca^ cosh w-^„sinlu<, we find that r = v^o'' v' = ^Vo\ r = »'(^o'^"'^l'"-csinhu), V (c cosh II — c,^' slnh u) = c, 1 = pl^, m = pm^, n = p (??, cosh u — slnh u) , p (cosh w — ?2 J sinh u) = 1 . * Cf. J. J. Thomson, Recent Heseai-ches. cTiap. i. t For the theory of differential equations of this type see C. Meray, Ann. de tkole nonnale, t. xvi. (18yy), p. 5u9 ; E. Cartan, il/id., t. xviii. (1901), p. 250; and A. C. Dixon, Phil. Trans, A, vol. cxcv. (1899), p. 151. If the equation can be written in the form d (v, w) = 0 the equations v = const., w - const, represent a moving line of electric force. ^[)\ Ramanujan^ On certain infinite series. 11 Hence, if ^s^ (1 + ?? J = ''o + ^'"o^ ^^'^ have s = 5^e", also fj, = v'/j,^ and X = pv\^. Again, since v = — -" , we have da ct'' = vX^' sinh if, V [c + (c - ^j') e" slnh it] = ce", v[c — [Z,^ + c) e~" sinh u\ = ce~", iience it is easily seen that the differential equation for s^ is which is of the same type as that satisfied by s. The differential equation for a can be transfornaed in a similar way. ON CERTAIN INFINITE SERIES. By <S. Ramanujan. 1. This paper is merely a continuation of the paper on ''Some definite integrals" published in this Journal.* It deals witii some series which resemble those definite integrals not merely in form but in many other respects. In each case there is a functional relation. In the case of the integrals tliere are special values of a parameter for which the integrals may be evaluated in finite terms. In the case of the series the corresponding results involve elliptic functions. * pp. 10-18 of vol. xliv. 12 Mr. Rcunanujan^ On certain infinite series. 2. Tt can be shown, by tlie theory of residues, that if a and /S are real and a0 = \iT'\ then (-[\ a _ 3a 5a (a + 0<^osha (9a + 0 cosh 3a (25a + ^) cosh 5a /8 3/8 5/3 + + (/3-Ocosh/3 (9/3-0cosli3/3 (25/5-<) cosh 5/3 4 cos\/(aO cosh\/(/^0 Now let inoina O^^p^ina Kn/>2bina ae _ doe _^ 5«6 cosh a cosh 3a cosh 5a 0e-^^P 3/3e-9'«/5 5/86-25!"/3 [cosh/3 cosh 3/3 cosh 5/3 Then we see that, if t is positive. (3) f e-2?«F(jO^?? = V-T-, ^ ^ J„ ^ ^ 4Goshj(l-i virtue of (1). J (4) /(«) = - •ij V(a«j| cosh }(l + ej Vl/^O} in virtue of (1). Again, let - /f-1 2n Y \2nJ X SS (- l)h{/^+'') {/u, (1 + z) Va - V (1 - /) v'/3} e-(7r/a./-j>"«+«V/3)'4n (/.= !, 3, 5, ...; v = l, 3, 5, ...)• Then it is easy to show that (5) f e-2i"/(n)(f« = ^ 4 cosli I (1 - 1 ) ^/iat) \ cosh {( 1 + 1 j \/(/^^j} " Hence, by a theorem due to Lerch,* we obtain (G) F{n)=/(n) for all positive values of n, provided that a& = ^7r'\ In par- ticular, when a = /3 = ^7r, we have (7) sin ^TTH 3 sin -|7rn 5 sin ^^^tth cosh^TT cosh|7r cosh|7r 4?i \Jn 22(-l)2(M+'')e-^M'':4' 4n 7r(/A'-'-v')' 4?i (/x=l, 3, 5, ...; v=l, 3,5, ...) (ytt + v) COS + (/u, — v) sin See Mr. Hardy's note at the end of my previous paper. Mr. Ramaiiujan, On certain infinite series. 13 tor all positive values of n. As particular cases of (7), we have sin(7r/a) 3sin(97r/a) 5sin(25n-/a) cosh i-TT cosh %tT cosh I TT a\/a { \ 3 5 "I (cosh ^TTtt cosh|^7ra cosh |7ra j 8 \/2 (cosh ^-na cosh |7ra cosh 4V2 if a is a positive even integer; and , sin('7rla) 3sin(97r/a) 5sin(257r/a) cosh ^TT cosh |-7r cosh §7r aV^ f 1 3 5 8\/2 (sinh |7ra sinhfTra sinhfTra 4 V2 if a Is a positive odd integer; and so on 3. It is also easy to show that if a;8 = 7r*, then a. 2a_ ^(a + 0 sinh a (4a + 0 sinh 2a ' (9a + ao) ir,^^^::^-,.^.::....^^,....:;,^,^-..] [ /3 23 3/3 I ((/8-«)sinh/3 (4/i^-«jsinh2/:?"^ (9/3-0 sinh 3/3 '") 2t 2 sin V(aO sinh V'(/i^O * From this we can deduce, as in the previous section, that if a/3 = 7r', then f^gina 2ae4'«« 3ae9'«« sinh a sinh 2a sinh 3a /3e-"'/3 2/Se-4'«/3 3>3e-9'«/3 "*" sinh/3 ~ sinh 2/3 "^ sinh 3/3 "'■" = ---- /f-l 2 72 Y V2?J/ X 22 {m ( 1 - 0 Va + V ( 1+ 0 Vy^i e-(27r/.^-i^=a+.v^)/4« (|tt=l, 3, 5, ...; v=l, 3,5, ...) 14 Mr. Ramanujan^ On certain injinite series. for all positive values of??. If, in particular, we put a = /3 = 7r, we obtain (12) . 1 ; — + . . . 47r sinli TT sinh27r sinli37r = — - 2S e-TM^/Sn i (a + v) cos — / N • ■""( + (fM — v) sm — in j (fM=l, 3, 5, ...; v=l, 3,5, ...) for all positive values of ??. Thus, for example, we have , , 1 co3(27r/a) 2cos(87r/a) 3cos(187r/a) (\3) r^ ^ - -{ : r-^ +... ^ iiT sinhTT sinh27r sinhSTr 3 5 — + ^-n^ — + -^-r^, — +• ira suih|^7ra sinhj^Tra if tt is a positive even integer; and 1 cos(27r/a) 2cos(87r/a) 3cos(l87r/a) r\^\ ,1 1 — : -I — !^ '. — i -J^ L — £ 4-,, ^■^47? sinhTT sinh27r sinh37r J f^ 3__ 5 ^ (cosh^Tra cosh|7ra cosh^Tra if a is a positive odd integer. 4. In a similar manner we can show that, if a/3 = 7r"j then 4- — H H 1-... = a -^ f^^* + /3 -:^ ; <^a)* - \ + * I showed in my former paper that this integral can be calculated in finite terms whenever nu is a rational multiple of tr. I take this opportunity of correcting a mistake : in the formulse (48) the first integral is — and not — . Mr. Ramanujan, On certain infinite series. 15 tor all positive values of n. Putting oi = /3 = TT in (15) we see that, it" n>0, then 1 cosTT-n 2 cos47^?^ 3 cos97rn •' a ax H ; 2 S e-^'^/^"/" 6-2- -1 2«V(2n) ^=1 ,=1 X I (^ + v) cos j-^-^^^ -^1 + (/x - v) sin j-^^^^-^^ -'I As particular cases of (16) we have . . 1 cosCtt la) 2 cos Utt I a) 3cos(97r/a) (17) — + o + r^ — ^^ + j^-^ '—^+... r X con (ttx' la) , /M n/ 1 2 3 \ if a is a positive even integer; 1 cosf7r/a) 2cos(47r/a) 3co3(97r/a) (1^) S^^ + ^^^^T + e^--l "^ e6--l "^- if a is a positive odd integer; and 1 cos(27r/ffl) 2cos(87r/o) 3cos(l87r/a) ^ ■> ^+ e2--l + ~e^^-l ^ "^ e6--l +' r a? cos f27ra;7a) , , , / 1 3 5 Jo e-^-«— 1 * Ve'^^ + l e37rrt_|.i e^'^"+l it a is a positive odd integer. It may be interesting to note that different functions dealt with in this paper have the same asymptotic expansion for small values of n. For example, the two different functions 1 cos?i 2cos4?2 3 cos9?i 1 1 1 • +... Stt e'^^^-l e^^-1 e^^-l 1 f°° X cos 71X^ ^ and — dx J„ e2^^-l have the same asymptotic expansion, viz. (20) l__!!L+_^__i!^ +....* ^ 24 1008 6336 17280 * This series (in spite of the appearance of the first few terms) diverges for all values of n. { 16 ) THE EXPANSION OF THE SQUARE OF A BESSEL FUNCTION IN THE FORM OF A SERIES OF BESSEL FUNCTIONS. By A. E. JolUffe, M.A. The square of tlie Bessel function / (a?) can be expanded in a series of Bessel functions with the argument 2x in the form + «..{'^.n..r(2^) + ^.„.....3(2'«)l+.-^ and -^ = (2>--l)(2n+2r-l) a,,._3 2r(2n-\-2r) (So far as I can discover, this expansion has not beeu given before). [Jn{^)Y sJ^tisfies the differential equation di X- d^' ^ " ^^ ^'" ^^ ^ ^'^ ^ di - ^' where ^ denotes 2x. If we write J^iX) ^OJ^ y in the left-hand side, we obtain By means of the formulse 2^;(^)=j..;(i)-j,,,(^), 2rJXl) = l\J..,{l)^J^^,{l% the result of substituting in the left-hand side of the differential equation can be reduced to 2aJ-l(2« + l)J„J+2a,[2(2;H2)/„..,-3(2a+3)J^„J + 2«j4(2n + 4)/,„^,-5(2n + 5).4,„j+..., which shows that, when g,^ _ (2r-l)(2» + 2r-i; o^ 2r (2?i + 2r) ' the series is a solution of the differential equation. When a^ is properly determined, it must be the square of J^(x). The value of c/g is determined as tliat given above, by considering the coefficient of a;"' in the expansions of the series and {J^(x)Y in powers of a). ( 1^ ) SOME PROPERTIES OF THE TETRAHEDRON AND ITS SPHERES. By T. C. Lewis, M.A. 1. So far as 1 am able to ascertain, no direct elementary proof of the following proposition has hitherto appeared. If A' be the vertex and BCD the base of any tetrahedron^ then the sphere which passes through the points B^ C, D and touches the inscribed sphere of the tetrahedron will also touch the sphere escribed on the base. This may be proved by means of the penta-spherical co- ordinates of Gaston Darboux, Take as a tetrahedron of reference any orthocentric tetra- liedron ABCD on the same base BCD. The equations of the faces of the tetrahedron A' BCD are p,^3-p,^,= hiPi^x-ps^'.) p,^,-p,^s=K{Pi^i-p,^,) Let ^a,.Xj.= 0 be the equation to the inscribed sphere, of radius r. Let the distances of A^ B, C, D from the ortho- centre be a,, a^, a^, a^. Then ^,p-o.,p,= K<^,^'^{KPx'+p.''r{i-k.yp:\ ^,P,-^,Pi=^-,<^x+K^ where /iT,, K^, K^ are the values of V}/CPi'+P„*''+ (1 -^'JVsl or ^J\l:J^a^ i-a^—2k^p^'\, when n is 2, 3, or 4 respectively. Therefore Sa*=l = ^. («5P5+ <^y+ ^. KP..+ ^-.^i + -^.1'+ •••+ ~^< p/ P\ Pi rs (Vpi p/ P3 P4 / P2 P, P* 3 \ P, P3 Pi / VP, Pz Pj Vs P3 P4/ VOL. XLV. C 18 Mr. Lewis, The tetrahedron and its spheres. therefore ni . '^ iV, />/ Pa P4V P2 P3 p. i + I-P5 — . + -I + ^ +<!-. + ->+ -J + «, ]-^ + -^' + -^ =0 (u). ( Pi Pz P^ ) Here the coefficient o( a^p. is seen to be l/»*; let // be the sum of the terms independent of a^. Then l/r.a,p, + Zr= 0, \ Thus a^p^ = a^ — Hr I a.^p,= k^a-Hr+K^ &c. = &c. The inscribed sphere Sa^£c^= 0 is therefore determined. 2. Let a sphere pass through the points B, C, D. Its equation will be -mp^x, + p.^x^+p^x^^-pp^^-^[m-\)p^x=0 (iv). Its radius R' is given by iR"=m-'a--2mp^'-^ P: ^ P:-^ P:+ P: (v). Let this sphere touch the inscribed sphere whose equation has already been determined ; then the following condition is satisfied, viz. - ^"a.P, + «,P, + Oa^a + «4P4 + ('" - 1) "sPs tiierefore { - w«, + [k^ + A-3 + /.-J a^-2Hr + K^ + A^ + /v j ' = m'a; - 2m p/ + p/ + P3' + p/+ p/, therefore - 2ma, |(^^ f /•3+ A-J o^ - 2^- + TT, -I K,^ K- ^1 -(V+ ';-.;^+ o «;^-p/-P3^-p/- 3p/ + 2 (A;,+ A'3+ A:J p;+ 4Zry (/r,+ ^-3+ /:J a, -2(/c.,+ A'3+A-Ja, [K^+K^ + K^ + 4Sr(^,+ ir3+ /g - 2h\K- 2K^K- 2K^ 7^3. ilir. Lewis, The tetrahedron and its spheres. 19 Therefore = p: + [K + K' + ^^:+ KK+ Kh+ ¥s) «/+ 2^''^ - (^'>+ ^3+ K) p: 3. This determines vi in order that the sphere through B^ C, D may touch the inscribed sphere. If it also touches the sphere escribed on the base, whose radius is r„ this equation must remain true if r is changed to r,, the sign of rt, being changed, and the corresponding value of E being H^. The necessary condition is that, whatever values be given to k^, k^, k^, 2HE^+ ^^K,K, + KJ<,+ K,K,-{k,k^ + k/;+k.^k,)a;+p:^ ~ If the known expressions for — , — , H, H. are substituted, T r^ the above identity may be demonstrated by a piece of work which, though lengthy, presents no mathematical difficulty. 4. By making use of this identity the equation to determine m so that the sphere (iv) may touch the inscribed sphere reduces to m \ (H H.\ ,,,,.7,1 I ■■'. rr, a, \'-, rj ' ' rr, which remains the same when r and r,, H and H^ are inter- changed, and the sign of a, is also changed. Therefore the proposition is proved. 5. But as a matter of fact we have proved more than we set out to prove, namely the following comprehensive theorem: Through the angular points of any face of a tetrahedron there may he drawn four spheres each of which touches two of the eight spheres which touch all the four faces of the tetrahedron. 20 Mr. Lewis, The tetrahedron and its spheres. In the above work if iT, be negative instead of positive, tlie place of the inscribed sphei-e therein is taken by the sphere escribed on the face A CD., whose radins is r^; at the same time let H become H^. The identity (vii) remains true, mutatis mutandis. Thus we obtain a sphere which passes through B, C, D., and touches the sphere escribed on A CD and also touches the sphere which touches A CD a,nd BCD on the reverse side, i.e. on the opposite side to that on which the inscribed sphere touches them — if there is such a second sphere. But if there is not a sphere touching the planes opposite A' and B on the reverse side, there will be one touching the planes opposite G and D on the reverse side, and in determining it K^, K.^, K^ will be opposite in sign from what they are for the sphere escribed opposite B. Now if m in (viii) be evaluated we obtain m _ 2^,g;g; 2lc^K,K, ^Ic.K.K^ ^">\ Pi P4 P* P-Z P2 Pi + terms independent of the sign of K.^, K^, K^. Therefore m also remains unaltered, not only if the sign of a, is changed, but also if the signs of K^., K^, or K^ are all changed ; and this whatever the signs of /ij, A'^, or K^ may be at tirst, the corresponding values of r and H being, taken. IMius the general theorem is established. 6. If A BCD is an orthocentric tetrahedron A must lie on the line and therefore it is clear from th« equations (i) that the necessary condition is "^2 ~ ^3 — %• But we may take this orthocentric tetrahedron as the tetrahedron of reference, so that k,^ — k^ = k^ = 0, and H = H^=\. Also^ K^ = a^, K^ = a^, K^ = a^ Therefore the identity (vii) becomes 2 + {a,a^ + a^a^ + a^a^ + p/) ~^ + (^ - ^j '^ = (a, + a3 + aj [-- + -j, which reduces to a^n^ +«,«3 + a^a^ + p^' - (a, + a^+ a^+a^) (r, + r) + 2a, r. = 0, Z?r. Mui)-^ Determinants and alternating mtmhers. 21 and there are tliree otlior similar identifies corresponding to the different vertices, therefore by addition the following symmetrical identity is found: — + 4p; + 2 (r«,r, + rt^r, + a^r^ + a^rj = (a, + a^+a^+a^ (4r + r, + r^^r^ + 0*...(ix). DETERMINANTS WHOSE ELEMENTS ARE ALTERNATING NUMBERS. By Thomas 3Iuir, LL.D. 1. In the Proceedings of the London Mathematical Societi/, VII. (1876), pp. 100—112, there is a paper by Spottisvvoode on alternating numbers viewed as the elements of a determinant. The paper contains a fairly large number of results, particular and general, some of which were, he says, obtained from notes by Clifford. Probably the most noteworthy theorem in the collection and at the same time the least satisfactory is that on multiplication, and to it I wish to direct a little attention. 2. Spottiswoode says (p. 103) : " the ordinary formula for the multiplication of determinants may be applied, namely, — \, /i, ... X', /u-', ... =(XX'), (\/x'), ..., 1, 2, ... 1, 2, ... (nX'),(fifi'), ..., if it be understood tiiat, after developing the right-hand side of the equation according to the ordinary rule, those terms which require an odd number of changes to bring them into P * If a^-<c,the following result with legard to a plane tiiaiigle is deduced, viz. 2 (rt, + O2 + flj) = 'i + '2 + '■} + 3r = ili + -ir, i.e. the sum of the distances of the vertices of a triangle from the orthocentre is equal to the sura of the diameters of the inscribed and circumscrihed circles, a property which may be easily proved by elementary trigonometry. Any distance a, is negative if drawn from an obtuse angle of the triangle. If 7a, r/j, Tc are the radii of the circles inscribed in PBC, PGA, PAB, respectively, the sum of the sides of an acute-angled triangle may be expressed thus— a + b + c = 4:R+r + ra + rb + rc - '"i + '■•i + ''a + '"« + '■* + '■«• If C is an obtuse angle, « + 6 — s = r + jv — J"a — ?"6. These properties of the triangle are appended because the author has not met with them before. C2 22 Dr. Muir, Determinants and alternating numbers. the form \iJ,...X'fjb' ... are to have tlieir signs altered, while tliose which require an even number of changes are to retain their signs." In regard to this we may note first that the formula can be expressed much more simply. The determinant on its right is, as conditioned by him, no determinant at all, but a function differing from a determinant in having all its terms positive, and usually known as a permanent with the notation "j His theorem thus is for the third order a, 7, ^2 ft. 72 "3 2A.a SX^ S\7 2/xa Syu/3 SAt7 Sm Sr/3 Si'7 In the second place it has to be noted that what Spottiswoode means by calling the elements of l^^/J-.v^l \oi^ft,'ys\ alternating numbers is simply that any two of the eighteen, X, and 7, say, are such that 3. Save for a verification of his formula in the case where the determinants are of the second order, there is no demon- stration given. Probably he verified it also for the third order; but further than the third he could not have gone, otherwise he would have seen that the sign preceding the determinant on the right is not always minus. Thus, taking the case of the fourth order, namely, I ^./^/3P4 1 • ! ^AJsK 1 = -l^^a . Sm/B . Sv7 . IpB I , we see that the product of the two diagonal terms on the left is and that the diagonal term on the right is -■^X(x.'2fi[i.1vy.'2p8. This latter being - (\,o(, +...+ X^aJ (^,/3, +...+ f^.S;} (.^,7, 4...+ .^,7^ (p,8, +...+ /),SJ, one of the tern)s of its expansion is which on the shitting forward of /u^, Vj, p^ becomes Dr. Muir, Determinants and alternating numbers. 23 l^lius the minus sign preceding the permanent cannot be correct. 4. Let us tlierefore examine the matter anew. When the elements are ordinarv imuibers we know that \ U V O I.Gt/S'VO = and that this comes about because every term on the left is matched by an equal term on the right, and because all the remaining terms on the right are cancellable in pairs. For example, it' we take any term of the determinant on the right, say where <u is the number of inverted-pairs in /357a, and select from the 4* sub-terms involved in it the sub-term SAa ^XS iA.7 v\S Syua '2fM0 V^ry Syu8 Sm ^y/S St^7 VrS Spa 2p/3 Spy SpS (-ir-Ma--«A-»'e7c.P/ (R) this sub-term is either matched on the left-hand side by the term in which case a, b, c, d is a permutation of 1, 2, 3, 4 ; or, it is one of the many cancellable sub-terms having a, ^, c, d not all different. With this before us let us now consider the effect of changing the elements from ordinary into alternating numbers. Taking first the case where a, i, c, d are all different, we readily see that identity is no longer ensured by the two terms R and L being merely composed of the same factors: the order in which the factors appear must be the same as well. We must be able therefore in accordance with the laws of alternating numbers to alter as required the order of i^'s factors without bringing about a sign-factor differing from that of L. Now, in order that the factors X,^, yu^, v^, p^ in R may be made the first four of the eight, the number of sign- changes necessary is 1 + 2 + 3, and in order that the remaining factors /3^, 5^, 7^, a^ may appear in the order oc^/3„7^8j the number of sign-changes necessary is 6). 24 Dr. Muir, Determinants and alternating numbers. Consequently, the index of ^'s sign-factor after being Increased by 1 + 2 + 3 + to must still be equivalent to the Index of Z's sign-f;ictor — a manifest Impossibility so long as jR's original sign-factor is reckoned according to the sign-law of determinants. We observe, however, thai if we make the original sign-factor of i2 not (-1)"" but (-ly""''^ the requisite transposition of factors would change it into as desired. It is thus suggested that for alternating elements all the signs on the right should be the same, namely, (—1)'"'"*'; in other words, that the right-hand side of the mnltiplicatioM-identity should be changed from + + |2Xa.2/u/3.2:v7.2p5; into (- 1)"^" | SXa.2/Li/S. 2v7. S/3S|. In order fnllv to justify the change it is of course necessary to consider the other case, namely, whei'e a, 6, c, d are not all different. When this is done, however, it is found that cancellation takes place on the right exactly as before, the requisite difference of sign, which does not exist in a perma- nent to start with, being provided by the law of transposition of alternating numbers. 5. A more direct and more generally satisfactory way of establishing the identity, now that its true form is known, is by l)eginning with the permanent on the right and deducing from it tlie factors on the left. 'Jo this end the more elementary properties of permanents require to be known ; namely. The numbers employed beiiKj alternating numbers (a) the interchange of tico columns of a permanent does not alter its value ; (b) the interchange of two rows of a permanent alters only its sign ; (c) if (wo roios he alike, the permanent vanishes; (d) if any tivo columns of a permanent be of the form it vanishes ; (e) any determinant is expressible as a permanent differing only in having two rows interchanged / Dr. Muir, Determinants and alternating numbers. 25 (f) if the p^^ row of an n-line permanent he multiplied hy (I), the permanent is thereby multiplied by {—ly^oi; (g) if the roivs of an n-line permanent he multiplied in order by (w,, oj^, Wj, ... respectively, the permanent is thereby multiplied by {—lY"'''~^^(o^w^Oli^... ; (h) if the columns of an n-line permanent he multiplied in order by o>„ &)j, tWj, ... respectively, the result is equal to the corresponding determinant multiplied by (— 1)*"'''"''(o,<Bj(»j... . These are readily established in every case by considering the individual terms of the permanent or determinant con- cerned, and always bearing in mind that in the formation of the terms the elements are taken from the rows in order. For example, in the case of (d) it is sufficient to note that every term ...|„a...^j3... is matched by another ...^..|3...^,a..., and that the two have necessarily different signs on account of a and /3 being alternating numbers. Again, in the case of (h) the given permanent being + + we have to note that any term of the permanent resulting after the specified multiplication is of the form where r, s, t, u, ... are the numbers of the columns from which the elements constituting the terms are taken. JNow, by the transformation-law of alternating numbers, this is changeable, first, into and thereafter into where rrr is the number of inverted-pairs in r, s, t, u, ... . But (— l)''a^/y,7j8„... is a term of the determinant \oi^fS^'y^8^...\] hence the result is as affirmed. 26 D)\ Muir, Determinants and alternating numhers. 6. These preliminaries being settled let ns now consider the permanent 4- + /*,a, + /*,«, + M^a, /i,/3, +/i,/3j + /ij^3 /^,7, + /x,7, + ^,73 the restriction to tiie third order being made merely for con- venience in writing. \\\ the first place the permanent is expressible as a sum of 27 permanents with monomial elements; and, this being done, it is seen that 21 of them are of the type dealt with in tlieorem (d) of the preceding paragraph, and therefore vanish. There thus only remains for consideration the six-termed expression + + + \a, \/3^ ^3^3 1 /A,a, ii^Q^ ^37,1 + ^^^^ ^^. "373 1 + + V, A-,/3, X,7^ Ai^a, /*,^, \lz V^3 ''^^^ »'n73 + V, ^3^3 \l.\ fx.a., /X3/33 ^l;i^ + ^a. "3^3 »'27, + + + |^3«3 ^.^, \72 + 1 /^a^^s /*.^i /^•.72 ! »'3a3 »'i^, ^7j + + + ^«3 M, ^.7,1 ^j^s /^s'^e /^i7, j ''3^3 »'2^J »'.7i i (-1/ M, ^^3 ^.7, /^2a2 Ms'^S ^l7, + ''2«2 *'3^3 ".7. which, by theorem (h) of § 5, Is equal to 1 ^I^.»'3 I • «r%3 + I \^^^'', I • a2^.73 + I ^3^.''2 I • «3^l72| + l^i^s^l •a,^372 + i ^2^3^ I •aA7, + 1 Vs^ \-^,^2lx^ ' and, by a law of determinants, is equal to a,/3,73-aA7, + a3^,%| aA72+aA7,-a3%,i ' and therefore equal to (-l)1V.V3|.|a,/3^7j. 7. Spottlswoode next considers the case where the two determinants to be multiplied are identical, and where there now comes into play the second half of the law of alternating numbers, namely, that the square of any such number is zero. (-l)'iV2'^3!- Dr. Muir, Determinants and alternating numbers. 27 He of course readily sees that in this case the permanent on the right is skew symmetric ; and from the examples \\^^.^^V = - \^j^3P4°"5r=f^' IXlL TKv i — Ykfju . 2/u.v = 0, he concludes that iftlie elements of an odd-ordered determinant he alternating numbers, the square of the determinants is zero. As 2\/z, 2A,i/, 2/Ltv, ... are not alternating numbers, this would seem to be equivalent to saying that a skew symmetric permanent of odd order is zero. The danger, however, of such hasty deductions and of excessive trust in analogy is here again apparent, for he goes on to formulate the result where, apart from the above-noted oversight as regards sign, there is a fault in the calculation. The process should, I think, stand as follows: — '^it^t^zPX' = — 2X,v — '2/jlv . Sv/3 — 2Xp — 2/i./) — ^vp = 2Xyu, (S/t/3 . 2\v - ^fiv . ^\p + 2Xfj, .lup) -2vp + 2Xp (- S\/* . Sj'jo + 2Xv . 2/iip + IfjLV. 2\p) '2p,v = (svy^svp)' + i^xvYi 2ppy+ (^xpy(^p.vy + 2 SX/A . Srp . -^Xv . SmP - 22Xai . Svp . 2\p . 2fj,v -f 2^Xv.2fip.'EXp.'l^iv, where the occurrence of the minus sign precludes the possibility of the riglit-liand member being expressible as a square. Capetown, S.A., nth March, 1915. ( 28 ) AN INEQUALITY ASSOCIATED WITH THE GAMMA FUNCTION. By G. N. Watson. The Weirstrassian definition of F (z), valid for all values of z (other than negative integers), is effectively equivalent to the formula of Gauss „, X ,. 1.2...n r (Z)=\\m ; -.11. «->«> z{z + l)...{z+?i) The difficulty which arises at the outset of the tlieoi y of the Gamma function is the reconciliation of tl/is result with Euler's integral definition ^ 0 valid when the real part of z is ^Jositivc. Write z = x + iy; then, when x>0, it is easy to see that the integral n {z, n) n converges, and, by integrating by parts n times [n being a positive integer), it is readily proved that \.2...n To establish the equivalence of the Gaussian product and the Eulerian integral, it is therefore sufficient to shew that hm ^(,--jr-<. = J^e-r' dt i.e. that lim \C \e'- (\-l\\e \It+ Ce-T'dt] = 0. This result is proved by Schlomilch {Hoheren Analysis, p. 243) by some rather elaborate analysis. Bromwich [Infinitp, Series, p. 459) has a simpler proof when U <^ < 1 ; his proof is that Frof. Wafsofi, On a certain inequality. 29 I e'W'^dt-^O as ?j -> co in virtue of the convergence of e^t'~^df\ and he sliews by means of the inequality* V n) 2n that the modulus of the first integral tends to zero when 0 <x < 1, and infers the result for other values of a; in virtue of the recurrence formula r(z + l)=zr{z) satisfied both by the integral and by the limit of the product. Further considerations are necessaiy when x is an integer and 3/7^0. It is, however, possible to obtain a much more powerful inequality than that due to Broinwich, by quite elementary methods; this inequality is 0<e-'-(l --) <e-'- \ n I n when 0 <«<n and n is a positive integer; and tMs. inequality is su-fficient to prove that for all values of z such that x>0 ; for, assuming the inequality, the modulus of the last integral < ■' 0 J n since the last integral converges and is independent of n. To obtain the inequality we proceed thus: It is obvious that whenf 0 < y < 1, ■i 3 l + V<l + v + ~ -h ^, +...<! ■\-v + v'+v^-i... _ =1/(1-.), writing v = tln, we see thai, when 0<it<n, 1 + -) <e«/«< (\ -- n 1 \ n * The inequality is established by Bromwich by an ingenious device based on the consideration of the integral j:E('-r--[-"('-y"j;- t The first portion of each of the following inequalities is true when v=\. 30 P^'of. Watson, On a certain inequality. so that (l + ^|)"<e'<(l-|y . Hence, when 0 <t<n, we have Therefore, when 0<t<n, Now if l>a>0, we have (1 — a)"> 1 — ??a, by induction Avhen 1 — no. is positive* and obviously when 1 — na is negative. Therefore writing f jii' for a we get \ n J n and so from the preceding series of inequalities vve get 0<e-'-fl--y<e-'-, \ 71 1 n whicli is the required result. The result is still true if n> 1, when n is 7iot restricted to be an integer, provided that 0<t<n; for the only place in which it was assumed that ?i was an integer was in proving the inequality (1 — a)"> 1 — ?«a, and this is easily proved when n > 1 and 0 < a < 1 ; for, by Taylor's theorem, (1 _ a)"= 1 - ?«a + in (w - 1) a'(l - ^a)"-^ where 0<^<1, and if n> 1 the last of the three terms on the right is positive. * For if (1 - a)" > 1 - na, then (1 _ a)'"'> (1 - a) (I - na) = 1 - (m + 1) a + Ha^ > 1 - (n + 1 ) a. ( 31 ) NOTES ON A DIFFERENTIAL EQUATION. By G. W. Walker, M.A., F.R.S. The difFerential equation which occurs in the problem of the two dimensional distribution of a gas under the influence of its own gravitation may be written in the form and a solution of this applicable to real cases is where </> and i// are any conjugate functions of ^;, y, so that An associated linear equation occurs in dealing with small motion of the system, which is of the form and which, on using variables ^ and ^, transforms to A particular solution of this is If we take new variables so that the equation takes the form 8 dx' 1 a a^ 1 ^'z 8z VT OCT OT ^ + — ^- = - dt!J 8cT ^' ^x"^ (ct'+1) •i ' 32 Mr. Walker, Notes on a differential equation. Hence we have solutions of tlie form ^ /. cos where /„ satisfies lA^M^K, ^^ .1/;. Let vT = e^, tlien |^=:(.^-^-2sech-'0/,. so that solutions are These may be recognised as associated Toroidal functions of the first and second type. We may note that for n>2 the solution of the first type fails, but the second form gives a solution for all integral positive values of n. The complete solution in terms of two arbitrary functions may be found thus : Writing (f)+i\p = ^ and (p—t\p = r], the equation takes the form By Laplace's method the solution of this is where F^ and F^ are arbitrary functions of ^ only and rj only, respectively, and F^' and F^' are their first derivatives. ( ^^ ) A SET OF CEITERIA FOR EXACT DERIVATIVES. By i:. B. Elliott. 1. Some time fip^o* I called attention to a criterion, dlflferent from Euler's, wliicli decides whether a rational integral function of a dependent variable and its successive derivatives is or is not the result of differentiating some other. The following is a more complete investigation of such criteria. Let X be an independent, and y, z, ... any number of dependent variables, and let y^^z^^ ... denote —4, -7-7., .... We consider, not all functions but an extensive class of such functions, and enquire when an i^ is a TJ^^ where r is any number, and D is the operator of total differentiation 9 3 8 3 9 8fl5 -"'91/ dijy dz ^ dz^ The advantages of the method are (1) that of applying as simply to cases of many dependent variables as to the case of one, (2) that of providing as definite a single condition for i^ to be a D"^ as for it to be merely a Z)(^, (3) that of exhibiting as a result of direct operation the associated ^ when the appropriate condition is satisfied. The wide class of functions F to which it applies includes all those which can be arranged as sums of parts that are homogeneous, and of degree not zero, in some one set 2/s yp 3/2' •••) ^"^ ^*'® fiioreover rational and integral in the derivatives y^^y.^-, •■• of the set. If such a sum is a derivative (first or r"') its separate homogeneous parts are so separately, for operation with D or D" does not alter degree in. y, ?/,, ?/,, ..., and conversely. The parts may be taken separately. Accordingly we confine attention to a function u = F{x) y,y^,y^, ...', s, 5;,, 2;,, ...;...; ...), which is (i) homogeneous of degree i (^ 0) in y, y„ y^, 7/3, ... ; (ii) rational and integral in ^,, y^, 3/3, — * " Note on a class of exact differential expressions,'' Messenger of Mathematics, vol. XXV., p. 173. VOL. XLY. D 34 Prof. Elliott^ A set of criteria for exact derivatives. Tlie limitation (ii) is imposed in order to secure tlie annihilation of u by some power of the operator &> about to be introduced. Other functions u which are so annihilated are really treated at the same time, 2, Mean by iv the greatest sum of ?/-suffixes in any term of u. As well as the operator i>, which is total^ we use the operator ^Gij, -^'dy, ^'dy, which is not total but, like i and w, refers to the set y, ?/j, ?/^, ,., only. If there is choice among sets we naturally choose for y, y^, y,, ... the set of smallest w. Repeated use will be made of the alternant 7) 7i r) the effect of which on u is to multiply it by its degree In y,y^,y,, ..., however u may also involve x and other sets The criteria to be obtained depend on compound operators of the type (r, n) = (Dq) - ri) {Dm- r + 1 .i) ... (Day - ni\ where r, n are not fractional, and 0 ^r^n^w. For instance (r, r) is Dco — rt, and (0, n) is Deo (Bco — i) ... {D(o—ni). The mam facts are as follows : — Lemma. For any n, (0, w) =/>"''&)"". Theorem (O). (O, iv) u vanishes identically. Theorem (r), for r= l, 2, ,.., ?<;. According as (r, to) u is or is not identically zero^ u is or is not an r"' derivative D^c at least. Theorem {w-\- l). u cannot he a {io-\- Ij"' derivative. The last of these theorems is at once clear. In fact, if u z=Dv' is integral in ?/,, 3/,, 2/3, ..., v cannot be fractional in them. Also the greatest sum of y-suffixes in u exceeds by 1 the greatest in v. Thus in a i)""v, where v involves any ot y, ?/|, ?/j, ..., the greatest sum of y-suffixes in a term cainiot be less than io+ 1, whereas in our u the greatest sum is xv. Another immediate fact is the half of theorem (l) that n is a Dv when (1, w)u = 0. For upon transposition this relation expresses io\.i"'u as a Dw {...}. Prof. Elliott, A set of criteria for exact derivatives. 35 3. We will now prove the lemma, and deduce theorem (0). Tbe lemma is true for a = 1. For an operation with w or D does not alter y-degree t, so that use of the alternant above gives (U, 1) u = Bco (Day - i) u = l) [(i)D . co — iw} u = D [{Doi -\-i)a) - {(i)\ II = D'oj'a. We have then only to prove that, if true for n - 1, it is for n. Now, on the assumption foi n — l we have (0, n) = i>"ft)" {Bco - ni) = i>"a)"-' {{D(o + i)(a- ma}\ = Z>"<u""' {I)a)-n-l.i\(o = i)''<u"-' \Da}-n-2.i} ca' : 2)". i>a).ft>" =/>"+'&>"'. The deduction of theorem (0) is immediate, u contains no term with sum of ?/-suffixes exceeding w. Now operation with (o lowers sum of ?/-.suffixes by 1, and annihilates terms free from 3/,, ?/„ Thus w"'"« = 0, and so (0, w) u, i.e. 4. We next prove the half of the general theorem (>•), that if u is a D'^v, then (r, w) ii = 0. It has already been seen that if w is the greatest sum of y-sutfixes in any term of u = D'v, then the greatest sum in any term of v must be iv — r. Thus <u" "*i' = 0, and so (0, t{>-r)u = 0. Now by repeated use, moving backwards, of {D(o - ni) D^B{Du) + i)- niD = D {Deo -n-l.i) we obtain, with u — D'v, (r, w) u = {D<jj — ri) (Deo — r+l.i)... {Deo — loi) D'v = D{D<a=r-\.i){D(o-ri)...{D(o-w-\.i)D'~'v = D{r-\, io-\)D''v = D\r-2^io-2)D''-'v = D''(0, w-r)v = ^. 5. Conversely, we have to prove that if (?•, w)u = 0, then u is a D'v. This follows from a fact, to be proved in the 36 Prof. Elliott, A set of criteria for exact derivatives. next article, that any u o given i and w can always be expressed as a sum ot" xo-\-\ ^larts, some perhaps vanishing, where the coefficients are definite numerical constants, and the (?•+ 1)"' part, for r=l, 2, ..., to, is expressed as an r"" derivative. The deduction from the fact is as follows. Given (/•, lo) u = 0, we have also ()•—!, lo) u = (i)a) - r — 1 . t) (r, w) M = 0, and in succession {r-2, iv)u = 0, {r-3, w)u = 0, ... {l, lo) u = 0. Thus (/•, w) 11 = 0 necessitates u = D'-{Ay[r + l, io) + A^^^Do>'-''{r + 2, w)+...-{ A^D"'-'-(o'"]u. 6. The precise theorem of separation of which a part has been used may be thus stated : Theorem. Ani/ u such as specified at the end <?/ § 1, and in fact any u of 7ion-zero y-degree i throughout which satisfies (t>^'''^\i = 0 for some 2V, can be expressed by direct oiieration as a sum of IV + 1 parts {some perhaps zero) each satisfying one of the equations (Deo — ri) u = 0, for r — 0, 1, 2, ..., iv ; and of these io-\-\ parts, the first, u^, is not a derivative, lohi/e generally the (r + 1)"', u,., is an r"' but not an (r + l)'* derivative. It is not stated (or true) that in all cases the parts Wj, M,, ,.., 11^ involve no higher derivatives than the sum ii does. 'J'iie separation, like others which 1 have considered else- where, is effected by use of the identity among polynomials s-n ( I I \ where F{z) = (z — a^{z — a^ ... {z — a^. Taking lo for «, particular values for a„, a,, ... a,, and the operator Day for z, this tells us that A, A, A.. A + Tr-^.+ ,. ' ..+.-.+ \D(ii Dw — i Doi — 'li '" JJt X Dot) ( D(a — i) ... (Dw — wi) u, where, for r = 0, 1, 2, ... iv, A={-ir-'-" ^ rl (w — r) ! * Prof. Elliott, A set of criteria for exact derivatives. 37 u is thus -wriitcn as a sum of ?o + 1 parts, of which the (r 4- 1)'*", for each ?•, is presented in a form whicli shows that it satisHes {Dw — ri) i(^= A^(0, to) u=0. Also, for each r, u^ = A^D(o(^D(a-i) ...{Dcd-r-l .i).{D(o-r\\ .%)... {Dw-ioi) u = A^D''t£)'{r + 1, to) u, where the final (io+ 1, to) denotes 1. Thus so much of the theorem as was required in § 5 has been proved. 7. The theorem, however, states further that the first part u^, when it does not vanish, is not a derivative, and that the (r+l)"" part u^, presented above as an ?•"' derivative, is notan(r+iy\ Now u^ satisfies {Doi - ri) u^.= 0. If it were an (r + 1)*^'' derivative it would also, by theorem (r+ 1), have to satisfy (Do}-r+ l.i) (Dco-r + 2.i) ... (JDo) -wi) u^=0. But this equation is inconsistent with the otljer. For on substituting in it, from the other, riu^ for Do) u^ we arrive at the unsatisfied (- «) (— 2 1 ) . . . {r — IV . i) u^ = 0. This applies for r = 0, 1, 2, ... r« - 1, but not for r = to. How- ever w,„, i.e. A^D'^co'^u, lias the same lo as u, and cannot be a [lo + l)"" derivative by theorem {w + 1). Corollary 1. Any u satisfying {Dw — ri)u = 0 is an r"" derivative, by theorem (?•), and not an (?*+ 1)"' by the above. It must not be assumed conversely that every r"*, but not (r+ 1)"\ derivative satisfies this equation. Corollary 2. (Dco — k) u = 0 cannot be satisfied except for one of the values 0, i, 2i, ... xvi of k. Corollary 3. If w?< = 0 (with i^'S) u cannot be a deri- vative. For if it were (l,t«)M = 0 would give io\i'"u = ^. [N.B. This does not apply with' 2 = 0. If i= 0 for v we have (&)/> — Dw) u = 0, and so coDv = 0 whenever wu = 0 or c. For instance o) annihilates all of i)" iyfy), I.e. i>-' \ogy, D'^' log [yy-y;% I)"<f>{x', ^,^.,^„...; B'hgy).] Corollary 4. (1, w) m, when not zero, is annihilated by Bo), and so, with i^O, by (o] and generally (r, iv) u, with i^O, is annihilated by &>''. D2 38 Prof. Elliott, A set of criteria for exact derivatives. 8. The process for examining a given u may proceed as follows: — Form Duiu. If this is a multiple of ii, then u is exhibited as a Dv. In such a case the multiplier must be one of i, 2i, ... tvi. If I)(au = riu, then m is an r"' but not an (j-^-iy*' derivative. In particular if I)(ou = ioiu, then u is a 10^^ derivative. [Example. ?/-V is an integrating factor of <\^'j^ _45y?/y^-]- 40^1^= 0, which makes it an exact Z)^y = 0.] it' no \D(o — ri) u = 0, take [Deo - loi) u = u\ and operate on it with JJo}. Jf the result is a multiple of u\ the multiple must be ri, with r one of 1, 2, ... w — 1 [not w because (iJ, IV - 1) i/=0 and [wi—i] {ivi—2i) ... [wi—w— 1 .i) ui^^\. If it be ri, then u is an r^^ derivative by theorem (r). If, however, no {Dco—rij [Dw-wi) u=0, take (Deo— to— I .%) [Dm - loi) u = u", and form Dtnu'. If this is an riu' , with r necessarily one of 1, 2, ... w — 2, then u is an ?-'^ derivative. If not, proceed again in like manner. Finally, if [Dw-'li] (Deo — Si) ... {Dm — ivi) u ^ 0, then u is a first derivative (only) or not a derivative according as this is or is not annihilated by Da) — I. 9. A few words as to the excluded case of i=0 may be added. The substitution of e^' = ^ in this case gives to the form free from y' u=/{x', y,',y,', ...', z,z^,z.^,...] ...), where notice that f would also be free from ?// if &)F= 0 were satistied. It is not to be expected that / will be homogeneous in ?/,', 7//, .... But if it is a derivative (first or r"') its various liomogeneous parts must be so separately, and conversely. Deal with them one at a time. If, according to § 1, we are still concerned with functions i^ which are rational and integral in y^, y,, •••, so that their liomogcneity of degree 0 arises from negative powers of _y as factors of terms, the functionsy will be rational and integral in y,', 3/j', ..., and they do not involve y'. Any part of/ whicli may be of degree zero must be free from all of ?/', ?//, ?/./, .... If it be a constant or a ^ [x], it is of course a derivative of any order. ]f it be a </)(x-; z, z^, 0^, ...; ...), it must be examined by consideration of another set s, z^, z^,... Our conclusions, however, witii regard to functions F with ii^ 0 have applied, not only to functions rational and integral Mr. Hargj-eaves, A transformation of central motion. 3U in y , y^, ..., but to other functions annihilated by some power a)"" of u). The functions y which are the transformations of these are annihilated by l^^j , and so are rational and integral in y,' though not in the whole set y^\ y,', .... A part of y of degree 0 in the set may now involve ?/,', 3/,', — and we may proceed by use of a second exponential trans- formation applied to ?/,'. To the homogeneous parts of f with non-zero degrees criteria such as have been developed apply, using y', ?/,', y.^', ... instead of y, ?/,, ?/.^, The applicability is complete in the ordinary cases of/ rational and integral in ?/,', y^', ...; but in the additional cases of / rational and integral in y,', while not so in the set, there is applicability only when /, or a part of it in question, is annihilated by some power of 8y, (^i/i A TRANSFORMATION OF CENTRAL MOTION. By R. Har greaves, MA. The title is used because the suggestion is derived from the problem of central motion and tlie examples given are connected with it, but the transformation is not limited to that problem. §1. If the expression i(f' + r'6J' + r'^' sin'^) is used for kinetic energy, and no preliminary argument is adduced to shew that the motion must be plane, then the kinetic potential on ignoring <p is L = W^r'0')-^^-F^r) (,). But this may be regarded as defining a problem of plane motion with d an angle in the plane, a problem in which the central force is supplemented by a repulsive force k' j y^ in a fixed direction y. We have then the assurance that this problem can be made to depend on that of L = \{P + r'n-F{r) by a transformation, which in fact is cos ^' cos a = cos ^, sin^'=sin^sin </>, cos^'sina=sin^cos^...(2). Here 6 is a polar distance measured from OZ, <p an azimuth 40 Mr. Hargreaves, A transformation of central motion. measured from a plane ZOX perpendicular to the plane of motion, & an angle in the plane of motion measured from its intersection with ZOX, and a the angle between OZ and this line of intersection. The new orbit is derived from the original by attributing to any radius an angle in the plane, which in the primary orbit stood as polar distance for the same radius. The general position for central forces is that the kinetic potential L = \{r- + r^&^)-F{r)J^ \ lakes r depend on L^ = ^i-' — F{r) — 2r' while h being r^6, K^ ^/i^) = k' (constant), and ^„ (constant) = i f f" + ^J -^rFij-) ..(3), ■(4: + F{r) where ru= 1. Also the connexion with the problem i' = 1 (;.•-• + ,.VO-F(r) is given by }idd' = kdd, or dd' = kdQl>^\k' -f{d)] A more general proposition is that if T^ and U^ be kinetic and potential energies for coordinates x^...x^, X a positive function of these co-ordinates, then T -T TJ ^-^ R- -^^^^ •(5), makes x^...x^ depend on L^=T^— U^ — —y ; while {Xdy-{-f{e) = k\ and E^ (constant) = 1\ +U^ + £^ . X ■ \ The connexion with the problem L' = T„+ Z7„ + ^ 6>" ) - [...(6), is given by Xd' = k, making dO' = kddjW[k' -/{O)} ) exactly as in (4). The 6 component of force in (5) gives — (XO) +----^^ =0^ the integral of which is used in (5). ut 2JL Mr. Hargrea'ces^ A transformation of central motion. 41 The x^ component of force has a term dx\2 *" X }~^\ ^ X' 1 dx~ ax, 2X in agreement with derivation from L^. It' f{6) is negative we may have - /:'' for k\ The most general form reducible to (5) IS L=T^-U^+^ ad^ + Xr,, where a and the coefficients of a quadric '1\ in Oj...d may be dependent on 6^, but not on § 2. For the problem (1) with which we started and F(7') = — fifr, the orbit is ,?/?•= 1 + {^ cos ^±^\/(sin*^- sin'a)}/cosa \ with k' = fjil, Rnd 2E/ 1 k' = A' + B' - 1, L.(7). f(e) = Jc' sin'a/sin'0, A sin ^ = k\/(s\n'd - sin'a)j The case B = 0 is symmetrical with respect to the line 0 = ~; the case ^ = 0 represents part of an ellipse or hyper- bola traversed from 0~a to 6 = it — a. and back. The condition for the existence of asymptotes is yl' + i?V 1. If A''->rB'<l then as A increases from 0 the single line becomes a closed curve with 6=- a. and 6 = it — a for tangents, and two points of inflexion on the + A section so long as A < cot' a, but for A > cot'a tlie oval is convex. For A'' + B^>1 there are two asymptotes which for B positive and <1 both lie on the —A section, the +A section connecting the tangents ^ = a and 8 = 7r — a. by a finite arc. For A' + B''>1, but B positive and > 1, one asymptote lies in the + A section the other in the — A section ; in this case the tangent 6 = it -a does not belong to the orbit, which only proceeds to the asymptote on the + A section, that asymptote for which 0 is nearest tt — a. The change of sign in B corresponds to writing tt — 6 for.^. Tiie problem of repulsion from each of two axes at right- angles is solved by taking /(^) _ cos^acos^/8 sin'asin^/3 ^ k^ cos^'t' sn\'6 ' which corresponds to the transformation l .(8), sin'^ = sin'/S sin'^^' + sin'a cos^^', I and makes h^ ^m'd cQd'd= k' (siuY3 - sin'^) (siu"'^- ain'a)/. 42 Mr. Hargreaves^ A transformation of central motion. where /?> a. The angles a and j3 may be found in terms of the ratios iy/F and M\U' when the repulsive forces are i/a;^ and M\}f\ and if L=M, /3=- -a. The independent solution is Z/ r = 1 + {± -4 V (siu^^ - sin'a) ±5V(sin'';8- sin'''^)}/V(siir/3 - sin'a)...(9), with F = /i?, and 2EJ'lk' = A' + B'-l as in the last case; and here also the condition for asymptotes is A^ + B'^ > 1. The typical form when A' + B' < 1 is a distorted Hgure of eight, and the lines d = a, 6 = ^ touch the curve, eacli at two points. In the case with asymptotes one or both of the njore distant of these points of contact may be excluded, the exclusion turning on A or B being separately greater than 1. If we follow the curve from each asymptotic end the sections cross and form a loop touching 6 = a and 6 = 13 at the less distant points. When the primary orbit is an ellipse to the centre, i.e. F{r) = iJLr^j2j these transformations give respectively for the orbits cos=a/r' = ^(sin'^-sin'a) + 5cos'^±2Ccos^V(si"'^-sin'a) (sin';3- sin'a)/»-' = ^ (sin^^ - sin'a) + 5(sin"'/3 - sin'^) ±2C^J\ {^m'O - sin-a) (sin'/3 - sin'^^)}, f with filk' = AB- C\ and 2EJk'=A-\-B ) in each case ; forms which present less variety than those derived from focal orbits. In these examples the centre of force lies outside the new orbit. But if we take f^B) = k' siu'e siu'^, we get 0' = [ (Z^/V(l-sin''esin"^^), and a complete circuit for 6 is possible. So also for f(6) = — 2ma cos 9 when 2?»a < k^, and dd' = kdei^{k' + 2ma cos 6) ; i.e. we have an exact solution for an attracting centre in combination with a doublet at the origin. ( 43 ) ON THE STEADY MOTION OF FLUID UNDER GRAVITY. By TV. Burjiside. The steady motion which corresponds to Rankine's trochoidal waves is an instance of a two-dimensional rotational motion of a fluid under gravity for which the stream lines are lines of constant pressure. It" ;r, y are measured horizontally and vertically down- wards, and xp is the stream-function, this moiion is determined, with suitable units, by the equations x = 6 ■\-e~'' cos 9, y = r — e '' sin 6, These equations in fact give fdxp\' . fd4^ It is not, I believe, known that the only steady two- dimensional irrotational motion of a fluid under gravity for which the stream-lines are lines of constant pressure is a uniform horizontal stream. Assuming the existence of such a motion, and taking the origin at a point in the fluid which is not a point of zero- velocity, the motion in the neighbourhood of the origin must be given by z = a^io + OL.^iv^ + a^iv^ + (i), where z —x + ty, ty = ^ + i\p, a, ^ 0, and the series on the right-hand side of (i) is absolutely convergent so long as |m?| does not exceed some finite positive quantity. If the axes are taken horizontally and vertically downwards the pressure equation- is p+C (iy-(i)'-^^=^('-+-'^=/w. ffy-W (''0. p and if the pressure is constant along each stream-line, the pressure in the neighbourhood of the origin must be given by where the series is absolutely convergent so long as |;/-| does 44 Prof. Burnside, On steady motion of fluid under gramty. not exceed some finite positive quantity. If xo and w are conjugate iraaginaries, 1 dz dz , ^ „ , . ,_ ^ „ , , q* dw dw 1 2i 1 y = -{z-z) = - {a^w-a,to + a.y-ay- +...), Now equation (il) Implies that, for sufficiently small values of |?('j, the relation . . ,» ^ dz dz \ with the preceding values entered, must be an identity. Writing the earlier terms at length this identity is [ p,4&^«.-?^'s-(2;> + ^.i,o 2i 2i 4 2i> ^f.^-(f 2ij \8i 2iJ 8i \8i 2iJ 8i 1 + — =0. Equating to zero the coefficients of ?<?, w, lo^, low, w\ and taking into account that a,^0, i^a, = 4i - 2^ - «! - 4;>„«A + ^^^ 2a3«. - ^ 2a, a. + f a, a. = 0 / ,(ui) Frof. Burnside, On steady motion of fluid under gravity. 45 These relutlous determine aj, a,, a,, oCg, and y^ in terms of a,a,2\ ^"^ Pi' When the coefficients of the terms of the third order in w and w are eqn;ited to zero, there are four additional equations and only three more coefficients, so that a rehition between a,, a„ p^, and p^ must arise. The coefficient of lo'w^ omitting terms which obviously cancel, gives the relation 2.1 ^ .(iv). From the preceding relations (iii) P,P, = \{Px-^d{V,-^x) and 2i ' 1\- + 2i a,+ V 4 ^ 2iJ = 0, 2«/^o L The above relation (iv) then becomes p. — a, [ 2a, or, entering their values for p, and a,, I.e. 6^V;73 - (;>,- aj (2^, - a,) (7^, + a, -a,) = 0. The constants [>^, p^, etc., are essentially real, and therefore a, must be real. Hence, at every point in the fluid which is not a point of zero-velocity, the fluid velocity is horizontal. This condition is satisfied only by a uniform liorizontal stream. Tlie following considerations give an independent proof of the same result. Jf ;// is the stream-function for a steady irrotational motion of a fluid under gravity, in which the stream lines are lines of constant pressure, while the force- potential is a function of ?/ only, then Since ^ satisfies (v), so also does \dx J \dy m hold simultaneously. + =fj{y)+/{4^) (vi). 46 Dr. Wilton, A transformation of the 'partial This gives the equation where /' and/" are the first and second differential coefficients oi f{\p) with respect to \p, and g' and g" are the first and second differential coefficients ot"^ (?/) with respect to y. The last equation is of the form ^=^'(^,y) (vil), Hence, from (vi), (f^)'=^+/-{^(^,?y)r (viii)- Differentiating (vii) and (viii), with respect to y and x re- spectively, d^ ^dF dF d^P of ~ dy d^ dy ' dx' ~ 2^ ^ dxp' Adding these equations, and taking account of (v) and (vii), there results -^ ci so that ^ is a function of y only. The motion is therefore a uniform stream. A TRANSFORMATION OF THE PARTIAL DIFFERENTIAL EQUATION OF THE SECOND ORDER. By J. B. Wiltun, 31. A., D.Sc. The equation r=/ix,y,^-p,q, s, t) (1) may, by the properties of equations in involution, be trans- formed into a partial differential equation of the second order which is linear in the derivatives of the second order. In general this new equation will involve five independent variables x, y, z^ p, and q\ but in particular cases it may involve only two, which may be either x and y or 'p and q. We assume t=^<l>[x,y,z,i->,q,s, ^)] ^^ and therefore r =f {x, y, z, p, q^ s, ^) J differential equation of the second order. 47 and we determine ^ by solving the ordinary equation of the tirst order 8^8/^9/3^-1 f3l in which «3, 3/, 2;, />, and q are treated as constants. The solution will involve an arbitrary function of x, y, 2;, 2^, and 5', which we call A. Putting f=e (x, y, z, p, q, s, \) 1 d9 d(p . ,.. . we have ;r- 7r-=l (4): ds ds and the elimination of X between the equations r=e, t = ^ (5) leads to the original equation (1). ( d\ _ d ^ a A A. ^ \dxj ~ dsG ^ dz dp dq ' /^\ ^ ^ ^ s - - Kdy) ~ dy dz dp dq' Then differentiating the tirst of equations (5) with regard to ?/, the second with regard to x, we have di _ (dd\ dO d\ ^ ^\ dx ~ \dy) 8a, dy ds 8y 8s _ /^\ 8^ ^ 8^ 8s dy ~ \dx) 8A, dx ds dx ) and, on account of (4), these lead to fd6\ dd dx c^ U<H\ 80 ^]_ \dy} "^ dx dy "^ as \\dx} "^ dx dx]—^-'' from which, since 6 and 0 are known functions, we derive s in the form , ^^ ^^. and then from (5) we have r and t as functions of the same variables. There are two conditions to be satisfied, namely 8>- 8s ,85 dt 7— = 77- and >r- = ;r- dy dx oy ox which, on account of (4) and (7) reduce to one. Substituting in the first of equations (G) we obtain da^ _dj^ da- _dl dx _ fdO\^^ dx da dy dX dy \dy) ^ '' 48 D7\ Wilton, Transformation of a diferential equation. wliicli, since er is a known function, is an equation, of the type indicated, to determine \. When any vahie of \ lias been determined from this equation, the result, on substitution in leads to consistent values of r, s, and t, and Iience by three quadratures to a value of z. When the original equation (1) does not contain z, p, or q, we may regard A, as a function of x and y only; and wlien (1) does not contain x, 7/, or z, we may regard X as a function of ;; and q only; but in general X is a function of the five variables x, 3/, z, ;>, and q. Tiie expanded form of equation (8) is extremely compli- cated, but it belongs to the particular class of equations in which the characteristic invai'iant* is resoluble into two linear factors. The factors are /du\ dd j(lu\ (du\ do j( \dx) da- \dy Bo- fdu\ da- /du\ y +.-.-( 3. 1=0, ^^, \dxj d\ \d]/ , ^ ^ d\ dX where X , X = — , — - . ^ " dx' dy If equation (8) possesses an intermediate integral of the first order, it must clearly be of the form u [x, y, z, p, q, X, 0-) = 0 ; and the equation itself must be the same as (du\ du dX dn da _dO ( /dn\ du dX du da) \dxj cX dx da dx ~ da \\dy / dX dy da dy ] ' Comparing this with equation (8), and making use of (7). we readily find that u must be a common integral of the two equations g /du\ dd /du \dxj da \dy The conditions of co-existence of tliese equations appear to be the same as the conditions that (l) should possess an intermediate integral of the first order. * Forsyth, Theory oj Difftrentud Equations, vol. vi., §§32$ and 334. The UniTersity, Sheffield. du dx^ dd ''da idd 1 dx' ^ti d(l> 8X' )\i dn dx'' -m ld<i> 1 dx dd 'da /dO\ 1 dd dx ( 49 ) FACTORISATION OF N=(Y^+i) AND {X^^ ^ Y^^). By Lt.-Col. Allan Cunningham, R.E., Fellow of King's College, London, [The author is indebted to Mr H. J. Woodall, A.R.C.Sc , for useful suggestions and for help in reading the proof-sheets. J 1. Introduction. Tins Paper is intended to give Rules for the factorisation of the four allied numbers N^Y^-\, N'=Y'^+\ (1), N=X^'''^ r^^ N'^X^''+ Y^^, [X prime to r].(2), and to introduce the Tables (printed at the end of the Paper) of the factorisaliun thereof up to very high limits. 2. Rarity of primes. The salient property of these num- bers is that they are nearly all composite, and are indeed nearly all algebraically resolvable. In fact the only form among them not so resolvable is that of " Fermat's Numbers " — ^V'=r^+l, where F=2<', [e=:2'"] (3); and, among these, the only known primes aie the lowest two, 2'+ 1 = 5, and 4*+ 1 =257, whilst the next two are known to be composite, viz. 16'«+1 = 274177. 67280421310721 ; 256256 4-1 has the factors* (2'3.39 + l) (2'M 19 + 1). 3. Notation. All symbols integers, p, q denote odd primes. £ denotes an even number ; <u denotes an odd number. 6 = 2"' ; «' denotes a square. m denotes any factor of n. F{n), F'{7i), ^{n'l, *'(«), see Art. 6; 0 («), 0'(«), see Art. 6. 4. Algehraic and Arithmetical Factorisations. The general process of factorisation of large numbers is naturally divided into two main, and very distinct, steps — I, Algebraic, 11. Arithmetical. The algebraic resolution is described in Art. 5—16; the * Discovered by the author. VOL. XLV. E 50 Lt.-Col. Caiiningham, Factorisation of N=Y ■+■ 1, dr. aritlnnetical resolution Is described in Art. 17-21/. The Factorisation Tables of iV, N\ wliicli are the outcome of this Memoir, are described in Art. 22-22c. 5. Ahjehraie Factors. An algebraic function f(x,y)^ which is ail exact divisor of the algeliraic expression F {x, y) for all values of x, y is styled an ahjehraie factor ot F{.r, y). U f{x, y) itself has no such algebraic factors, or is in otlier words ir reducible, it is styled an algebraic prime factor of F {x, y), and is denoted (for shortness) by A.P.F. The maximum al-cbraic prin)e factor of i^ (a, ?/) is denoted (for shortness; by M.A.P.F. These are denoted symbollcully thus f{x,y}is an A.V.T. oiF{x,y) (4rt), (p{x,y) is <Ae M.A.P.F. oi F{x,y) (4&). The most important property of these algebraic factors, in relation to factorisation, is that F(x, y)= the continued product of all its A.P.F. (including the M.A.P.F.). ..(5). [The Hrithmetical factors of the various A.P.F. are generally of quite different forms, involving different modes of search (described in Art. 17-21/)]. The resolution of F{x,y) into its A.P.F. is therefore a most important — (usually the first) — step in the factori- sation of large numbers. 6. Algebraic Factors of Binomials. The number and nature of the A.P.F. of Binomials (.t" ? ?/"), where x, y have the same exponent (n), depend chiefly on the nature (prime or composite) of that index (/() : so that a short notation exhibiting distinctly the relation to the exponent {it) is con- venient. Let F{7i) = x''~y, F'{n)=x''+y» [.v, j both + ] ...(6a), /(n)= an A.P.F. of 7''(n), /'(«) an A.P.F. of F'(«) (66), 0 {n) = the M.A.P.F. of F(n), 0' (u) =the M.A.P.F. of t" (»)• -{Gc). Hereby F(l)=/(l)=.0(l) = (.v-:y), i^'(l)=/'(l) = 0'(l) = (.v+j) i&d). It will now be shown how to obtain the M.A.P.F. as the quotient of the products of various A.P.F. of F{u), F' {n). Here five types should be distinguished according to the form of n. i. K = e = '2'-'; ii. rt = w; iii. n = eui; iv. .t = $'", j>' = ij'" ; v. nxy = n- Lt .-Col. Cunningham^ Factorisation of N^Y -i-l.ctr. 51 7. Type!. ?? = e = 2^; [.r^^r, yi-rf'\ F{e)=F(\e)\\F'{le); 0(e) = F'(ie) (7"), F(2) = F(1)1|F'(1); 0(2)=F'(I) (76), F{4.) = F{l)\F'{i)\\F'{2); 0(4) = F'(2) (7f), F[S) = F{l)\F'(\)\F'{2)\\F'{^); 0(8)=F'(4) (7^0, F(<;) = F(I)|F'(1)1F'(2)|F'(4)| \\F'[le) (7^), and the factors F' (e) are irreducible for all values of e, so that 0'(1)=^'(1), 0'(2)=i^'(2), 0'(4) = F"(4), ...0-(e) = !-(*) (8). [The single bars (|) used above are merely special multiplication symbols used to separate distinctly the various A.P.F. ot' Fin) ; the double bar (|j) is used to separate all the minor A.P.F. from the M. A.P.F. of F{n). These symbols are most useluL iu arithmetical work : see tlie Tables ou pages 72—74]. 8. TypeH. n = oy; [xii^'\ yi^i"]. Let a, 6, c denote unequal odd priines \a <h <g]. The values of ^ (?*), ^'(") ai-e shown in the scheme helow, for all the (odd) values of n required in this l\Ietnoir. n <p(u) </>'<") 1 1 ^'(1) a 1 F(a)-F{\) F\l) ...(9a), F' (a)-F' (\) ...(96), rt» i F(a:-)-F{a) F'(a:-)-F' (a) ...(9c}, o" '■ Fia'^\'- Fia'^-i) F' (a'^)-F' {a"--^) ...(9({}, ab a'b SF(ab)F{\)}^{F{a).F{b)} [F{a^).F(a)\-.-{F{a').F{ab)} F(nbc).Fia).F{b\.F(c) F{bc).F{ca).F(ab).F{\) {F'{ab).F'{\)\^{F'(a).F'{b)} \F' (a:'b).F' {a)]^\F'{a-').F'[ab)] ... F'(abc)F'{a).F'{b).F'(r) ...{9e), ...(9/). ■ ■.((/), F'{bc).F'{cu).F'(ab).F'(\) and the values of F(ii), F' (j{) are shown in the scheme l>elow, expressed as the continued product ot" their A.P.F., fur the same values of n as above : the A.P.F. being arranged in order of magnitude (the smallest on the left). F(«) = n {/(«)} F'(»)= n(/'(rt); 'l'(l) <1>(\)Ma) (t>{l).<i>{a) <^(a2) <P(l}.i{,(a).<p(a^-)...<p{a"-') rt'{\).<P(a).'l^ib).<l,{a'j) a-^b <p{ 1 ).0(ft).(^(«').</>(6).</)(a6).^(a26) abc \ <^(l).!/>(a).'/.(6;.<^(c). <lj[bcj.(p(ca).(f){ab).<t>{al>c) f'ii) (10a). '/''(l).<//(«) (106), </)'(! ).<^'(a).'/>'(a-) (10c), <p'{l).<p'(a).<p'{a'')...(i>'{a'''') (lO(^), <t>'(\) 't>'{a).(t>'(by<p'^ub) (lOe), (t>'i\).^'(n).fl.'{a-).cpUb)(p'[ab).,p'(a'b).{U]f}, 0'(l).0V)-'/''(6)-('/>'(<:) <p'(bc).<j,'lca}.<p'{ab).(t>'{abc)...(\Off). Com})aring the above formulse, it is seen that, when n = co, (pin), 0'(") are of the same form (I la), F(?^), F'(?i) are of the same (orm ('16j. 52 Lt.-Col. Cunningham, Factorisation of N = } T 1, &c. 9. Type iii, ?? = e<w = 2". <» ; [a; ^t ^""j 3^ 7^ t?"]. The reduction ot" i^('O) -^ ('0 '•'^ treated separately iu Art. 9a, %. 9a. Type ilia. ?< = e&>, [a;^^^'", y:^rf"^. Reduction of F{n). The function i^('0 nv.xy be tirst resolved, as far as the quantity e = 2'' is concerned, in a way similar to that used tor F{e) in Art. 7, and with similar notation; thus F(*a,)=F(i*w)||F'(ie<o); F{2i,i)=F{w)\\F'{io); F{i..) = Fia,]\F'lu,)\\F'{2u>)i F {Su,) = F(w) I F' (00) I F' (2a,) 1 1 F' (4<«) ; (P(ew)=4>'{}eio) (12a), ^(2a;; = <^'((o) (12i), 0(4a>)=«^'(2«.) (12o), ^(8e«) = <^'(4w) (12rf), F(e(>i) = F('«)iF(a<)|F'(2w}|F'(4a.)|...||F'(iew) (12e). The factors F{u3), F' (co) above are of the Type ii. of Art. 8. This A.P.F. may be found by the Rules of that Article; and they may tlien be expressed as the continued product of their A.P.F. by tlie Rules of tiiat Article. fThe use of the single bar (|) and double bar (\\) is s-imilar to that explained at foot of Art. 7]. 9h. Type Vub. n=e(o', [x^^"\ y^ztf]. Reduction of F' (n). Here F\ea)) is irreducible, as far as the tactor 6 = 2"* of the exponent is concerned (compare Art. 7). The reduction of the factor (o is simitar to that of the n = a> in Art. 8. The value of (f>' (n) is shown in the scheme below for all the values of n required in this Memoir: those of F' {)i) are also shown alongside, expressed as the continued product of their A.P.F. ; tlie A.P.F. being arranged in order ot magnitude (tiie smallest on the left). '2a 4a ea 2«2 2a6 ,p'{n) f{-la)^F\'2) F'(ia)^F'{i) F'(ea}-^F'{e) F'{-la'']-^F'(2a) F'(4a^)^F'(4a) F'(2ab).F'{2) 'y'~C2^)Tr(2b} F'{2a^b).F'{2a) F\2a*).F'{2ab) F'{n) ^>'{2).cp'(2a) .I>'(i).'t>'(ia) (j)'(e).(p'{ea) ^,'{2).(p'{2a}.(t>'{2a-) <^/(4).<^'(4a)<^'(4«*) <l)'(2).,t>'(2a).<p'{2b).4,\2ah) ^'{2).<p'{2a).(t>'{2a'^).(i>'{2b).<p'{2ab)<p'(2a^b). .(13«), .(136), .(i3c|, .(13r/), .(13e), .(13/), .(13^). 10. Type iv. x = ^"\ ]/ = v"- When x, ?/, the bases appeiiiing in F(;n), F'(n), i\re them- selves both m"' powers of smaller bases s, r) — a case excluded Lt.-Col. Cunningham, Factorisation of N=Y + 1, cLr. 53 from Art. 7, 8, 9— then tlie A.P.F. of F{n)^ F'(n), viewed as functions of a', ?/, are further reducible, as functions ot ^, r]. For, writing <t>, $' as functional symbols ot f, ??, wlien a*, y are changed to ^"j ?;'", then F (n) = -r" ~jy" = ^""i ~ r)™" = * (nm) ( 14«), F'(n)=.v'»+jy" = ^"'''4-»j'"'' = <l>'(mJi) (14<^)) and tiie A.P.F. of *P(inn), ^' (vin) may now be found by the Rules of Art. 7, 8, 9, where inn now takes the place of the n of those Articles. And <P{mu), <!>'(»*/<) may also be expressed as the continued product of their A.P.F. by the Rules uf those Articles. 11. Type V. nxy=u. The development of this case (for tacttorisation purposes) depends on the expression of the M.iV.P.F. of F(ii), F'Qi), i.e. of ^(n), 0'(»O in one or other of the impure* 2'" forms, (P'^^nxy Q'), or in some derivative thereof {P'+mxyQ^) Avhere m is some factor of n, in the general case {i.e. inde- pendently of the condition n xy, or m ory = D). The necessary and sufficient condition for this is w = o), or 2aj only, \_n^im~\ (l''5)- A preliminary discussion of the general case occupies Art. 12, 12a-o?. The application to factorisation occupies Art. 15. 12. Impure 2'" Forms (n^ms'). When n has the form )i = w, or 2&», and =t= m«- (16), the M. A.P.F. ^ (>0, (j)' {n) are always (algebraically) expres- sible in one or other of the impure 2"" forms as below — n 4i-l P--nxyQ' P^ + nxyQ- 0'(") P'^ + nxyQ- '17a), P^-nxyQ^ (176). P* + nxyQ' (176), where P, Q are certain functions of n, x, y explained in Art. 13 and tabulated up to ?i = 46 in the Table A therewith. 12a. Impure 2'° Forms {n composite). If n be as in Art. 12, but composite, i.e. if >» = w, or 2tt}=km, and 4=ju.>*, [A = w, and prime to ni] (18), then «^(?0, 0'('O, the M. A.P.F. of F{n), F' (n) are always (algebraically) expressible not only in one or other of the impure 2"" forms {P'^nxyQ') as in Kvi. 12, but also in one or other of each of the derivatives thereof, {P'TrnxyQ'), {P'^kxyQ'), * In this Memoir (/''-+ nxyQ^) is styled an impure *2'<= form ; and (/"-T^Q") is styled a pure 2'<^ form when n does not depend on x, y. J, i;, Ac. Ji2 r 54 Lt.-Col. Cunningham, Factorisation of N= F + 1, &c. similar to (17ni,Z>,c), Avliere ???, h now take the place of n in those foriuulse, the si{,Mi (T) being determined hy the forms of m or k: and this is possible for every way in which n can be resolved into two cofactors {n = hni): but the P, Q are quite different. The process of finding the P, Q in this case from those of Art. 12 is rather troublesome. One way of effecting it is shown below, for the case of (^ (n) = {P'+m xy Q'): it depends on expressing <p (») as a quotient of two 2"= forms of same determinant (D=±m.ry). The case of ^'(??) is precisely similar. Write a;*^=|, /=->?, [_k = a»\ whereby F{n) becomes a function of S, V, which may be written 4>^.("0, with ^,.(va) as its M.A.P.F. Then, by Art. 8, _ F{km).F{\) ^ %[m) PCO ^ <\>Jj^ '^^"- F{k).F{m) 4>,(1) - F(m) '^.(»0' and, by Art. 12, (p,^{m) = one of (P/T m E.V QJ), = one of (P;/+ "2 xij Q^'), M'here Q,' = [xyY^"-'^ .'q, [k being odd], and <^, (m) = one of (P,'T »« 3:2/ (J,') therefore (^f «) = one of ,/.,^"'"^"^ ^V » [same sign in both] (19). As these 2'" forms are of same determinant {D±inxy), their quotient ^ (/«) is (algebraically) expressible in the reduced forms, (p{n) = F--mxyQ- {19a), bv the process of conformal division * Here P^., Q^. and'P,, Q^ are given by the Kules of Art. 12. The carrying through of this whole process is rather trouble- some where n is large. 12h. Impure 2'" Forms of Suh-Factors {n = km) : When n is composite, as in (181, every A.P.F. of F{n), P ("), s:iy '^(^•), ^("0, <t>'i^), </>'('") 'S also (algebraically) expressible in its own impure 2'" forms of determinant I> = ±kxy, mxy under Rules similar to those of Art. 12, 12a. Here k, m take the place of n of these Articles. 12c. Impure 2'" Forms. n = ms\ [no square factor in rn]. When n = 7ns'; then, by a suitable substitution for .r, y. * i.e. division with preservation of 2^" form. See the author's Paper on Connexion of Quadnitic Forms in I'roc. Land. Math. Soc, vol. xxviii., 1897, pp. 289 et seq. for a full explanation of the process. Lt.-Col. Cunningham, Factorisation of ]S'=Y^+l^ &c. 55 the quantities F{n), F' (n), wliicli are functions of x, y, may be brought to the forms ^{in), ^\vi), where 4>, <t>' are functions of s, r] (and m is free from squares). Hereby <t>(j/0) *'("«) and their A.P.F. now fall under the forms of Art. 12a. The only cases required for this Paper are Type!. n=^a-,a', a*, ... ; Type ii. n=a-b; Type iii. w = 2a', 2ff*. The reduction of F(^ii), F\n) to the forms ^(ii), ^'(») is here first shown for each type and followed by the expression of ^ ("0, ^'("0 '" ^^'^ ^""^''" {P'^'n ^v Q')' Type i. n=d\ a\ a\ ... «". Take ^ = 03"-", 77 ^t/"-", 7t = «'* gives F(n)=.v"~y" = ^"- yf = ^(a) ; m = a ('20«), F'{H) =.v" +_)•" = ^^ + ij" = ^'(a) ; f)i = a (206). Type ii, n = a^b. Take ^ = x", r} = if. n—arb gives F(») = .v"-j)'" — 5"'' -!)«* = ^ {ab) ; m=ab (21a), f'(»)=.v''+j"' = ^«'' + »j''«' = <l>'(a6) ; m=ab (216). Type iii. n= ^ci", 2a', ... 2a". Take ^ = a;"--«, Ji = ?/«-2a^ ?i = 2a" gives f '('^) = -^'"+J'" = ^""+''"'' = *'(2«) ; m = '2a (23). And the reduced function (j) (vi), (p' {m) now fall under Art. 12 (since qh is free fiom square factors), so that the A.P.F. 0(?>i), <j)Xvi) may now be expressed in the form {P' + m^rj Q') as in the scheme below. Here P, Q are the same functions of ^, 7] as shown in Art. 12 for ft', y. Type 71 S V i. a2 a;« ya i. «■" ,-a ii. a-b Z" ya iii. 2a- X" y" « = 4i-fl .a = 4*-l a = 4i+l a = 4i — 1 a6 = 4i + l a6 = 4j-l 2a <j)(»i) r-i-?nb,Q'- P^ + 7nb,Q' <p'{m) r"-mbiQ- (236), P^ + Hi5'/(?2....(24«), V-mlnQ' ....C2\b), P- + vibiQ- ....(Iba), P*-mbiQ\...{lbb), P^-vtK'iQ' (■16). 12d. Examples. A few examples will serve to illustrate the principles of Art. 12, l2a,h. The quantities P, Q^ are of course different in each of the 2'" forms here shown. Ex. 1°. F(15), F'(15) have 0(15), ^/(15) as M. A.P.F., and have also 0(5), 0(3), 0(1) ; 0'(5), 0'(3), 0'(1) as lower A.P.F. 0 ( 1 5 ) = P^ + 1 5.VJ' Q' = P-- o.vyQ- = P- + Z.x-y Q'- ; (p'(\5):=P--\oxy Q- = P^ + oxy Q-=P-- Zxy Q^ ; (p{5)^P^-5xyQ^; ^{}i) = P- + 3.vyQ^; (p'(o) = P' + 5xyQ-; . <p'('i) = P'--3xyQ'- Ex. 2°. F'(30) has 0'(3O) as M.A-P.F., and has 0'(1O), 0'(G), ?)'(2) as lower A.P.F. 0'( 30) = P2 ._ 3o.v_y (JJ = p2 _ 1 O.vyQ" =P^- Gxy Q- = P- - 2.\y Q-, 0{1O) = P-- 10. vyQ', 0'(6)=:P--(Uj(?-, 0'i2) = -'"-^-^J^'- 56 Lt.-Col. Cunningham, Factorisation of N= Y^ T 1, &c. Ex. 3°. f (45), r(45) have 0(4o), 0'{45) as M.A.P.F., and have also 0(15), 0(9), 0(5), 0(3), 0(1); 0'(15), 0'(9), 0'(5), 0'(2), <p'{\) as lower AP.F. (p'{l5) = P- + 5xyQ'=P^- 3xy Q' = P^-\ 5xy'' ; 0(15), 0(5), 0(3) end 0'(15). 0'(5), 0'(3) have the forms shown in Ex. P. 0 ( 9 ) = P2 + 3.13' Q\ 0'( 9 ) = P' - 3.VJ ^=. 13. Values of P, ^. Tlie quantities of P, Q are homo- geneous symmetric functions of x^ y so tliat tliey may be written P = J„.v'=^ + ^,.v^ '^ + J,.v'=^-y+... + ^,.vj-=^-> + ^„j-=r (27a), Q= ^^.v" + i^.-v-j- + A-v'^-=/ +...+ i?,.vj^'' + ^^j-' (276). Here, using t {£) to denote the Totient of 2, then — 7i = aj has tij- = iT (a>), « = ■»•— 1 (28a), n = '2a* has ■nr = ^T (2(ti), 'c = -nr— I (286). And .4,, ^r are functions of 71 only, (not ot .v, j) (29a), also ^0 = 1, Bf,= \ alwai/s (296). ^1, = |(h+1), when ii=ii+\=p; J,=i(?«~I), when n = ii—\ =pq..(29c), Ai=^(n—'[), when ;i = 4i— 1=;?; --1,=^(k + 1), when )t=ii+l=pq..i29cl}, ^1 1 = |h , when m = 2w (29e). The rest of the coefficients (J^., B^) are complicated functions of n, and are difficult* to calculate, so that (for the present purpose) they are most conveniently taken from Tables. U'ablef A on page 57 gives the values of the coefficients A^^ B^ which appear in the terms A,x'^-^y% AyX'^y'^-'-, B,.x^'ryr^ ^...v'/'" (30), for all the values of the index n up to n= 46 (except n = 43), the middle coefficient — (or the middle pair of coefficients, if equal) — being enclosed in brackets. They are tabulated tn the same order as in the formulae 27a, i, so that it should be easy to allocate them to those terms, as the Table gives the value of CT, K also in the same line. "^J'he Table is drawn up really for those values of ^{n),<p'{n) which are ex[)res3ible in the form 0(«) or (p'{u) = P'-I).Q^ "...(31). where D=nxy, as in Art. I2a,b or =n|»j, as in Art, 12^, this being the form required for the important Auiifeuillian notation (Art. 15). * The method of calculation is explained in Ed. Lucas's Memoir Stir lex fnrimilts de Cauclnj et de Lejeutie Diricklet in the Report of the Association Francdise ponv P Aoancement des Sciences, Congress of Paris, 1878. t This Table is extracted (with some alterations and corrections by the present author) partly from Ed. Lucas's Memoir Suv la Serie recurente de Fennat, Rome, 1879, and partly (with some cliiintres) from the Memoir above quoted. Both these Metnoirs are out of print, and difficult to obtain : (see Appendix IL for err.ita in the originals). F_ Lt -Col. Cunningham, Factorisation of N=Y +l,(ir. 57 Tab. a. ■^ 71 0' 1 Coefficients Ar in P K 0 Coefficients Br in Q 2 1,1 I 3 0' 1 1,1 c I 5 '/> 2 i.(3).i 1 •il 6 (/.' 2 i,i3).i 1 1,1 7 </)' 3 i,(3.3),i 2 i,(l),l 10 (// 4 i,5-(7).5,i 3 I,(2,2),I 11 </>' 5 i,5,(-«, -0.5,1 4 M.(-i),l,l 13 "/' 6 i,7,i5,(i9),i5,7,i 5 i.3.(5.S).3.i 14 <(>• 6 ',7,3-(-7),3,7.i 5 I,2,(-I,-I),S,I 15 <P' 4 i.8,ii3)-8.i 3 I.3.3.I 17 <P 8 1,9,11, -5, (-i5),-5. 11,9,1 7 i,3.i,(-3.-3),i.3.i 19 '/>' 9 1,9. 17, 27, (31, 31), 27, 17,9,1 8 i,3.S.7.(7),7,5»3.i 21 '/' 6 i,io.i3.(7),i3.io,i 5 1,3,(2, 2),3,I 22 -/>' 10 I. II, 27, 33, 21, (II), 21, 33, 27, II, I 9 i,4.7.6,(3.3).6,7,4,i 23 f/* 11 1,11,9,-19,-15,(25,25), — 15,-19,9,11,1 10 i.3.-i.-5.i.(7).i. -5.-1,3.1 26 '/>' 12 1,13 i9.-i3.-"-i3.(7), 13,-11,-13,19,13,1 11 1.4.1. -4.1.(2,2),!, -4,1,4,1 29 <^ 14 i.i5 33.'3,i5.S7,45-(i9), 45,57,15.13,33.15.1 13 1,5.5.1.7,11,(5.5) ",7,1,5.5,1 30 <^' 8 I.i5.38,45.(43).45.38.i5.i 7 i,5,8,(8,8),8,5,i 31 '!>' 15 1.15,43.83.125,151,169,(173.173), 169,151,125,83,43,15,1 14 1,5,11,19,25,29,31,(31), 31,29,25,19,11,5,1 33 0 10 1, 16,37, 19, -32, (-59), -32, 19.37,16,1 9 1,5,6,- i,(-9,-9),i,6,5.i 34 '^' 16 1,17,59,119,181,221,243,255,(257) 255,243.221,181,119.59,17,1 15 1,6,15,26,35,40,43,(44. 44).43.40,35.26,i5,6,i 35 </>' 12 1, 18,48, 1 1, -55, -1 1,(47), -I I, -55,11,48,18,1 11 1,6,7,-5.-8,(5,5^ -8,-5,7,6,1 37 '/> IS 1,19,79,183,285,349,397,477.579.(627), 579,477.397.349,285,183,79,19,1 17 1,7,21,39,53.61,71,87, (101,101), 87, 71,61, 53, 39.21,7,1 38 <P' 18 I, 19, 47, -19, -135, -57, 179, 209, -83, (-285), -83,209,179,-57, -135,-19,47,19,1 17 1,6,5,-14,-21,10,39,14, (-37.-37).i4.39.io, -21,-14,5.6,1 39 <P' 12 1,20,73,119.142,173,(193), 173,142,119,73,20,1 11 1,7,16,21,25,(30,30), 25.21,16,7,1 41 0 20 1,21,67,49,7,35, 15,1 1, -23,- 65,( -31), -65,-23,11,15,35,7,49,67,21,1 19 i,7.".3.3,5,i,i,-9,(-7, -7),- 9,1, 1,5.3.3,", 7.1 42 0' 12 1, 21, 74,105, 55, -42, (-91), -42,55,105,74,21,1 11 i,-i3.-45,68,83,(-i20, -i20),83,68,-45,-i3,i 46 '^' 22 1,23,103,253,469,759 1131,1541,1917, 223 1,2463. (2553),2463, 2231, 1917, 1541. '131.759.469,253. 103. 23. 1 21 1,8,25.52,89,138,197,256, 307.348,(373,373),348,307, 256,197,138,89,52,25,8,1 58 Lt.-Col. Cunningham y Factorisation of ^= ^ + 1) <^'^ 14. Sam of coefficients A^, B^. Let 2 (^), 2 (i?) denote the sums of llie coefficients in Let (t'j v), (t, v)* be solutions of T'' — nv'-= - 1, T2-Jil;2=+1 (31a), ■whereby {nu')" — nT'- = n (316). Now take x=\, y=\. These values reduce P, Q to P = Z(^), <2 = I,IB) (32), ■wherebj- 0(«) or 0'(«) = 'P'-«^' = {S (^)}--w {-(^)}* (33), But (p (n) or ^'(»0 are at same time reduced to either n or 1, as follows : — i. « = 4«+l=r;) gives 0(«) = =w, whence Z(A)=nv', 1(2?) = .(34a) ii. ?j = 48+l=;)y gives 0()<) = whence (l»._lm)(]l_l.) 1, (1^- P)(i«-1«) L{A)=^T, 2(i?)=u (346). iii. ?t = 4i— 1 =p gives 0'(n) = 1, always, whence 2(.4) = t, Z(2?) = i' (34c), iv. n = 2i=2p gives (ji'{m)=\, ahoays, whence 1.{A)^t, I.{B) = v (34rf). These Results (dia-d), giving the values of ^{A), ^(B) in terms of the known solutions (t', v'), (t, v) of (31f/, i). are very useful — (being easily applied) — test^] of the correctness of the tabulated coefficients (^^., B^). A Table is subjoined giving the values of <p (n) or ^'(»), 2(^), 2 (i5) — as in the formulae (34a— c?) — for ready appli- cation of the Test, for all the values of n in the general Table of A^, B^. (Table A). n 2 3 5 6 7 10 11 13 14 15* 17 19 21 22 23 26 29 0.0' 5:(^) 2(^) 2 I 2 2 I I 5 I 5 5 2 2 I 8 3 I I 19 10 6 3 13 65 18 I 15 4 I 17 31 17 8 4 I 170 39 I I I I 29 55 197 24 51 377 12 42 5 10 70 n 30* 31 33 34 3.5* 37* 39* 41 42* 46 0.0' Z{A) I 241 44 I 1520 273 I 23 4 I 2149 420 I 71 12 37 53^5 882 I 1249 200 41 205 32 I I 337 24335 52 35«8 * This symbol t, as here used, must not be confused with its use as the symbol for "Totient," as used in Art. 13. t In fact it was by this Test that several Errata in each of Lucas's printed Tables were discovered. A List of these is given in ApiJcndix II. Lt.-Col. Cunningham, Factorisation of N= Y ■+■ 1, d:c. 59 In most of tlie cases the minimum values of (t', i>'), (t, v) are the ones to be used in the formulEe (34a— of) for 2 (^), S (Z?), but, in the cases marked * the solutions next greater than tlie minimum (t', v), (t, v) are the ones required, viz. n ; T- — 7iv"— +1 15; 31- - 15.8-= + 1 35; 71''-35.12==-1 39; 1249--39.2002=+i 771 ; t'- — Jill'- = — I 37; 8822-37.145*= • VI ; r-—nv"= + 1 30; 241--30.442= + l 42; 337^-42.522= + ! 15. AurifeuilUans. The formulas of Art. 12- 12c show that when the determinant {± D) of the impure 2'° forms (17rt-c, 19a, 23a-26) has the form + Z) = ?j .vy or m xy = 4*, a perfect square ( 35) , so that one or other of the R.P.F., i.e. 0('O) ^ ('0? [oi* 0 ("0? '/>'("0] '•'5 (algebraically) expressible as a difference of squares, viz. i. 71, or m = 'ii+l gives (f>{n or 7n)=P^-(sQ)- (36a), ii. 71, orm = 4« — 1 gives (p'(n or }n)=P' -{sQ)'- (36i), iii. M, or7« = 2w gives (p'{/i or m) = P- -{sQf (36c), and is therefore at once (algebraically) resolvable into two cofactors (say L, M) so that q)[7iorm), or (p'{n or vi)== L.M (37a), where L = P-sQ, M=P + sQ (376). 'J'he functions 0 {n or m), (^'(n or «0, which are resolvable in this way, are styled* AurifeuilUans, and are described as off order n or m. The two algebraic co-factors are styled Aurifeuillian Factors. Tlie condition of this resolution (35), styled the Auri- feuillian condition, may be satisfied in the following ways: 1°. When y=\, so that f («)=y'-l, F'(«) = v»-M, then y = 7in- gives D = {nnf=s-...{Z%a), 2°. When x and j'> 1 ; then .v = ?-, y = 7iii- ; or .^• = »t^ >' = ')^; give X) = (»*?'/)' = s2... (386), * From having been first tabulated, and used for factorisation, by M. Auiifeuille of Toulouse ; see Lucas's Memoirs above quoted. They ai'e obviously of great use in factorisation : thu.s nearly all numbers N'= 1'^+ 1, in which Y^\n, are of this kind, or else some of their algebraic factor's are of this kind. t Special names are applied to distinguish the orders. Thu.'j those of order 2. 3, 5, 0, JiC, are styled Bin-, Trin-, Quint-, Sexi-, iic.—AurittuiUians Ihese names are due to the pi-esent author, who has made a special study of them, see his Papers in Lond. Math. Soc. Proc, vol. xxix., 1898, and MesAemjer oj Math<^- mutigs, vol. xxxix., 1909, and many places in tlie Educational Times Ueprints. 60 Lt.-Col. Cunningham, Factorisation of N= Y +1, <lr. 3. "When a- and^'>l ; and n — nitin, then .v = ?i,S^, j = n2'r; or .v = ??2?"> y = '>hif; give Z> = ())5t))' = s^...(38e). [In the formulae (38a, b, c) m may be substituted for «, when required]. It will be seen (Art. 12, 12c, 15) that Aurifeuillians can only occur among the A.P.F. of F{n)^ F' in) ; and that, when n is composite {n = hn)^ the M. A.P.F. ^(»i), ^'(»0 'ii^y have an Aiiriteuillian ot" any order [h. m) for wliich the Auri- feuillian condition (35) is satisfied; hut. yor particular values of a*, y, this condition can only be satisfied in one way. Similarly, when n is composite (u = /:««), the lower A.P.F. of i^(??), F' [n), say (t>(vi), <f)' (m) niay possibly each have one Aurifeuillian form (of order m) for particular values of x, y. 15a. Tabulation. In the Factorisation Tables at end of this Paper the Aurifeuillians are recognisable at sight, because their Aurifeuillian Factors (L, M) are shown separated by a colon (:), thus 0(m) or (p'{m} = L:M (39). The 07'der (7n or m') of Aurifeuillians (p (wi), <p' (m) occur- ring in each F(n) or F" (ii) is shown by the figures in the columed headed Aur. Ex. \. 7? = 27. iV' = 27-' + 1=381 + 1. Here iV' = 0'(l)0'(3) . 0'(9) . 0' (27) . 0' (81), by Art I2c. And each of the A.P.F. after 0'(1) is a Trin-Aiirifenillian (>n' = Z, Art. 15). Ux. 2. ?ir=45. iY=45<5-l. Here A^ = 0(l).0(3) . 0(5) . 0(9) .0(15) .0(4.j). And each of the algebraic factors 0(5), 0(15), 0(45) is a Quint- Aur i- femllian, (m = 5, see Art. 12^, 15) ; but 0(3), 0^9) are not Aurifeuillians. Exceptional Cases, {L = \). In a few cases, when x, y are small, the lesser Aurifeuillian factor {L) reduces to Z, = l. In these cases the niimber ^y or 3' may be written i\ or ^¥' = 1 : M, to show that JV or -V is tlie limiting case of an Aurifeuillian. Ex. ,v = 2; iY'= 1-+(2.P)2 =1 :5; [L = l, J/=5]. x = 3; A'' = {P + (3.12)3} + (1 + 3.P) = 1:7; [L=l, M = 7]. 16. Quotient- Aurifeuillians. If -4, = Z/,il/j and A^=L^M^ be Aurifeuillians of same order (m), and A^ he a divisor of A^ with quotient !E, thus 21 = jLi- Ai is an AurifeuUian of same order («») (40), and n = 1L.m = ~= ^^' (40a), ^1 LiMi Lt.-Col. Cunningham, Factorisation of N= Y T 1, ctr. 61 and the co-factors H, iH of E can be found directly by tlie preceding E,ules, or by the property of sucli Quotients, viz. 5f4 L2 Lo IfVV M'y Mn ^ = l,''m.^ •^ = m°'Z7 (^'^ These divisions can be performed algebraically; but, it is often more convenient (in practical factorisation) to find the (numerical) values of Z«,, il/,, L.^, J/^, and then perform the divisions (arithmetically). [When either L, or Mx contains a small divisor p,, the proper divisor L, or 3/, ot either Lo or M2 is easily found, by determining Hrst (by trial) which ot Ao, 3/2 contains ^1- Then the Hule is "The multiple Lx or Mi of px is a divisor of the multiple L2 or Mn of /),"... (42). This process is most useful when 0'(w,) = /^,. (p'{m.,) — Ai are both Bin- Aurifeuillians. Ux. n = 2w^ gives iV' = (-'w')-'""+ 1 = 0'(2).0'(2w).0' (2^-^), [Art. 96]. Here 0'(2.) = ^ , 0'(2.^) = ^^^ , [Art. 96]. Now f (2), F'i'Ioj), F'i'lio-) are all Bin-Aurifeuillians so that 0'(2w), 0'(2«j') are Quotifiit Bin-Aunfeuillians, and may be resolved as abovej. ^x. ;i=18; A"' = (2.32)2-^'=0'(2).0'(6).0'(18); 0'(2) = F'(2)-l^'-l-l = (l^+l)'-2-18=(19-6):(19 + 6) = 2o: 13; =Z,,3/, ; F'(6) = (182)=+ 1 =(i83 4-l)=_2.183 = (58o3- 108) :(5833 + 10S) = 5725 :5941 =L..M.,; f '(18) = (18^)2+1 =( 18»+ 1)^-2. 18» = (18" + 1)- -(25.3')- = 198358660513 : 1 98359920225 = £3 • -'-^a ; = 33358093: 37.37.25309; [Here the small factor 5 in Lo and il/3 shows that Lj is the divisor of 3/,]. Pakt II. Arithmetical Factors. 17. The finding of the arithmetical prime factors {p) of the various A.P.F. of F{n), F' (n) is the most difficult, and most laborious, part of this research. The Theory of Numbers is the guide in this part of the work. The five following Articles, 18—205, apply equally to all forms of <p, <p'. 62 Lt.-Col. Cunm)}gham, Factorisation of N= Y -+- 1, dec. 18. Linear forms of factors (p). All prime factors (/)) of the A.P.F., whicli occur in this Memoir, must be of the following* linear forms, For0(«). 0'('O> 7' = "'nr-t-l; for 0(?»), (f>\m), ;j = ww+ 1 .-.(43). 19. 2"= forms of factors (p). The A.P.F. of F(}i), &c., can always be (algebraically) expressed in the following forms, 71 or m = ii+l gives 0 (?t or w), 0'(» or w) = S- — mT- (44a), 71 or wi = 4i-l gives (Z!)(« or mj, 0'(?4 or tn) = S-\-mT'^ (446). , n or m = 2/jL, [m = w] gives 0'(« or m) = a* + b-= both i>- + ;u.7'=...(44c), norvi = efi, [« = 2'', «> 1, /i = w], gives 0'{«or»n)=a2 + b* = c' + 2d« = S' + MT==S'^ + 2/x7"-...(44r;). [The S, T in the above forms are of course different]. Each of these 2" forms involves in general a set of linear forms of factors; but the fact of their being representations of A.P.F. of F, F' reduces these to the single type (43) already quoted. These 2'" forms are thus of little use in factorisation, so will not be further treated of. 20. Quasi- Aur if euillian Forms. The impure 2" forms (P-+nxy Q') treated of in Art. 12-12r/ form an important help in limiting the linear forms (45) of the arithmetical factors of the various A.P.P". of F, F'. When the determinant ±D = n xtj, or m xy of those forms, has the form ±D = n.\y, or m.vj=/u,s' (45), then the impure 2'" form, in which such A.P.F. would be in general (algebraically) expressible under Art. 12— 12c^, reduces to a pure 2'" form of determinant ±D = fi, for then The A.P.F. = P- + »«.v}'(?-, or P'* + »i a;)' ^- becomes P- + iM{sQf (46). These functions are here styled Quasi-Aurifeuillians, and the conditions (45) which lead to them is styled the Quasi- Aurifeuillian condition (from the analogy of the functions treated of in Art. 15). The algebraic values of P, Q may be taken from the Table A of Art. 15. It will be seen that the pure 2''' forms thus produced — depending on particular values of the elements u;, y — are * Except that n or in itself is a divisor of </> [n or in) when n or in = {x~y), and of (/)'(» oi' "*) when n or ?«= [x-¥y)\ but these cases do not occur with the foims (1), (2) of F{n), I<'(n) of this Memoir. Lt.-Col. Cunningham, Factorisation of JY -^ Y^^ 1, dtr. 63 additional to, and usually of difftrent determinant {±T) = fx) to, those of Art. 15, wliicli are common to A.P.F. in j2;enenil {i.e. for all values of x, y). Some examples will make this clear. Ex. Taking »i = lo, the two examples below show for 0(15) and (ji'dH), when ^.v, y) = {l\ 3),^), or (5-, 5),'). Toji Line; The three impure 2>"= forms, common to all x, y. 2nd Line ; The Quasi-Aurifeuillian 2i<= forms of the above for certain x, y. 3rd Line ; The normal pure 2'« forms, common to all .r, y. (x, y, \.ixtj ; F^+\hxvQ\ P'-^-^xyQ^ , P^-5x>,Q^ ; Im2mre2^<^. 0(15) \P,[W,o[HuY; P^ + b{sQY , p-'+{sQ)' , P--\5{s(2)^; Quas-Aw. [x, y. ; t^-ou- , . , T' + loV'' ; formal. (X, y, \bxy ; P^-l5xvQ\ P'-3x>/Q- , P' + .ixr/Q'' ; Impure 2''^. 0'(15) r, 5.;^ 3(5?.)/^ P--3(sQ)-, P^-\5xyQ\ P- + {sQ,^ ; Qiias-Attr. [x, y, ; A- + 3B^ , r^+15f7'^ , . ; ^-or7nal. [Note that the P, Q are of course different in each form]. 20a. Case of IX = I. The important case of the A.P.F. — F''—{sQ'), which gives immediate (algebraic) factorisation, has been fully tieatcd of in Art. 15 under the name Auri- feitiUian. Referring to that Article it is seen that, under the condition fx — 1 (whicli is that of Art. 15), When 0(7!) or (p\n)=^P--{sQf, then 0'(h) or (j){n)= P'^ + {sQf...{M). The latter form, conjugate to that of Art. 15, is styled Ant- Aurifeuillian. An example will exhibit this property (47) clearly. Ex. Take « = 27; i^(?0 = 27-"-l = 38'-l=*(Sl); i^'(")=27"+l=3"+I=*'(8I). *(81)=0(1).0(3).0(9).0(27).0(81); 4>'(81) = 0'(1).0\3).0'(9).0'(27).0'(81). Here, except 0(1) and 0'(I) — All the 0(3)... 0(81) are Trin-Ant-Aurifeidllians, (algebraically) ex- pressible in form P'^+{sQ)''. All the 0'(3)...0'(81) are Trin-Aurifeuillians, (algebraically) expressible in form P- ~{!.Qf = L.M. 20b. Quasi- Aurifeuillians, Linear Forms of Factors. The pure 2'" forms [P'^ jxisQ)'] arising from the Quasi- Aurifeuillian condition (45) involve certain definite linear forms of all arithmetical factors (p), usually different'^ from that given by (43), which is connnon to all A.P.F. And, wiien n is small, these linear forms are simple and few in * This considerably limits the linear forms possible in such cases. 64 Lt.-Col. Cunningham, Factorisation of N ^ i^^T 1, &€. nuni1)er. Tliose for sniall^* values of n or m are tabulated below, up to n or m "^ 10. 2'"^ form P = 4-gr+l ; f'-ar+],^; fi-tg-+l ; 20tB-+ 1, 3, 7, 9 ; 24-nr + l, 5, 7, 11 H-BT + l.Q.ll ; 40-or+l, 2, 9. II. 13, 19. 23, 37; . ; 8'nj-+ 1 P = A'*— SB'-; t--bu^; g' — <-'h^ ; t' — ~iu^ ; t-—l{)u^ V2-m+\ ; 10^+1; 24-tir+l,5; 2S-nr+1,3. 9; 40w+ 1, 3. 9, 13 21. Residuacitij . The following five Articles (21a— e) apply only to the A.P.F. of F(n)^ ^'('0 wherein A! = 1 ; i.e. only to the forms (1^ -F l)t of this Paper. 21a. 2"^ Residaacity. When /( or m=2}—\, then 0(n or m) = (ji{hn or ^m) 110'(i« or ^m) ...(48a), then 0(g« or ^m)~Q (mod;j), when (t/Jp).,= + 1 (486), and 0'(^« or5w) = 0 (mod ;:>), when (ijjp)„z= — l (48c), Here these cases are at once determinable by the simple laws of 2'" llesiduacity. 21^. Residuacity of order v. Let ^ be the hast exponent satisfying the Congruence / ^yp-ir"^! (mod;;) (49), where p = v%-\-\, ^ = (p — \)-^v (49a). Here ^ is styled the Haupt-Exponent of y (modulo p), and y is said to be a Residue of p of order v. this last relation is often expressed thus {ylp)i, = l, which means j'^ ' "^ = 1 (496). and here it is clear that r,i=t, = [p—\)-^v, with (j/p)v=l (49e), is the condition that ^(«0 or ip,' {m) = Q (mod p). 21c. Case of {Y T 1). As a result of Art. 18, taking a; =3/, 0(«) or0'(;O = M-A.F.F. of (y^+l) = 0 (mod ja) requires p = kY -^\ (50), whence ( — A-F) in + 1 (mod p). Hence F^ + l = 0 (modp), if (--6)^= ±1 (mod p) (50a). * For the linear forms when n or vi> 10, see Legendre's Theorie des Nombre.i 3rd Ed., Paris, 1830; t. i., Tab, 111. to VII. t And therefore not to the form (A'-^-^ + Y^^). Lt.-Col. Cunningham, Factorisation of N=Y + 1, dc 65 When k is small (compared to 1^) tills affords an easy way of testing whether ^ is a divisor of ^ (») or ^' {n). 21d. Residuacity-Ruhs. The above test may be written ^(p-i).i-_ +(_i)r [mod^j], or (/c/;;)*= ±(-1)^ (506). Hules are known for determining whether ■(^z/p)^=l for the cases of small indices (A), viz. k = 2, 3, 4, 6, 8, 12, 24, but they are dependent on the theory of complex numbers, and are too ditficult for ordinary use. They have been reduced to really simple forms tor the eight small bases (z) z = 2, 3, 5, 6, 7, 10, 11, 12, but these Rules* are too lengthy to quote here. 21e. Simple Cases. A simple application is when ^- = 2, 4, 8, 16; the reduced results are shown in the Table below: p Y Y^^+\ {modp) r^=-l (mod jo) 2r+l . p = Sto-+1,3; p^S-xir+0,7 (51«), 4F+1 . j9 = S-cr+l,5; . (516), fw (2//;)s=-l; (2/;^)s= + l (-51^). "^ + ^ \. {2lp),=. + i; {■i/Pk=-^ (old), Ifiy+l i*^ (2»,= -l; (2/;^),= + l (ole), ^"^^^ \. (2lp),= + l; {2IpU=-1 (51/;. To apply these Rules, note that p = 8'rsr+l=a.- + {i(3y- gives (2//;),= (T)/3 (52a), p = S^+l=iU + \y- + iSliy- gives (2/p), = (T)"^'^ (526). 21f. Table of Roots y {mod p)- The short Table B following gives the Results of the above Art. 21— 21e, i.e. the proper' roots (y) of the Congruences M.A.P.F. of{yy-\) = 0, {y''+l) = Q (mod p &;/> 1000) [y< p&p"]... {53), omitting however (for shortness' soke) all primes (p) of forms p = 2y + 1, 4_?/+ 1, where y is a prime : as the roots (y) thereof can be at once inferred by the simple Rules of Art. 2lZ», „=2y + l= 1^'^ + ^' gives jyi' - 1 = 0 (modj*), [y prime] (53rt), 1 8tsr + 7, gives3'»+l =0 (modp), [jy prime] (536), p = 4y + l gives j'y-l EEO, and (23')«s'-l=0 (modp), [y prime]... (53c). * See two Papers On the mimei-ical factors o/"(a"-l) by the late C. E. Bickraoic in Mesteiiger of Maths., vol. xxv., 189G, pp. 1-44; and xxvi., 1897, pp. 1-38. VOL. XLV. I" 66 Lt.-Cd. CuiiningJmm, Factorisation of N =Y^ -v- 1, &c. Proper Roots [y] of if = ± 1 [mod p d' p"), [;/ </>]• Tab. B. p + 1 * 4, 8 ' -1 P 4-1 2 ' -1 P + 1 2 / -1 17 331 683 »9 9 337 42, 56 21 691 31 6 349 87 58,174 701 175 35,350 37 9 18 3S3 . 44 709 59, 177 354 4' 20 367 183 727 121 43 373 93 31,186 733 6l 15 30 379 189 63 739 <^7 33 397 743 371 ~i 5 35 401 25,100,200 75' 125 375 73 18 409 204 757 9, 189 378 79 39 419 209 761 380 8() 22 11 421 105 210 769 40,64,192,384 48 97 24, 48 6, 8 431 . 787 3s)3 101 25 50 433 809 40 4 103 51 439 73 811 109 443 34, 221 821 205 410 "3 7, H 449 112, 224 28 823 411 127 457 . 827 413 131 65 461 23, 115 230 829 207 414 137 34, 68 487 243 853 213 426 5 39 69 23 491 245 857 428 151 499 83 859 429 143 157 .521 52, 260 5 877 219 146,438 Jb3 81 27 523 58 881 110 55 181 45 30, 90 541 135 270 883 49, 441 191 38 95 547 42, 273 907 453 193 48, 96 12 5^9 284 911 7 '97 49 98 57' 95 919 199 11, 99 577 36 929 232, 464 58 211 105 593 37, 74 937 26, 264 223 599 J 299 941 235 470 229 601 947 473 43 233 58 29 607 loi 303 953 239 119 613 153 306 9b 7 483 241 8 617 77, 154 971 . 251 25, 60 619 309 977 61, 244,488 257 4, 8 63. 991 . 495 271 30,54 135 641 16 32 997 83, 166 277 643 107 281 28,35,70 647 323 283 659 329 ■17 P" + 1 ^' -1 307 311 34 51 165 65 1 673 165 33, 330 8 292 14 3^3 52, 156 677 169 338 37' 18 Lt.-Col. Cunningham. Factorisation of JS =Y T 1- &c. 67 22. Factorisation-Tahles. At the end of tliis Paper follow three Tables (I,— 111.) — the principal outcome of this ]\len)oir — giving the factorisation into prime factors as com- pletely as practically possible with the means available. Tab.I.,II.; F(r) = (r^-]); F\Y)={Y^ +\); [up to r= 50]. Tab. III.; f (Xr) = (X^^- F^^); F'{XY) = {X^^ +Y^^^); [uptoXr=30]. The following is an Abstract of the degree of complete- ness of the factorisation attained; tiie larger numbers are of course very incomplete. y F(Y); F'[Y); F{XY); F'iXY); Bi'Ses Cotnpltte Good Pi ogress r= , 1 to 16; 18,20,21,22,24,25,30; 27,28,82,84,35,36,40,42,44,4.5,48; r- XY= XY= 1 iol6; 18,27; 21,23,24,25,28,30,33 35,36,45; 1 to 2i; 24, 15.2,6.5; 28,10.3 1 to 18; 21; 22,24,15.2,10.3,6.5 Limit 50 50 30 30 The Tables themselves are described ia the Articles 22 a-c, following : 22a. Arrangement of Factors. Each number i^ or F" is shown resolved as far as possible into its A.P.F., and those A.P.F. which are Auiifeuillians are (usually) resolved into their twin co-factors (L, M). Each A.P.F. and each L, M are shown resolved as far as possible into their arithmetical factors. The A.P.F. are arranged in order of magnitude, the lowest on the left, and the highest (the M. A.P.F) on the rigl.t; and the L precedes the M. Within each A.P.F., and within each L or J/, the arithmetical factors {p and p*-) are arranged in order of magnitude of the primes (;j), the lowest on the left, and the highest on the right. In incomplete factorisation a blank space is left on the right (of the incomplete A.P.F., L, or M) to admit of the insertion in MS. of new factors. 226. Special multiplication symbols (• | || ; :)• These are used to separate various kinds of factors in such a way as to indicate the nature of the factors. Use of dot (.). This is used between arithmetical factors in the same A.P.F., (but not between the A.P.F. themselves). A dot on tlie right of an arithmetical factor, followed by a blank, indicates the existence of other unknown arithmetical factors. Use of bars (| and ||). These are used between the A.P.F. of (X*- F«), where e = 'l'^, (see Art. 7), thus — X'- Y'={X- T) I {X+ Y) I (X2+ Y^) I (X^+ F^) I ... II (x''+ Y^"), the double bar (||) being placed just before the M. A.P.F. Thus the arithmetical factor, or group of factors, between a pair of bars (I ... I) is always an A.P.F. of above form. Use of semi-colon (;). This is used between A.P.F. not of form (X'— Y'). This occurs in both F(>0, F\n) when n = w, (see Art. 8), and also in F'(") when n = eu), (see Art. 96). A semi-colon on the extreme right indicates the complete factorisation of theM.A.P.F. Use of semi-colons (;) between bars (! ... |). This occurs in the case of 68 Lt.-Col. Cunningliam^ Factorisation of N^ Y -f- 1, &c. F(ew), which is first resolved into its A.P.F. with respect to the exponent e = 2'\ (see Art. 9a), thus — Each of the above A.P.F. of form (X*'*'=y*"), where k = 2", is further resolved (see Art. 96) into its A.P.F., which are separated by semi-colons (;), thus taking the form Use of colon (:). This is used between the twin " Aurifeuillian Factors '* (L, il/) of an Aurifeuillian. These Aurifeuillians occur as complete A.P.F., so that their ends are marked by either bars (|) or semi-colons (;)— [see above]. Use of queries (?). These are used in two ways : — (1) A query (?J on right of a large arithmetical factor (>10") indicates that this factor is beyond the powers of the Tables available to resolve or determine primes. i'i) A query on right of the (small) arithmetical factors of an "Auri- feuillian Z-Factor " indicates that it is uncertain whether this belongs to the Z- or il/-factor. Blank spaces. In the incomplete factorisations blank spaces have been left for the insertion of the (as yet unknown) prime factors in MS. 22c. Special column-headings. The entries in four columns on the right, headed Fac, Aur, Lim., In., have the following meanings : — Fac. This column shows the number of A.P.F. in F (I ) or F'{Y). Aitr. This column shows the order of Aurifeuillians (if any) in F (Y) or F\Y). Lim. This column contains symbols (f, J, If, §) which show — (in case of incomplete factorisation only) — the limit to which the search for divisors {p andp") has been carried, thus t to 1000; * to 10000; •[[ to 50000 ; § to 100000 ; [or a little further]. [It will be seen that the search has been carried to at least lOOOO throughout Tables I., II., and in all but three cases in Tab. Ill ]. In. This column indicates by "initials (B, C, &c.) — according to the list below— the names (so far as known to the present author) of the original workers who have effected, or have materially contributed to, the various factorisations — B. Bickmore, Chas. E. Lo. LoofF, Dr. C. Cunningham, Allan Lu. Lucas, Ed. E. Euler, L. Where no initials are given, the present author is responsible. Appendix. In this Appendix is given a short description of the extensive* Tables which were available tor the Factori- sations of this Paper. 23. Tables for factors of {Y T 1). It will be seen from Art. 2lh that the search for factors (;>) of y"=pi = 0 mod j) is involved in that of finding 'proper roots (?/) of the Congruence >'"'-l=0 (modi!)). [m = ?=(/)-l)-^j/] (51). * These Tables have bten over 20 years under preparation. Lt.-Col. Cunningham, Factorisation of X=Y -\- 1, d:c. 69 Tlie search is llierefore dependent chiefly on Tables of the sohitions, i.e. pruper roots {y) of that Congruence. The Tables available are described in the Art. 2Za—d following. 23rt. Benschles Tables. These Tables* give the completef set of roots {y<kp) ol the Congruence j.m_i = o (mod />> 1000), for the following values of m : — m — every odd prime, and prime power < 100, every odd composite up to 69 (except 65), every power of 2 up to 2'= 128. every multiple of 4 up to 100 (except 88, 92) and 120. Tlie roots (>') of y"'+l ==0 are not especially mentioned, but appear as follows ; — The roots (y'<hp) of y'^+l rEO, with v)=io, appear as negatirs roots (—jV) of y— 1=0, with ?«=((> The roots {y' <\p) of j'"' + lEEO, with ;« = €, appear as roots (j) of ^"■-1=0. It will be seen that these tables give a very extended range of the index (m); but the range of the modulus {p) is so restricted (/;;:)> 1000) that their use in factorisation is very limited. 236. The auth:>r'' s Tables. The author has had extensive Tables of this sort J compiled, giving the complete set of proper mots y, y' :— When m = lo ; of j'"' -1=0, and y'"' + 1 = 0 (mod ;; and p"). When m =t; ot v'"' + 1 = 0 (mod p and p" ) for the values of m stated below, and up to the limits of ;j and j)'^ stated : — ?« =2, 3, 4, 6, 8, 12 I 5,7.9, I 10,11,13,14,15; j9 and i&«> 100000 | 60000 | 50000 (or a little over). 23i:. Creak's Tables. Mr. T. G. Creak has compiled TablesJ§ of tlie same sort as the above for the values vi, stated below, and within the limits of p and'p" stated : — »«=i6 to 50, 52, 54, 56, 63, 64, 72, 75 (mod ;; and /;''> 103 up t^ iq'). 23d. Small bases j';^12. Besides the above, the author has — in conjunction with Mr. H. J. Woodall, A. R.C.Sc, compiled Tables giving the Haupt-Exponents (^) and Max. -Residue Indices (i/) of the following Bases (_>■) : — 2/ =2,11 for all primes and prime-powers ;f> 100000. y = 3, 5, 6, 7, 11, 12,^ for all primes and prime-powers ;;|> 15000. y = 3, 5, 6, 7, 11, 12,** for all f = w> 105 and = i :)> 210, for all primes and prime-powers ^ 50000. * Tajchi complexer Primznhlen, by Dr. C. G Reusclile, Berlin, 187.>. t Some errata have been found in this part of these Tables. A list of these will be published heieafter. I These fables aie now in course of publication. § These Tables Mr. Cieak has kindly placed at the author's dispo.=al. [] In five papers on " Haupt-Expoueuts of 2" in the Quarterly Journal of Math., vol. xxxvii , xlii., xliv., xlv. ; 1900-1914. ^ In course of publication. ** At present only iu ilS. f2 70 Lt.-Col. Cunningham, Factorisation of N= Y^ + 1, dec. 24. Tables for factors of (Z^^? Y^^). The Tables available for this purpose are described in Art. 24a below. They differ in use from the Tables des- cribed in Art. 22a— d for factorising the simpler forms (F^+l) in that the particular A.P.F. of F{n) or F' {n) to which the divisors formed belong cannot always be iden- tiHed from the Tables themselves. These Tables are also not nearly so exte»»slve as the XT' xr previous ones, so that the factorisation of {X ^ Y ) cannot be carried to such high limits as in the simpler case. 24a. Cano/i. Aiithmelicus. This Canon* gives two kinds of Tables for every prime (p) and prime-power (;;") as moduli up to p and p" ^ 1000. One Table gives the Least + Residue (R) of all the powers gP of the base (j, up to the limit p < p or ;/. {i.e., it gives R to Argument p). The other Table gives p to Argument /{, with same limits. Here g is in every case some prirnitice rooff of the modulus {p or p'^). Use of the Table. The right-hand Table gives the powers {g", g^), such that g" = x, and g^^^y (mod p or p"). Hence .r"':?:j"' = «/'»"qrff'"/' = ^'"/^{5r™»-»'/3q:l), [mod p or p«]. Hence .v"'-j"» = 0 (mod p), if m(a-/3) = 0 [mod (p-1)], .v™_y' = 0 (mod ;;"), if m («-^) = 0 [mod -r], vm^^,m = o (mod p), if m(«-/3) = 0 [mod i(P-')]' but not =0 [mod (/; — !)], x'"+y"' = 0 (mod ;/), if m {a-fi) = 0 [mod ^t], but not ^0 [mod x], where x = (p—l). ;;""'. This Table suffices for finding all the divisors p and ;)''5>1000 for all bases (.v, y) whatever, because the Base (^f) of each Table is always a primitive root of the modulus {p or ;/). [The use (in factorisation) is very limited on account of the lestricted limit of the moduli {p and /;" '^ 1000)]. 246. Binary Canon. This CanonJ is quite similar to the Cduon Aritktneticus (Art. 24a), and has the same scope. It differs only in ihat the Base 2 is used in e\ery Table throughout (instead of a primitive root, g). It can be used in precisely the same way as described in Art. 24a : but its use is of course limited to bases (.v, y) such that real values (a, /?) exist giving 2** = ^;, 2P=y {raod. p or p'<). * Canon Arithneticus, by C. G. J. Jacobi, Berlin, 1839. This Canon hag unfortunately many Errata ; "the Appendix contains five 4to pages of these. The present author has'found a few more : a list of these will he given hereafter. t The priuiiLive roots (g) selected are frequently so laige as to be very inconvenient for numerical calculations, eg. with ;) = '.'97, the chosen _<7 = 6oi). Fortunately this is of no importance when" (as is usual) only the Residues of g" are required. X Binnrij Canon, London, 1900, by tlie present author, prepared for the British Association. Lt.-Col. Cunningham, Factorisation of N= Y +1, dLr.. 71 24c. Other Canons. The author has had Tables* prepared giving (at sight) the Least Residues {R, R'), both + and — , of the powers [zl') of the small Bases {z) named below on division by all primes (;;) and prime- powers [pi^) up to the limits of p, ;;, named below: — Unse s = 2t 2t 3, 5, 7, lot, 11 Poivers of z ; p> 100 3(5 30 Moditli; p Ik ?"> 10000 12000 10000 These Tables suffice for finding divisors {p and 7/) of (a"^_)"") where •v, y are any of the above-named Bases (s), or small powers thereof, up to the limits of m=p, and p, jj'^ named. 24rf. Special Congruence Tables. Two sets of Tables of the same kind and scope werej available, connecting the auxiliary Bases 2 or 10 with each of the Bases j' = 3, 5, 7, H by the Congruences : — i.§ 2^" = ±y'«, and 2^''>'. y"" ~ ±1 (mod p or p'^:j> 10*). ii.§ 10"^" = ±y"\ and lO*"'. y"" = ± I (mod 2> or ;>*^> i0<). The Tables give (at sight) the solutions {x,,, .v^', «, and the + sign) of the above Congruences for each of the small Bases J' = 3, 5, 7, 11. In both Tables i., ii. «4 denotes the absolute mininwm exponent possible for the Base j. •Ao, .To' mean the least exponent of the auxiliary Base y going with the exponent a» of y. From these Tables may be formed, by aid of the Haupt-Exponents (^2. ?io) of the Base 2, 10, all possible Congruences connecting the auxiliary Bases 2, 10 with the other Bases r = 3, o, 7, 11 : — i. 2^=+j", 2*.j'°=±l; 10^= +>'^, 10^3'"= ±1 {nxoA pSi p''>\^*]. In all such Congruences the tabular exponent a^ is a necessary factor of the exponents a possible to y. Hence these Tables are suitable for finding factors (p and />" > 10*) directly of numbers of following forms, [j/=3, 5, 7, 11] : — i- (2^= + >'"), [l^.y^+D; ii. (l0-+j"), (10^v"+l). They may also be used — with some additional trouble — for finding factors {p and p"^ lO'j of the forms (r« + «c*), (ij«.«<!«'+ 1), vi'nere i", w are any of the Bases 2, 3, 5, 7, 11, or any of tlieir powers, or any products thereof. * All at present only in MS. t 'I'he Tables of Bases 2 and 10 were prepared by the author and Mr. H. J. Woodall, of Stockport, jointly (but independently). % these are now in course of publication. § I he Tables i. were prepared by Mr. H.J. Woodall and the present author conjointly (but independently). 'Ihe Tables ii. were piepared by Mr. H. J. AVoodall and Mr. T. U. Creak conjointly (but independently). 1 2 3 4 5 6 ( 8 9 10 11 12 13 14 15 16 17 18 19 20 21 24 2.5 26 27 28 29 SO 31 32 33 34 3o 36 37 3S 39 40 41 42 43 44 45 46 47 48 49 60 ; 2 Lt.-Col. Cunningham, Factorisation of N= Y +1, d:c. Factorisation Table of F{Y] = [ Y^ — 1). Tab. 1. o; I 3; 2; 13; 3!5l'7; 4; 11:71; 5;43ll7;3J; 2-3; 29.4733; 7l3;3!5; i3lli7;24i; 2; i3;7S7ll4; 1:7; 19:37; 9;4i.27i||ii; 9091; 2-5; 15797-1806113; 11; i57|i3;i7:i9ll5-29; 20593; 4.3; 1803647:53.264031; 13:8108731113.5; 7027567; 2.7; 241; 1 1. 4931; 61 39225301; i|3l5l'7|2.S7lt)5537ll64i-67oo4i7; 16; 2699538733.^:19152352117.? i7;343; 991-343271119; 307; 73465841; 2.9; 19; 251:11.61 13. 7; 15238111401; 41. 2801. 22236 1; 4-5; 463; 43-631-3319; 4789-6427:227633407; 3-7; 67.353-'i76469537l|23; 89-285451051007; 2.t i; 461.1289. 23; 60. I25; 7. 791577; 349: 13,7311331777; 97- 1 '34793633; 4; ii-7i;938425i:'Oi.25i.40i||2.3; 521; 1901.50150933101; 25; l|27;937-6449.38299-397073. 2; '3; 757; 109.433.8209; 3889. 27; 113.4422461I29; 13007:35771115.137; 281. 4-7; 59-^ 29;49.i9;83793i;i22ii.5i94ii6i||3i;i3 67;ii.7i26t;27i483i.5i783i; 2-3-5; 3 '13; I '!5;5-4i| 17; 61681I257; 4275255361 II65537; 32; II23; 2113. ; 67. 3. II; 103.137. 115-7; 307-443- 153 II 12643.28051. 4708729; 2.17; 31.49831; 43.44007727; 281. 5;43; 19.2467I7; 31; 46441I37; 13.97; 73-54i:55ii7 II1297; 1678321; 577. 3313. 2478750186961.? 4.9; 149.1999.7993.? 37; 113-13; 191. 2.19; 7.223; 53. 131. 157. ;3I2l. 3.13; 2625641 141; 121.20641I1601; 281.5501:241.1758111 II769-3329; 8.5; 83? 41; 13.139; 3851. 1460117; 1009. II II43; 1723; 29.337.548591; 547.19489. 2.3.7; 173.6709. 43; 6337- 19-5; 23.43i6489:89-99i.3037|| 1113-149; 4.11; 19.109; 1471:2851; 10009.829639; 2891101:31.183451; 181. 9-5; II47; 2.23; 1693.? 47; 13.181149; 37:6115.461; 53o6ii3!53o84i7; 8929. 3155927939II II17-H3- ;97-'93- 2-3; 29-4733; 3529. II 8; 113:911; 197.883. :3823. 49; 6377551; 151- II 113.17; 11.557041; 251. ,c «.. s j 1 T ►J 2 2 r> 0 2 5 4 2 8 6 3 4 2 6 3 2 13 4 4 7 2 17 + 6 1 2 ■t i + 1 6 5 4 21 4 2 + 8 6 1 G 0 1 4 H 5 H 6 H 2 29 -t- 8 2 ;j; 12 § 4 33 f 4 ■t- 4 i + 12 6 1 'XI ■h L ■U 4. 4 + 4 + 8 10 t 2 1 1 + s <- + 2 6 1 11 6 5 z 4 t 2 t 10 3 t G 7 + 6 J E E ]•: Lo B B C Lu C E Lu C Lu Lu C C BC LuC Lu ELu Lt.-Col. Cunningham^ Factorisation of iV= I'^q: l, &c, 7H Factorisation Table of F' [Y) = [ ri'+ 1) . Tab. II. )' F'{Y)=^Y^ -VX 1 s »^ "^ 1 2; 2 1:5; 2 2 ;{ 4; I ••7; 3 3 •1 257; 1 •'' 2.3; 521; 2 (1 37; 13:97; 2 6 7 8; 113:911; 2 7 s 257; 97 673; 2 <» 2.5; 73; 530713; 3 c id 101; 3541:27961; 2 10 Lo 11 3.4; 23.89.199:58367; 2 11 B \-l 89.233:193.2227777; 2 B Vi 2.7; 13417. 20333. 79301; 2 C \\ 197; 2929.3361:113.176597; 2 14 Lu 1.') 16; 211; 31. 1 531; 1923 I; 142 1 II; 4 15 C !(■ 2741 77.672S042 13 10721; 1 Lu 17 2.9; 2 t IN 13:25; 229:457; 33388093:37.37.25309; 6 2 BC ]9 4.5; Ii363i4669i9.?:870542i6ii2i.? 2 19 X 20 I 6000 I ; 2 t 21 2.11; 421; 81867661; 337. 4 11 BC 22 5-97; 2 22 1 23 8.3; 47. 139. 1013. 52626071 :2.i98o7766i567473.> 3 23 X 24 17. 2801. 2311681:33409. 2 n 25 2.13:41.9161; 3 n 2<i 677; 53- 2 26 n 27 4; 1:7; 19:37: 19441:19927; 163. 208657. 224209:i297-58794'578i; 5 3 C 28 614657:449.23633. 2 t 29 2.3-S;233- 2 X 30 17.53:809101; ; 61.181.21872881:1784464680181.? 4 30 1 C 31 32,373.1613..? : 2 31 t 32 641.6700417: 2 ^ E 33 2.17; 7. 151; 23.1871. 34544013769? ; 661. ■1 t 34 13-89; 2 34 X 35 4.9; 1 1.132631; 29.5209.11831; 71.701..? 4 35 X 36 17.98801; 5953 473895897; 3 t C 37 2-19; 593- 2 I 38 .s- 17-17; 4 38 t 39 8.5; 1483; ; 79..? : 5 39 r + 40 I7I7-H3-337-64I-929; 2 t C 41 2-3-7; 2 + 42 5-353; 6734621; ; 4 42 + -1- C 43 4.11:947.1291..? : 2 43 X 44 45 41.113.809:353.9857. 2.23; 7.283; 41.97841; 7309.II36089; 61 ; 2 X 6 X C 4G 29.73:1013..? : 2 46 X 47 16.3:659..? 2 47 X 48 ;'769. 2 I 49 2.25; 13564461457; 16073. 3 X C 50 41.61:5122541:7622561; 101..? 5 2 X C Y — 74 Lt.-Col. Cunninghcun, Factorisation of N=Y + 1, ttr F9 ^ XUl'J J71V «o ^ • z :2 c^ .CD . O C O .9»^ I <M(MlMC^T(<CO<M-»<ca(M<MC4-*-<l<TJ< + 1^ H >1 o ~ ••• 0\'^ vO f^ o^So " 2:^ 2 O ro"^ M -^ M c- ;;, " ■r^ '■'> ir< ?J LO ,^ '^ ^ h- "^ «^ ;.- ro i^ ro r^oo tOOO J.^-"" «- LO ro OO "^ CI rOc>3 o ?" <M<MCOClCOC^l^CO(MCClM-^C-4COU-; CO ic •* i^ «C C3 >o 1^ ^ cc CO i^ >r^ o co '"?7 1 ++ -1— -n- ■*"K 1 "Oj^-co ov^ 1 •<j.-(<<oM'-*cocc-*i-*cc-^iii;oooooo O o o „• O ,^ •- • - ro p. " ro .. . r^ ' 1 ..CO • o- n ?: r^ -^ O p „ rmy. ~ — ^ Tj- r^ ►- ro I-' OO ,._ ro vO >/-) r^ t^ .» O --O L/ N M- •- O ^) ■~ CO ro ^00 o ~ '-J "^ ■£ >-. ^4 ►- M OO - W ^ - "• 5- " :? - -1 "^00 o . s C^ " LTj ro t^ 1 -- " rc " ;,' Lt.-Col. Cunningham^ Factorisation of N=:- Y + 1, &c. 16 Appendix II. Erraia in Ed. Lucas's Tables of {Y" — nzZ-). \°. Comptes Rendus de V Association Fraiti-aise pour V Arancement dcs Sciences, Paris, 1878, pp. 164 — 173. Sur les Formides de Cauchy et de Leiennc-Dirichlet. By Ed. Lucas. page 168. Col. of " Coefficients de F," : Lineof ?j = 29. i^o>- 33 + 15, ^«<rf 33+ 13 + 15. Lineof;i = 33. For-\^], Read +\^]. Lineofrt = 41. For —bl, Read +67. 2". Separate Reprint of above Paper, Paris, 1878 : page 5, line 13. For z,. Read Z,. page 5. Coi. of "Coefficients de }',''> Line of w = 29. For 33 + 15, Read +33 + 13 + 15. 3°. Sur la Serie recurrente de Fermat, by Ed. Lucas, Rome, 1879. page 6. Table of " Formules de Mm. Le Lasseur, &c." In the formulae for Y : Line of ^ = 22. For +.v^>'^ Read +Ux^yK Line of ^ = 29. For +I5.v"j'3, Read +13x'^y^. Line of ^ = 33. For -19.v3y^ + , Read -59x^y^-. The Tables in tlie above Memoirs are not identical : they differ only in the signs (±) of the coefficients .4,., B, when « = 4? + 3 for all odd values of ?•. The signs in the Table of Memoir 1° apply directly to (p [n) : those in Paper 3° apply diiectly to (/>'(«), when 7i = -ii-^3, this being the factorisable Aurifeuilliau form : these signs are adopted in Tab. A of the present Memoir. ( 76 ) NOrE ON CLASS RELATION FORMULA. By L. J. Mordell, Birkbeck College, London. Let F(m) be the munbi-r of uneven classes, G (^m) the whole number ot classes of tonus of deterrniinint —m, the classes (1, 0, 1), (2, 1, 2) and their derived classes being counted as i and -^- respectively; to F{0) we attribute the value 0, to 6^(0) the value — j^^- Let a be any divisor of vi which is <\/m and of the same parity as its conjugate divisor d. In any summation involving a, we take ^a instead of a when a = \//n. Further, call any divisor of in, h or c according as its conjugate divisor is odd or even. Put Q=^F{n)q\ where as usual <^ = e"^'". Then II where n — 0 is omitted from the right-hand suninialion, and e'(x)=^, ^o. = ^oo(^-)- We also have more or less similar equations when 0^,, [x, qj) on the left-hand side is replaced by manv other functions, e.g. ^00 (ic, men), or again when Q ami R are replaced by series in which the coefficient of g" is equal to the class number of particular kinds of quadratic forms of determinant — 9i, e.g. taking amongst those reduced f^rms whose third coefficient is odd the excess of the number of those whose Mr. Mordell^ Note on class relation formulce. 77 first coefificlent is odd over the number whose first coefficient is even. The derivation of formulsfi of this kind is extremely simple, and will form the subject of a paper entitled '' Class relation formulse," which 1 hope will appear in due course of time in the Quarterly Journal of Mathematics. I may- notice, however, a few obvious applications of the formulae A and B. Putting x = ^ in (A), we find 11=0 V 1 +q''^'' w=0 1=0 •(!'), where if t is zero, we Avrlte ^n for the coefficient n. Equating coefficients of ^"', we have F{m)- 2F{m-l'') + 2F(m - 2') -. . .= S a (-1 )5(n+'^)+i...( 1 ). But, putting rw = 0, we have 8^/9„ =-4 2 ng^"- fiZJL ] i 1 ^,0^00^0," The second term on tlie right-hand side can be written as 1 1' = 1 i: it' e^^ -n' dx and this is found to be 00 8 S KM , when 03 = 0, i2k-1 so that equating coefficients of 5"', we find i^(w) + 2F(m-rO + 2i^(wi-20+...=-2a + S5....(2). Putting now 03 = ^ in (^), «> /I _o2»«x \ Q 6 '6 =^J»(-'^""'^"HrT?=' + --- 78 M7'. Mordell, Note on class relation formulce. and adding this to equation (l'), we have = S [24F(»)- 12 G^ («)]?" (3)- M=0 Putting rc = 0 gives \11B - n=V ^ VI- $2"..' TT ^oi^.n"^00 ^•^ '^on ^.' 10 9 9 01 on '0 '_ 1 ^,;'_ 1 d tt'' t^^^ IT' dx CO = 1 + 8 S q2n when aj=0, SO that equating coefficients oF §"', 8 S i^(w - r') - 6 S (7 (»i - r') = 4 S a + 4 2 (- 1 )'''+'c, but 8Si^(7n-»-') =-82 a + 82 ^', r where tiie summation for r extends to all positive, negative, and zero values for which m — r' is not negative, as is customary in such summations; so that by subtraction = -6Sa + 4SZ' + 22(-l/c (4). We can find other formulae by putting x = ^co, ^ (1 + &>) in equations A, B. though we have to differentiate these formulae bef)re we write x = ^{l -f <u). Equations 1, 2 were given by Kronecker in a slightly different form; as was equation (4), only in the particular case, however, when m is odd.* Equation (3) seems worthy of a moment's consideration, for it gives us instantly the fact that the number of solutions of x^ + 7/ + z^ = ti is 2iF{n) —12G{n). And multiplying the equation throughout by ^gg, and making use of equations (2, 4), we find that the number of solutions of x'^ + y' + z' -^ t^ = m is 24 (- Sa + 2y) - 4 {- 6 2a + 42i + 22 (- \'fc\ or 8{S6-2(-l)'c}. * See H. J. S. Smith, Report on Theory of Numbers, CollecUd Works, vol. i., page 343 ; and page 324 for equation 3. ^[r. Mordell, Note on class relation formulcR. 79 But more than this, it contains Implicitly tiie following well- known equations, due to Kronecker,* 4ii^(4« + l)2K4»+i) = ^^^^'^^, 7i=0 00 8 i F(8» + 3) ^J(8»+3) = B\^. For remembeiing that F{n)=G{ii) if ?j = l, 2mod4, 2F{n) = G {n) „ n = 7 mod 8, 4:F{n) = 3G [n] „ n = 3 mod 8, and putting successively iq, I'q, L*q, t*q tor q in equation (3], multiplying in older by t~\ t"', i"^, t"*, and adding, we have 48ii^(4n + l)$4«+l= 2 r[l+22*+ 2(2"+. .. + t'-(25 + 2r/+. ..)]', J- 1 or if ^ = l + 2f/+22''+..., B=2q + 2q^ + ... 4 the right hand is S t *■ (^ + t'-^)^ which is easily found to be »=i 12A^B. Writing now q for q* we have the first result, and similarly foi' the others. We also notice that it" we write a;=l/^, p an integer, in equations {A, B), we can find formula for ^ F{r/i — r'^), r ^G{m — r'), where r takes all positive values =±/;modp, )• where k is given, for which w — r" is not negative. These t'ormulse involve finding the coefficient of 5'" in expressions such as — — ~ 6\, (lip) ^„„. We may notice that formulae for expressions of this kind have been found by Hurwitz.f They involve the coefficients of q'" in the expansion as a power series in q' of the integrals of the first kind belonging * Cf. Heimite, Collected Workg, vol. ii., pages 109, 240; Acta Afathematica, vol. 5, page 2'J7 ; Kronecker, Monatsberichte, 1802, page 309 ; 1875, page 229. t See Klein-Fricke, Modul-functionen, vol. ii., p. 6S5. 80 Mr. Mordell, Xote on class relation formulm. to the fundamental polygon defined by the linear fractional group of order /?. Tliere is no difficulty in identifying products and quotients of theta functions as integrals of the tirst kind whenever it is possible, but tliis need not trouble us here. But we find interesting results of this kind from the ex- pansion of Qd^J\00., mw). For instance 2^^„(0, 2co) = S n (- l)«+'-+l $«-2,-^ where r takes all integral values from —\{n — X] to ^», both included. From the coefficients of 5"' we find 2\F{m)-lF{m -2.1') + 2F(m - 2 . 2') - 2F(m - 2 . 3') +...] = Sa: (— 1)^+2'+^ where m = a;'''— 2/, aj> 0, and y is included in the range \x, — ^(o; - 1). Similarly 2g^„„(0, 20)) = - la;5"'+H„^„„(0, 2c.) ^=,,(0, 2a,). in—\ Again F{2m) - 2F{2m - 3 . l') + 2i^(2?/? - 3 . 2") - 2F{2m - 3 . 3=)+ ... = (-l)"'-2x-, where now x^ ~?>y' = in, with x> 0 and -\{x-l)<y< \x. These appear to be results of a new type. In the formulae due to Hurwitz, the rpiadi-atic form is taken a.s ax"^ + hxy + by' . A very simple expression for the class number of such forms of given discriminant has been found by Kronecker. I may add that, when tlie discriminant is negative, 1 have found Kronecker's expression for the class number in a simple and apparently general way without evaluating expressions such as ^ (ax'' + hxy + by')'^ wherein x,y ^ p->l. This investigation is contained in a paper entitled "The class number for definite binary quadratics," which will, I hope, appear in the Quarterly Journal of Mathematics. ( 81 ) SOME FORMULA IN THE ANALYTIC THEORY OF NUMBERS. By 5. Ramanujan, I HAVE found the followinj^ formulae incidentally in the course of other investigations. None of thetn see»n to be of particular importance, nor does their proof involve the use of any new ideas, but some of them are so curious that they seem to be worth printing. I denote by dix) the number of divisors of x, if x is an integer, and zero otherwise, and by ^ [s] tlie Riemann Zeta-fauction. where v (») = 1"'- 3"'+ 5"'- 7-'+... . (3) d\\) +ff (2) +ff (3) +...+ d'{n) = An (log »)'+ Bn (logn)' + Gn logn ^Dn + 0 (/«*+'),* where IT' IT 7 is Euler's constant, C, D more complicated constants, and e any positive number. (5) in-'d'-{n) = [^[s)f<i>{s\ 1 where (^ is) is absolutely convergent for R [s] > i, and in particular * If we assume the Riemann hypothesis, the error term here is of the form 0 («*"'). t Mr. Hardy has pointed out to me that this formula has been given ah-eady by Liouville, Journal de Mathematiqaes, ser. 2, vol. 2, 1857, p. 393. VOL. XLV. G 82 Mr. Ramanuja7i, Some formulce in d{l) d{2) d{3) '" d{n) and A,, A^...A^ are more complicated constants. More generally (8) d'(l) + d'(2) + d'{3)-\-...+ d'{n) = n{^.(lognf-i+J,(lognf-2+...+ A,,| + 0(n^+*)* if 2' is an integer, and (9) d'(l)+d'(2) + d'{^)+...+ d'(n) if 2' is not an integer, the ^'s being constants. (10) d{l)d(2)diS)...d(in) = 2''^'''^''^''^''^-''^^''\ where 0=7+ ||log,(l+-^)-|;| (2-^ + 3-^ + 5-^+-.. .)• Here 2, 3, 5, ... are the primes and *W = ^-^' + ^1^ (7 +,,- 1) +^.(^ + -y,+V -1)+... n iogn (logy</^ ' O^g'O 1 ., 3 where ^(1 +s) = — + 7" 7,« + 72* "Ta*' +••• or r!7,= Lim|(logiy+i(log2r4-...-f^(log.)'---^ (logvr'l (u) <;o„) = s^(»).;(|)<^(^) = SM(S)i(|)rf(|), * Assuming the Riemann hypothesis. the analytic theory of numbers. 83 where S is a common factor of it and v, and 1 ^~At(n) If Z)„(n) = c?(v) + c^(2y)+...+ ^(»?;), we liave (12) i)„(») = S/.(8)^(|)A(|). where h is a divisor of v, and (13) DXn) = a(v) nQogn + 2y - l) + ^(y) « + A„(»0, , "a(v) r(-^) r.^(v) 1^(5)^1+5) where 2 — ;— = ^ -^ — r , i, — t" = >^^7TX7\~ ' 1 f' ^(1 + s)' 1 V 4(1+5) and ^„(»0 = 0(n^ logn).* (14) (/(y + c) + d{2v + c) + d{3c + c) -^...+d (nv + c) = a/y) n (log« + 27 - 1) + l3^(v) n \^ {n), r v^ r(i-H^) kc-^-) r(i+-'') <^-.(ici)i' a/j<) being the sum of the sth powers of the divisors of n and a\{H) tlie derivative of cr^{n) with respect to s, and Ajn)= 0{n^ logn)t. The formulae (1) and (2) are special cases of ^^^ ^{2s-a-b) = 1-aJl) cT,(l) + 2-V„(2) cr,(2) + 3-V„(3) c7,(3) +...; . .X »,(s)77(s-a)»7(s-^')»7(s-a-J) ^ ^ (l-2^^'""''')^(2s-a-^) = rV„(l) (T,(l)- 3-(r„(3) (7,(3) + 5-V,(5) .7,(5) -... . * It seems not unlikely that A„ («) ia of the form 0 [n***). Mr. Hardy has recently shown that A,(m) is not of the form o{(?j log w)' log lop; ?j}. The same is true in this case also. t It is very likely that the order of Aj,_<.(w) is the same as that of A,(«). 84 Mr. Ramannjan, Formula in theory of numbers. It is possible to find an approximate formula for the general sum (17) ^„(l)^i(l) + <^„(2)<r,(2)+...+ o-„(n)cr,(n). The general formula is complicated. The most interesting cases are a = 0, J = 0, when the formula is (3); a = 0,6=1, when it is ^^^^ ^^^^^0og« + 2c) + «^(^)» r(2) r(3) where 0 = 7-4 + jr^ - YT^. ' and the order of E{n) is the same as that of Aj(?0; and a= 1, J= 1, when it is (19) %n'^{^)^E{n), where E{n)=0\n\\ogn)% Ein)i^o{n'\ogn). If s> 0, then (20) <T, (1) <T^ (2) (T, (3) 0-. (4) ...a, {n) = 6 c' (w !)\ where 1>^> (1 -2"*) (1-3'') (1 - 5-')...(l --=7"'), m is the greatest prime not exceeeding »j, and If (i + 2 + 2* + 2' + 2" +-}'^ = i + S r (n) 2", so that K («) '/ («) = 2 >• (v<) /«"', 1 then (22) /(l) + r^(2) + r'(3)+...+/^(H) = - (log«H-C)+0(n'*+^), 4 where C= 47 - 1 + I log2 - logTT 4- 4 log r(|) - ^, r (2). These formulae are analogous to (1) and (3). ( 85 ) AN INTERPRETATION OF PENTASPHERICAL COORDINATES. By T. a Lewis, M.A. 1. In Cartesian coordinates the position of a point is at once determined by its coordinates ; but if the system of coordinates due to M. Gaston Darboux is employed (called pentaspherical in 3-space, and by extension available for 7«-space in general) the position of a point when its coordinates are given has not, so far as I know, been investigated so as to lead to a similarly easy geometrical determination. This can however be done. 2. It is known [Messenger of Mathematics, vol. xliv., 1915, p. 161) that if a:, ?/, 3, ..., be the rectangular Cartesian coordinates of a point, and a;,, a?,, x^, ..., x^^^ its Darboux coordinates; and if a^., Z>^., ... be the Cartesian coordinates of the X;"' vertex of reference of the Darboux system, then 2a; + 2 _i_t = 0, Pk 2^+S Pk &c., &c. x^"^ = 2"^"\ Pk Pk &c., &c. Therefore Therefore the point is the centre of gravity of masses proportional to xjp^. at the k^^ vertex, k having all values from 1 to 71 + 2. In other words, the point Is the mean centre of the vertices for a sj'stem of multiples The position of the point Is therefore determined by a simple geometrical construction. 3. If it Is desired to find the system of multiples corres- ponding to only 72 4 1 of the vertices, this may be done; for g2 86 ilir. Leivis, Inter pr elation of pentaspkerical coordinates. any point whatever may be regarded as the mean centre of the ?i + 2 vertices for the system of multiples ^n.^PnJPx^ '"n.JPn.JPjj &C., &C., since the sum of these multiples is zero. Therefore the system of multiples for the n + 1 vertices from the first to the {n + I)"' is ip»^X-Pn..^nJpi'^ (P,»'v-P«.20p/) &C., &C. 4. This leads to an equally simple method for determining geometrically the centre of a circle or ??-sphere. If the equation of the circle, &c., be given in its general form the coordinates of the centre are given by ^'k = P {p-^^kPk)IPv where p is the radius of the circle or ??-sphere. The system of multiples at the vertices for which the centre of the circle, &c., is the mean centre is therefore p7/)/ - 2a, p/p,, />Vp./ - 2a, p/pj, &c. Since —.,+ —,+...+ -^r- = 0, Pi Pi P «+» this is equivalent to the system Thus the centre is at once determined. Jfor instance, take the equation to the circle P,a;, + p,:r, + P3«^3 = 0. The centre is seen to be at the centre of gravity of the triangle of reference. ( S7 ) A CIRCLE SIMPLY RELATED TO A TRIANGLE. By T. C. Lewis, M.A. 1. Let ABG be a triangle, P tlie ortliocentre. On AD^ BE, CF, the perpendiculars from the angular points on the opposite sides, measure distances AB, BK, CL equal to p^, p,, Pj respectively, i.e. AH'=p^'= 1 (6*+ c'- a') = he cos A • &c. Take points at one third of the distance from A to K, and from A to L, from B to H and from B to Z, from 6' to // and G to K. A circle passes through these six points, and its centre is at G, the centre of gravity at the triangle. It has interesting geometrical properties. Through B and G draw lines parallel to AC, AB, meeting at A' Then A'IC=\[d'+V+c'') = A'L\ Therefore a circle with centre at A', the square of whose radius is pi'+ P,^+ P^\ passes through the points K, L. Now AA'=^AG. Therefore -4 is a centre of similitude of this circle, with centre at A\ and of a circle with centre at G the square of whose radius, /o, is ^ (/3,^-f/3/ + /3j"). And the latter circle will pass through the points at one third of the distance from A to ^and to L. Similarly it passes through the remaining four of the six points mentioned. Therefore the proposition is established. The six points determine a hexagon, three alternate sides of which are parallel to, and two thirds of the length of the sides of the triangle ABG] while the remaining sides are parallel to, and one third of the length of the sides of the triangle HKL. Six other points on the circle are similarly determined If AH, BK, GL be taken in the opposite direction. The remaining centre of siijiiiitude of the two circles is at D', the middle point of BG. Therefore if any one of the four points K, L be joined to D' , and produced to a point whose distance from D' is one third of the joining line, such point 88 Mr. Leivis^ A ch-cle i^elated to a triangle. also lies on the circle with the centre at G. Thus altogether twenty-four points on the circle are deternained, and the circle might be called the 24-polnt circle. 2. The square of the tangent from any angular point A is aq^-p'^Ip: whence the Intersections with the sides can be determined. So the product of the segments of a chord through P Is AP.PD,ox-pI. 3. Using Darboux coordinates the equation to the circle is for this circle has Its centre at G and Its radius equal to p. So also the circle with centre A passing througii K and L is The interpretation of the former equation is that the circle is the locus of a point Q such that qa:'^qb'^qc'=p',\-p:\p'. It may be noted that the circumscribing and nine-point circles have equations and corresponding geometrical relations which are not so simple. Their equations respectively are and P,a;, + Pv^'2+P3^3+P.^4=0, and they are the loci of a point Q such that for the cir- cumscribing circle QA^QB-^^QG^-Qr - p: ^ p:-^ p:- p:, and for the nine-point circle QA^QB'^qc'^Qr=p:^p:^p:'rp: when i?, B' are the radii of the circumscribing and nine- point circles. ( 89 ) THE BROCARD AND LEMOINE CIRCLES. By T. C. Lewis, M.A. 1. Tf K is tlie isogonal conjugate of the centroid of a triangle ABC, and 0 the centre of the circumscribed circle, the Brocard circle is the circle on OK as diameter. The equation to the circuracircle in Darboux coordinates is and Its centre (x/, a?,', x^, x^) is determined by E' = p, x; + p,' = p,x; + p/ = pX + Pz = P, < - P'- K is the mean centre of A, B, C for the system of multiples a*, b\ c\ It is therefore the centre of the circle d'p^x^ + b'p^x^ + d'p^x^ = 0. Let R' be the radius of this circle, then {a' -^ b'' -t c' f 4^. ^ p^jp.'+p.y+p^jp'+py^Ps'jp^+py {p' + P^'+PsV _ {p:+ p:+p:) jp.w+p.w+p.W) + ^p:p:p: (pZ + pZ + p/)' •i -i 'i P\ Pt Ps ^„ •■! I ^ 2 1 « * q^ '^ (p, + pj + Pa J p, iP, +P, +Pz) P, But the perpendiculars from K on the sides of the triangle are proportional to those sides. Let the perpendicular on BG be Jca. Then '^~(a=' + 6^ + cV 90 Mr. Letvis, The Brocard and Lemoine circlea. therefore 1^ = Pi' + P,^-^ P3)' 2 -2 -i -Pi P2 Pz ^{P' + P' + P^jpy Therefore R" = 4/c' [R - p^) . Also OK' - R" = ^V.^Z + ^V.^a' + ^Va^/ d' + b' + c" a' + b' + 6' therefore ^ (P/ + P/ + Pz) = (1-12/;') 72', The equation to the circle on OK a.H diameter therefore is i (P. ^, +P.-.A P.^. - p.^j + i^- + ^'^- ^ j^'- +y''^^^ + ^' or p.a^i+p^a'.^ + p.a^j-p.x^ ^ aV,a;, + /;V,ft;, + cV3a;3 '^p'p'p" _q Pi' + Ps' + Ps' (P,' + P2' + P»'0 P/ Making this homogeneous it hecomes P,a;, + ^,a3, + p3a'3-p,aj^ + Hi-'' p. +P, +P: (Pi'+pZ + Ps which reduces to + , . ' V ' ., . p + -' + -' + -M = 0, ' )Pa V, P, Ps pJ P^^Pi^P'^^ I ^. •'^o ^, s'A Pi +P, +P3 Vi p, p, pJ Mr. Letvis, The Brocard and Lemoine circles. 91 From this it appears tliat the Brocard circle passes through the intersection of the circumcircle and a circle whose centre is jfiT which cuts the circumcircle orthogonally, this last circle being, however, unreal. 2. If LK be drawn through K parallel to 5(7, and meet AB in Z, the Lemoine circle is the locus of a point such that the sum of the squares of its distances from 0 and K is constant, viz. the same as for the point L. Now OU = W-AL.LB _ „3 a' {1/ + c*) c" and LK* = ^-^, — j-- j-7, therefore OU f LIC = R'- Therefore the equation to the Lemoine circle is ^ \Pi + P-i + Pa ; = (1 - ik') R\ that is, when rendered homogeneous, a^p.x.+h'p.x.+c'p X P,oc^+P,x, + p,x^-p,x^+ ' \^p\lp. ■ where the distance of the symmedian point from any side is k times the length of that side, this value having been already determined. ( 92 ) SUCCESSIVE TRANSFORMS OF AN OPERATOR AVITH RESPECT TO A GIVEN OPERATOR. By G. A. Miller. Let Sq and t be any two non-cominutative operators, and let s,, 5,, ..., s^ represent the n commutators obtained as follows : If these n commutators are all commutative with each other then it results directly that where the exponents of the symbols in the second member are the binominal coefficients in order. If t"" is any power of t wiiich is commutative with s,, it results that o s" <fi(«C«-l)} «« — 1 Hence it follows that t -(a+/J) .a+l3. - */3 */3-] Mm -1)}. ..»?.v and that *a+/3 *a+/3- -1 *«+/3-2 1)) = 1. When the connnutators s,, s^, ..., s^ are not assumed to be commutative the forn)ula3 which correspond to those just found become much more complex. To tind such a formula by induction we may proceed as follows: It is evident that each of these five transforms is the product of two expressions which differ from each other only as regards the subscripts. In the former of these two factors each subscript is equal to the corresponding subscript of the latter increased by unity. The number of the linear factors in the a'^'' transform is evidently 2". As each of the two factors of the a"' transform is similar to the (a — 1)"' transform, Di'. Miller^ Successive transforms of an operator. 93 the (a + l)"" transform can be deduced directly from the a"* transform according to the law involved in deriving the a"* transform from the (a— ly*". Hence the following rule: — To obtain the ?/"' transform multiply the [n — 1)"' transform on the left by the e.rpression obtained by increasing each subscript of this transform by unify. When n is even the first half of the former of these two expressions and the last half of the latter are identical, in order, and hence this part may be written in the form of a square. From the given rule to write down the Ji'" transform if the (n — 1)"' transform is given, it is easy to derive the following rule to find the n"" transform directly: when n>5 write the expression -s^s'„_,*„_2? then multiply it on the right by the square of the expression obtained by diminishing each of the subscripts in s„s^„.,s,_3 by unity, then multiply this product on the right by the result obtained by diminishing each of the subscripts in s^^s\_^s^^_^ by 2. The product thus obtained is again multiplied on the right by the square of the result obtained by diminishing each of its subscripts by unity, and this latter result is multiplied on the right by the expression obtained by diminishing each of these subscripts by 2. If s^ has not been reached by these operations, the last product is to be treated in exactly the same manner as the preceding product was treated, and the operations are repeated until s^ is reached. The expression in which s^ occurs is never squared even if the above rule would require that this factor be squared. Tliis rule clearly gives rise to an expression consisting of two factors which are such that the second can be obtained by merely diminishing each of the subscripts of the first by unity, and if this expression is the ?*"" transform the second of these factors is the (« — 1)"* transform. Hence this rule is equiva- lent to the one given above. It should be observed that even the first factors s' ,s , is formed from s„ according to this rule, and that when « = 1 this rule gives s^s^, when n = 2 it gives SjSi'Sft, when ?* = 3 it gives s^s.^'s^.s./^''s^, when ?2 = 4 it gives s^s^'s^is^s^'s^s^s^s^', etc. The n commutators s^, s.^, .. , 5^ cannot be independent unless the index vi of the lowest power of t which is conmiu- tative with s^ exceeds ?«, since In particular, when m = 2 it results that -2 -2 -2 S. = *. , S, = S„ , ..., 6- =S ,. 94 Mr. Neville, Systems of particles Hence these commutators generate a cyclic group whenever 1^ is commutative with s^. It" s^ is also commutative with t then Sj must also transform this cyclic group into itself, and hence we have the known results that if the square of each of two operators is commutative with the other operator these operators generate a group whose commutator is cyclic. The successive transforms of an operator have been employed frequently, especially in regard to prime power groups where each of the n commutators s^, s,, ..., s^ may be assumed to be contained in a smaller group than the preceding, with the exception of the first of these commutators which is contained in an invariant subgroup of the original group. The special formula when each of these n commu- tators is commutative with all of the others is also known, but the general rule of finding the ?*'" transform and the method of proof here outlined are supposed to be new. UniTereity of Illinois. SYSTEMS OF PARTICLES EQUIMO^IENTAL WTTH A UNIFORM ^J^ETRAHEDRON. By Eric H. Neville. Simple systems of particles equimomental with a uniform tetrahedron have long been known, but the methods given in the standard text-books for demonstrating their property leave much to be desired. Though as far as I am concerned original, the following method may well have been known to the teachers of the last generation, but there is evidence to the contrary in its absence from the pages of Routh. Let PQ, RS be opposite edges of a uniform tetrahedron of mass M, and let their mid-points be U, V and their lengths 2a, 2b] let the length of UV be 2c, and let G be the mid- point of UV. A plane parallel to FQ and BS, cutting UV m a point whose distance from G towards U is ct, cuts the surface of the tetrahedron in a parallelogi-am of sides a (1 + t), h{l — t)j with angles independent of t. Since a parallelogram of mass m is equimomental with particles of masses in j 12 at the vertices and a particle of mass 2m/3 at the centre, the equimomental with a uniform tetrahedron. 95 tetrahedron Is equimomental with a distribution of varyinj^ line-density along the Hve lines FR, PS, QE, QS, UV, the density in each line being proportional to 1 — t\ and the total mass of each of the first four lines being 71// 12 and of the fifth line being 21//3. Since J k{l-t')dt=4:kld, [ k{l-t')fdt = 4kl\5, the part of a line of density k{l — t^) which corresponds to values of t between —1 and 1 has mass n if ^ is equal to 3h/4, and the line is equimomental with three particles, one of mass nj 10 at each end and one of mass 4n/5 at the mid- point. It follows at once that A uniform tetrahedron of mass M is equimomental with a si/stem of eleven particles, one of mass J// 60 at each vertex, one of mass Mjlb at the mid-point of each edge, and one of mass 8il//15 at the centroid. The deduction of the familiar systems with five particles, of which four are at the vertices, and with seven particles, of which six are at the mid-points of the edges, requires only applications of the theoreirj that a system of three equal particles of mass «?/3 at the mid-points of the sides of a triangle is equimomental with a system of four particles, one of mass 7/2/12 at each vertex and one of mass 3?/i/4 at the centroid. It is evident that the method used here is applicable to many other problems, and it is interesting to use it in the case of a triangle. A uniform line of mass ?n is equi- momental with particles of mass w/6 at its end-points and a particle of mass 2m j 3 at Its raid-point, and the integrations of t, i\ and «'' from 0 to 1 are sufficient to show that a line PQ of mass n whose density is proportional to distance from P has its centroid at the point of trisection nearer to Q and is equimomental with three particles, one of mass »/l2 at P, one of mass ??/6 at Q, and one of mass 3»/4 at the centroid. It follows that a triangle ABC of mass M is equimomental with a system of seven particles, one of mass Mj\2 at A, two of mass J// 36 at B and G, one of mass i//9 at the mid- point of BG, two of mass Ji/8 at the points of trisection of AB, AG which are the further from A, and one of mass i//2 at the centroid of the triangle. Supei posing three distri- butions of this form each with total mass il//3, we find a syujuietrical system composed of thirteen particles, one of mass 5/1// 108 at each vertex, one of mass J// 27 at the mid- 96 Trof. Nanson, Note on an elimination. point of each side, one of mass Mj 24: at each point of trlsection of each side, and one of mass 7lf/2 at tlie centroid, and this system can be replaced imniediately by a system of seven particles, one of mass Mj IS at each vertex, one of mass i]//9 at the mid-point of each side, and one of mass J// 2 at the centroid. NOTE ON AN ELIMINATION. By Prof. E. J. Nanson. Professor Steggall having recently, Messenger, vol. xiiv., p. Ill, recalled attention to a verification by Cay ley, that if «+ Z> + c = 0 and «; + ?/ + 2; = 0, then reference may be made to a proof by Leudesdorf, Messenger, vol. xii., p. 176. The following verification, although not so elegant as those of Leudesdorf and Steggall, may also be put on record. Since a + h + c = 0 and x-'ty -\- z = Q we may take a, J, c to be the roots of X^ + qX+ r = 0 and put x=Xn + iju{h—c), &c., so that y — z = X (b - c) — 3/j,a, &c. Then, since ^a [a — h) (rt— c)= -9»* and 2&c(6 — c)=— S, where S = n(6— c), so that — S^ = 4^* + 21r\ we have xyz — — r)^ — SX-V + ^r\y? + S^Lt", Y\\y-z)= 8\^- 27rVV - 98\fi' + 27?y, so that n(b-c){y-z) + 27abcxyz = X(X' - 9fjt,') (8' + 27r'). Also 2 ax = X 2 a' = - 2 j\, and Ix' = X'la' + iM'^{b- cf = -2(X' + fi') q, so that 4(2aa;)''-32a^.2a'.2a;* = -8X.(\'- V)^'. Thus the relation to be proved is seen to be true because 8'+27r''+42'=0. Melbourne, August ^th, 1915. ( 9^ ) TIME AND ELECTROMAGNETISM. By Prof. H. Bateman. The interval hetioecn two moving 2Joints. § 1. For descriptive purposes a system of rectangular co- ordinates {x, ?/, z) and a time variable t will be used to express the ideas of motion and the propagation of light in mathematical language, but the observers whose experiences we are about to discuss are supposed to have no direct know- ledge of this system of coordinates. To fix ideas we shall also assume that with the above system of coordinates the velocity of light is the constant quantity c and is independent of the motion of the source and the state of the observer.* It will be convenient to regard this system of coordinates as the standard system, and to define motion in the usual way as motion relative to the axes of coordinates. Now consider two observers, A and B, each of whom is provided with an ideal clock which can be regulated so as to indicate at time t any arbitrarily chosen continuous mono- tonic function of t. The two observers are supposed to have no direct means of ascertaining whether they are moving relatively to one another or not. They are supposed to set their clocks so that they are 'together' according to Einstein's criterionf, and the problem is to find what function oft each clock must indicate in order that the criterion may be satisfied, when the two observers are moving relatively to our standard set of axes in an arbitrary manner, which may be specified as follows : {A) x = x{t], y = y[t), z = z[t) | [B) x = l[tl y = v{t), ^ = ?(0 ) We shall suppose, however, that each observer is always moving with a velocity which is less than that of light so that at any Instant t, B sees only one position^ of A by means of light sent out from A at time t^, and the light sent out from B * The foundations of an optical geometry of space and time, in which the above condition is satisfied, have been laid by A. A. Eobb, A Theory of Space and Time, Carab. Univ. Press (1914). t Ann. of Phys. Bd. 17 (!905), pp. 891-921. . I This follows fioin a theoiem due to Lienard, L'eclaircfie ekctvtque t. 16 (1898), p. 5. See al.so H. Bateman, The physiail aspect nf Time, Manchester Memoirs (1910); Electrical and Optical Ware Motion, Camb. Uiuv. Press (191o), ch. 8, p. IIG ; A. W. Conway, Proc. Loud. Math. Soc. scr. 2, vol. i. (1903). VOL. XLV. H 98 Frof. Bateman, Time and eUctromagnetism. at the Instant t reaches A at only one Instant t^. Tlie two instants <„ ^^j which satisfy the inequality f, <T<f„ are the two real roots of an equation F \t, t) =0, which, according to the usual theory of liglit, is [«'(0-^W+[y(0-'7W7+[^(0-nT)r=c'(«-Tr...(2). Let ^'s clock indicate at time t the number /(^ and C's clock the number «^(f), then ^'s clock will be said to be running uniformly* with reference to B it /{Q-/{t,) = 2T„, (3), where Tj, is a constant. This means that a signal always tMkes the same clock-time to go from A to B and back again. If now <j) (t) be defined by the equations /(g-^c.-^w=/(0 + ^«^ W' the two clocks will be '•synchronous'' or '■together.'' It is easy to see that tlie function ^ (t) satisfies an equation analogous to (3), for if t and t' are the two real roots of tlie equation i^{<,, t) = 0, we have <t>{r)=/[t,)+T„,, consequently ^(t'j — ^ {t) = 2T^^ (5). Hence whenever a function of type f{t) exists there "is a quantity 7'^^, symmetrically related to the two moving points A and B, which remains constant during the motion. This quantity will be called the interval between the two moving points. Determination of the interval in a particular case. § 2. Let us consider the case when the movements of the two observers A and B are specified by the equations [A) x=l {a + lit) , y ■-= m {a + ut), z = n [a + ut) [B] x = X{a + ut<, y = iJ, [a + ut), z = v{a + ut) ^"' '' where I, m, n, \, /i, v, a, u are constants. Equation (2) then gives a + ut,=^'^PlR-[R^-FQ)^] \ f7l a^nt.=''-^[R+[R'-PQP, * Cf. E. V. Huntingdon, Phil. Mag. April (1912), for the case in wliich the two observers are not moving relative)}' to one another. Prof, Bateman, Time and electi'omagnetism. 99 wliei"e u a R= -^ — IX — mfi — nv. W Tiie condition (3) may now be satisfied by writing /(<)=^ + -log(l + -)+-l„g(P^) (8). where A and B are arbitrary constants. Equation (4) tiieu gives , ^ A aB , I UT\ aB , / ^u'\ , ^, *(T)=^+-iog(i + -) + ^iog(e-)...(io). It is evident from the sj'mmetry of tiiis result that we can find a set ot" k observers A^, A^, ..., A^^ whose clocks are all together by specifying their motions as follows: (^^) x = l^{a+ut), ij^vi^ia + ut], z = n^[a + ut)...p = \,2,...h. P 21 P * ^ * U * * * P = —.1 I —vi m — V n , P2 11^ P 1 Pi P <V the clock belonging to the /j"* observer should indicate at time t the number The Interval between the p^^ and 5"' observers is then aB P +(P "" — PP)^ ^~2u ^ P -{P '-P FY' Pt ^ pg p g' When u->0, f^{t) reduces to the form A + Bt and T,^ becomes simply {Bjc) r , where r^^ is the distance between the two observeis A , A . A set of observers at constant intervals from one another who are provided with clocks which are all running together will be called an organised set of observers. It is clear from the above example that the different observers may or may not be at rest relatively to one another. 1 00 Prof. Bateman, Time and electromagnetism. Reflexion in a moving plane mirror. § 3. The imaj^e of a point source (cc, ?/, z, t) in a plane mirror moving, witli uniform velocity v in a direction per- pendicular to itself, may be obtained by means of the transformation* 2c' t = t-- y =!/' C — V 2v c'— V vt) -, [x - vt) .(11), where x = vt is the equation of the moving mirror and (:/,"', y\ z\ t) the coordinates of the image. These equations may be obtained very easily by noticing that the locus of the points in which I'ays of light, issuing from the point x, y. z at time ^ strike the moving mirror is a quadrlc of revolution having the source of liglit as one focus. Tlie image of the source is at the other focus of the quadric, and it is easy to calculate the time at which light must leave the image in order to coincide with the rays from the source which have been reflected. We sliall say that the above equations give the image whether the velocity v is less than or greater than the velocity of light. It is eas3' to see that the moving mirror is completely and uniquely determined when a point source and its image are given. To prove this we shall make use of a representation of a point source [x, y, z, t) by means of a directed spheref of radius ci, whose centre is at the point [x. ?/, z). Jn the first place it should be noticed that c[t-t) x'—x Hence if 0 is the semi-vertical angle of the tangent cone, whose vertex is at the centre of similitude of the two directed spheres representing the point source and its image, we have the relation sin^ = t'/c (12). ♦ H. Bateman, Phil. Mag. Dec. (1909), May (1910). V. Varicak, Fhys. Zeitschr, Bd. xi. (1910,, p. 586. t H. Bnteman. Phil Mag Oct. 1910), Amur. Jour. (1912). H. E. Timerding, Jahrest d. Ueutsch. Math. Vereiii, Bd. 21 (1912). K. Ogura, Scitnce Reports, Tolioku Univ. Vol. II. (1913). Prof. Bateman^ Time and eleciromagnetisin. 101 It is clear that tlie velocity of the mirror is greater than or less than that of liglit according as the centre of similitude is inside or outside the spheres. Tlie plane through tlie common points of tlie two spheres can be identified with the initial position of the mirror; for since the equations of the spheres are [X-xr + {Y-yY ^[Z-zY=c't\ {X-x'y+{Y-y'r+{Z-zr=c'i\ respectively, the plane through their common points is [x'-x] {2X-X-X') +c"{t'-t) [t'+t) = 0. Now X + x =v {t + 1') and c' [t' —t) = v {x — o:), consequently the above equation reduces to X = 0. The initial position and velocity of the mirror being known, its motion is com- pletely determined.* The determination of the time and position of an event fro^n the recorded times at tohich it is loitnessed hy an organised set of observers. § 4.' Consider an organised set of four observers A^,A^,A^,A^, whose clocks indicate the numbers 2',, T^, J!,, T^ respectively ■when the event is witnessed. If T is the required clock-time at which the event occurred, the quantities T-T, T-T, T^-T, T-T are the intervals between the event and the four observers. We shall denote these by the symbols T„,, 2^^, T^^, T^^ res- pectively, and the intervals between the diflerent observers by the symbols T,,, T3,, 2',.^, 2',,, 2;,, T^, respectively. The problem is to express T in terms of 2',, T^, T^^ T, by means of a relation of type r=/(^., T,. T^, T^) (I-) If we consider the special case in which the event occurs at a point occupied by the observer ^4,, we have, when 2'= 2\ and T=T^^T^^, T^= T^+ T^,, T=T^+T^,; hence the function f nmst necessarily satisfy the functional equation ^,=/(^., ^.+ ^,.v T,+ T,3, T,+ TJ. * When we use Minkowskis representation of a space-time point x, i/, z,t hy a. point with rectangular coordinates (x, y, z, ict) in a space of four dimensions ^4, a reflexion in a plane mirror moving with uniform velocity is represented by a reflexion in an hyperplane, and the theorem becomes obvious. 'I'his represen- tation of a ppace-time point ought perhaps to be associated \¥ith the name of Poincare. Gf. Eend. Palermo, t. 21 (1906j, p. 168. h2 102 Prof.Bateman, Time and electromagnetism. Similarly it can be shown that It must satisfy three other functional equations ot" a similar character. The relation (I.) can be found very quickly when It is known that the ten mutual intervals rprpnirnrnrprprprprn -'oi' -^02' -^0.1' -'04' -'-•al 31' -'-Ml -^14' ?4' 34 are connected by an identical relation. It should be noticed, moreover, that since the last six quantities are constant, the fourth quantity can be legaided as a function of the other three, and is consequently constant when the other three quantities are constant. We may regard the first three intervals, or three independent functions of them, as co- ordinates tixino; the position of the point at which the event occurred. ]f we call these coordinates X, Y, Z, the time T is given by the equation T=T-T^^=T^-^{X, Y,Z). It is of course important that a good choice of coordinates X, Y, Z should be made. Let us consider two events which are witnessed by the observers ^,, A^, A^, A^ at times T„ i;, T3, T^ and J\-^hl\, T^^IT^, T^^hT^, T^+hT^ respectively, where hl\^ hT„ ST^, 81\ are small quantities. Let us suppose, moreover, that the second event consists of a signal from an observer Q indicating that lie has just witnessed the first event. When this is the case the four increments 82\, 8T^, 8T^, hJ\ will not be independent, but will be connected by an identical relation which will be assumed to be of the form ^B ST .BT =0, m, « = 1.2.3.4, Avhere 7?,^^^ is a function of 2\, T^, T^, 7\. This relation can be expressed in the form ABX' + B8Y' + CBZ' + DST' + 2F8Y8Z+2GSZ8X + 2HBX8Y+ 2 U8X8T+ 2 V8 Y8T+ 2 W8Z8T== 0, wliere the coefficients are functions of X, Y, Z^ T. Now since this quadratic equation is of fundatnental importance in the description of the propagation of light* by means of the * It can be regarded as the equation determining the form of the wave front of an elementary wave issuing from tlie point X, Y, Z at time T. An attempt to formulate a sclieme of elecr.roiniij,'iietic equations consistent with the above equation has been made by the author. Proc. Lund. Math. Soc, ser. 2, vol. viii. (1910). Prqf.Bateman^ Time and electromagnetism. 103 coordinates X, Y, Z, T it is natural to endeavour to clioose X, Y and Z so tluit the above equation takes a simple form such as A hX' + BB Y' + C8Z' + Dhr = 0. Let us now consider a simple case in which this can be done. If the four observers are stationary relative to one another and space is Euclidean, and the configuration of the four observers A^, A.^, A^, A^ is either at rest or is moving unitbrmly without rotation relative to our standard set of axes, we may assume that the ten quantities 2' are propor- tional to the nmtual distances of five points in space. They are consequently connected by the identical relation* ^0,-^ T ' 02 T ' rn 2 04 1 0 1 1 1 1 0 1 T.: T.: 'i\: 0 1 T ' T.: ^;/ 0 rn 'i 1 TJ T..: 0 T ' T * S4 1 T..: 0 T' i\: i\: 1 TJ = 0. (II) This is a quadratic equation for the determination of T. We may discuss it geometrically by considering the four directed spheres representing the points A^, A,^, A^, A^, at times J",, T^, 2\, T^j respectively, and a sphere S of radius cT whose centre is at the point where the event occurred. The equation then expresses that this last directed sphere touches the tiist. Now there are two directed spheres S and S' which touch the four given directed spheres, and they are the repre- sentative spheres of two point sources which are images of one another in the moving plane miiTor which passes thi'ough the points A^, A^, A^, A^, at times 2\, T^, 2\, 2\, respectively. This follows at once from the fact tiiat rays of light starting from the two point sources will either arrive at A^, A^, A^, A^, at times 1\, 2\, 2\, 2\^ respectively, or can be supposed to have passed through these points at the respective times. The plane containing the centres of similitude of each pair of directed spheres of the set of four, with centres at -4,, ^j, A^, J^, respectively, is the initial position of the mirror. The velocity of the mirror can be found from the * Scott and Mathews "Theory of Determinants" (1104), p. 239. The relation is due to Cayley. 104 Prof. Batemmi, Time and electvomagnetism. ratio of the radius of one of these spheres to the distance of its centre from the plane just mentioned. If the plane does not cut the spheres in real points the velocity of the mirror is greater than that of light. If the velocity of the mirror is greater than that of light the centre of similitude of the two directed spheres S, S' lies within the two spheres, and it is easy to see that the radius of any directed sphere such as ' T^\ which touches both, is intermediate between the radii of S and >S". In this case there is only one value of T less than each of the quantities 2\, Tj, T^, T^ which satisfies our quadratic equation. The position and time of the event can then be determined uniquely, for when T is known the distances of the place of occurrence from ^,, A^, A^, A^ are known and the ordinary rectangular coordinates of the place can be found without difficulty. On the other hand, if the velocity of the mirror is less than that of light, the centre of similitude of the two directed spheres S, S' lies outside the two spheres, and it is easy to see that the radius of a directed sphere which touches both is not intermediate between the radii of the two spheres iS and S'. Hence in this case there are either two solutions of the problem or no solution at all. Let us now consider an organised set of five observers whose mutual intervals are connected b}' the identical relation corresponding to that between the mutual distances of five points in space. If a value of T, calculated from the observations of one set of four observers, agrees with a value of 2' calculated from the observations of another set of four observers, all is well. If, however, the value of T calculated in the different ways do not agree, a reason must be found for it. Several possible causes of the disagreement may be suggested. 1. The observers may be moving relatively to one another. 2. Space may be non-Euclidean. 3. The observers may be at rest relatively to one another, but the configuration of four observers may not be moving uniformly relatively to our standard axes, and consequently the assumption that T is proportional to the distance between A and A is unjustifiable. Geometrical representation of the interval between two moving points in certain particular cases. § 5. In the motion considered in § 2 the two points A and B pass through the origin at the same time t = — alu and Frof. Bateman, Time and electromagnetism. 105 travel along straight lines with constant velocities. If we adopt Puincare's representation of a space-time point {x, ?/, z, t) by a point with rectangular coordinates {x, 3/, z, ict) in a space of four dimensions, the two moving points A and B will be represented by two intersecting straigiit lines whose direction cosines are proportional to (/, m, ??, ic/u) and (\, /A, V, icju) respectively. It' 6 is the angle between these lines we have cost/ = - ^I[J^Q] ' hence „ iaBd ., - T,, = ±—- = ±ikd say. When we have an organised set of five points, which are moving according to the equations X = l {a-\-ut)y y=m^[a + ut), z = n^{a + ui), p=1.2...5, we may deduce an identical relation between the mutual intervals of the five points from the well-known relation between the mutual inclinations of five points in a space of four dimensions.* If c',.^=cosh(Z:2'^J the identical relation is c,. 21 31 3J c.. c. c 23 3i 1 C.. C J4 35 1 C.. = 0 .(III. This may be regarded as an equation for the determination of L If now we have observers at the moving points who witness an event at the clock-times 2\, T^, ..., 1\ respectively and 2' is the required clock-time for the event, the differences T -T, 1\-T, T^-l\ T-T, 7;- Twill be the intervals between the event and the observers. With the aid of the identical relation of type (111.) between the intervals of five points we may deduce an equation for T from the obser- vations of each set of four observers obtained by leaving out one of the five observers. The different values for T which are found in this way ought to agree ; if they do not there may be several possible explamvtions of the discrepancy just as in § 4. * Scott and Mathews, he. cit. 106 Prof. Baieman, Time and electromagnetism. A more general type of motion, in which the interval between two moving points can be expressed in a simple form, and interpreted geometrically, may be obtained by transforming the uniforn) rectilinear motion considered in § 2 by means of a transformation of the coordinates (a;, ?/, s, C) which leave the equation unaltered in form. Such a transformation corresponds to a conformal transformation in Minkowski's four-dimensional space 8^^ and consequently makes two intersecting straigiit lines in 8^ correspond in general to two circles interse(;ting in two points. Either or both of the circles can degenerate into straight lines. Since a conformal transformation in S^ leaves the angle between two curves unaltered, it follows that the interval between the two moving points corresponding to two doubly intersecting circles is proportional to the angle between the two circles. A circle in S^ corresponds to a point moving along a conic in our space*. If we I'epreseiit each position of the moving point by a directed sphere we obtain a chain of directed spheres with some notable properties. To discuss the motion it will be sufficient to consider the transformation which corresponds to an inversion in 8^. This is specified analytically by the equations where s^ = o6' ■\- y' + z* — cW and ^ is a constant. It may be specified geometrically by saying that the repre- sentative spheres of two corresponding space-time points [x, y, z^ i), ix' , y\ z\ t') are transformed into one another by an ordinary inversion with respect to a s[)here of radius b whose centre is at the origin. f Let us now apply this transfornuxtion to a system of points moving along straight lines in the manner specified by the equations x = a. + lp{a-\-ut], y=^ + m^{n + ut)^ z = y -]■ n^{a + ut), p=l, 2, .... * This type of motion has been considered by Born, Ann. d. Phys., Bd. 30 (1909),- Somnierfeld, Ann. d. Phys., Bd. aS (1912), p. 673; Kottler, Weinnv BerichU,'\iA I'Jl (1912); Hasse. PVoc. Loud. Math. Soc, ser. 2, toI. xii. (1913), p. 181 ; Schott, Eltctromaqnetic Radiation (1912), p. 63. t H. Bateman, Proc. Lond. Math. Hoc, ser 2, vol. vii. (1909), p. 84. Frof. Bateman^ Time and dectromagneti&m. 107 The representative spheres of one of tlicse moving points have their centres on a straight line and have a common centre of similitude Avhich lies within all the spheres if the velocity of the moving point is always less than that of light. It is easy to see that two spheres whose radii have the same sign do not intersect, also that there are two spheres through each point in space and therefore two through the centre of inversion. The inverse system of spheres must therefore contain two planes, and so the curve descrihed by the moving point extends to infinity and is consequently an liyperbola. Since the line through the centre of similitude and the centre of inversion cuts each member of the first set of spheres at the same angle, it also cuts each member of the second set of spheres at the same angle. The system of spheres obtained by inversion thus consists of spheres whose centres lie on an hyperbola and which cut a chonl of the hyperbola at a fixed angle, which is the complement of half the angle between the asymptotes of the hyperbola; the chord is the major axis of the hyperbola. The motion may be specified analytically by substituting the above expressions for X, ?/, z in the previous equation. Since a circle in S^ cuts a space i = const, in two points it follows that the circle really corresponds to the motion of two associated particles describing different branches of the same hyperbola. To obtain an organised system of observers moving along hyperbolic paths the representative circles in S^ must have two points in common, consequently each hyperbolic motion and its associated motion must be such that either the moving particle or the associated moving particle passes through two fixed points at specified times, it is possible of course that one particle may pass through one of the fixed points and the associated particle through the other. The cases that have been considered are clearly not the only cases in which a simple expression for the interval can be found. Let us suppose for instance that the observer A is at rest while B moves along a straight line through A according to the law x — xi*-)-, tl'^^" ^ and t, satisfy the equations c(t-<,) = xW' C(f,-T) = X(T)- Hence the function /(/) must satisfy the functional equation r 1 1 r 1 1 C ^ J 108 Prof.Bateman, Time and electromagnetlsm. If the function /(<) is given it is generally easy to find the function x(0) for instance \if{t) = t'^ we must have A geometrical representation of space-time vectors. §6. The four-dimensional vector analysis introduced into tlie theory of relativit}' by Minkowiski* admits of an interesting geometrical treatment with the aid of directed spheres. f A 4-vector whose components are A^, A^, A^, A^ may, for instance, be represented by the relation to a sphere S, whose centre is at the origin and whose radius is G of another sphere whose centre is at the point ^,, A^, A^ and whose radius is A^ + G. It is more convenient, however, to lepresent the 4-vector by the relation to the sphere S of a point P and an associated number v. This point P is the centre of similitude of the two directed spiieres just mentioned; its coordinates x, y, z are determined by the equations vx = A^, vg = A^, vz = A^, -vc = A^ (13). If V is regarded as analogous to a mass, the point and number representing the sum of a number of 4-vectors are found by determining the centre of mass of the masses at the different representative points of the vectors, and associating with it the sum of the masses. If the representative point P of a 4-vector lies within the sphere S, the vector is said to be time-lihe, if it lies outside the sphere, the vector is said to be space-like. If (w, V, w) are the component velocities of a moving point, the four quantities (w, v, w, —c) may be regarded as proportional to the components of a 4-vector which is time-like or space-like according as the velocity of the moving point is less than or greater than that of light. If the |)oint moves with the velocity of light the representative point is on the sphere and the 4-vector is special. * Gijlt. Nachr. (1908). Phys. Zeitsch: (1909). See also G. N. Lewis., Proc. Amer. Acad, of Artsand Sciences, Oct (1910). A. Sommerfeld, Ann. d. P/ii/.i. Bd. 32 (1910), p. 7t)5; Bd. 33 (1910), p. 651. L. Silberstein, The Theory of Relativity (1914). E. Cunninsjham, 'The t'rinciple of Relativitii (1914). B. Cabrera, Revida d. R. Acad. Madrid, t. 12 (1913), pp. 646, 738 E. B. Wilson and G. N. Lewis, Proc. Amer. Acad, of Arts and Sciences, vol xlviii. (1912). t H. Bateiuan, Phil. Mag. Oct. (1910). Frof. Bateman, Time and electromagnetism. 109 The angle between two 4-vectors can be defined In the following way. Let P and P' be tlie two representative points and let a circle be drawn to pass throngh P and P' and to cut the sphere S orthogonally. The angle subtended by tiie chord PP' at a point of this circle is then the angle between the two vectors. If P and P' are conjugate points with regard to the sphere S, they are at the extremities of a diameter of the circle and the angle is in this case a right angle. Hence two perpendicular 4-vectors are represented by points which are conjugate with respect to 8. A set of four iimtually perpendicular 4-vectors are thus represented by the vertices of a tetrahedron which is self-polar with respect to S. It is clear that one, and only one, of the four vectors can be time-like. If two 4-vectors (^„ A^, A^, v4J, (5,, B.^, B^, 5J are represented by numbers v, /x at the points P, Q, respectively, the special 6-vector whose components are A^B^— A^B,, A,B-A^B^, A^B,^-A.^B^, A,B^-A^B,, A.^B^-A^B,, A^B^ — A^'B^, may be represented geometrically by a force* of magnitude fJ-vPQ acting in the direction PQ along the line PQ. The two reciprocal G-vectors whose components are (FFFFFF)(FFF -F F —F ) respectively, are represented by forces acting along lines which are polar lines with regard to the sphere S, provided of course that the relation F F +F F +F F =0 2J 14 ' -^ 31 ^ 24 ' IJ 34 is satisfied. When this relation is not satisfied, the 6-vector F cannot be represented by a single force. It can be repre- sented by a wrench, but it is more convenient to represent it by means of two forces acting along lines which are polar lines with respect to the sphere S. Let (i/, H, Z/, E^, F^, F^) be the components of a general 6-vector, and let (h , h , h , e , e , e) he the com- ponents of a special 6-vector which is the vector product ot the two 4-vectors A and B. Using the ordinary notation for vectors in a space of three * The first three components of the 6-vector are equal to the moments of the force about the axes, and the last three to the three components of the force multiplied by c. The sum of a number of special G-vectors may be represented geometrically by a system of forces or its simplest equivalent obtained by the composition of forces. 110 Prof. Batenian^ Time and electromagnetism. dimensions we shall endeavour to find vectors h and e such that* H = ll + /<:e and E=e-/cll (14), whers k is a scalar quantity. Since (eh.) = 0, the constant k. is determined by the equation (1 - /c") (EH) - « (EE) + /f (HH) = 0 (15), and the vectors e, Jh., may then be determined from the preceding equations. Now let us consider the case when E and H are the electric and magnetic forces at an arbitrary point in an electro-magnetic field. We shall regard the 4-vectors A and B as proportional to velocity 4-vectors, so that their com- ponents are (ay^, av^^ ay^, —ac) and {hu^^ hu^^^ bii^, —be) respectively. We then have a representation of the vectors E and H in terms of two velocities U and V, H = ai Fvu] + CKab (U — v) 1 (16). E = cab (U - V) - Kab [vu) i Similarly if the 6-vector (e^, e^, e, — /?^, h^, 7?J is the vector product of two 4-vectors A' and B', whose components are (a'y'^, a'v' , a'u'^, —a'c) and (^V^, b'v'^, b'v\^—b'c) respectively, we have a second representation of E and H, viz. H = -a'&'c(u'-v') + ««'^'rv'u'] ^ (17). E = a'b' fv'u'] + CKa h [Vl - v') J The G-vector (H, E) is thus represented as the sum of two special 6-vector.s (h, e), (/ce, - «ll), and these can be repre- sented geometrically by forces acting along lines L, L\ which are polar lines with respect to the sphere S. The 4-vectors A and B are represented geometrically by numbers a and b associated with two points A and B on the line L, while the 4-vectors A' and B are represented geometrically by numbers a and b' associated with two points >4' and B' on the line L' , Since one of the two lines, say L, cuts the sphere S in real points we can choose either one or both of the points A and B so that they lie within the sphere S, consequently we can choose the velocities U and V if necessary so that they are less than the velocity of light, but the velocities u', v' will be * Cf. E. B. Wilaon and G. N. Lewis, Amer. Proc. Acad, of Arts and Sciences, Tol. xlviii., Nov. (1912). H. Batemun, I'loc. Land. Math. Soc, ser. 2, vol. x. (1911), p. ye. Frof. Bateman, Time and eleclromagnetism. Ill greater than the velocity of light, except perhaps in the case Avhen the lines L and L touch the sphere /S, one of the velocities u', v' can then be equal to the velocity of light. We shall now show that the velocity V is a possible velocity for the aether in the electromagnetic field (H, E). To do this we must prove that V satisfies Cunningham's relation* c'g + E (vE) + H (vH) = V {2z<; - (vg:)| (18), where g = [EH]/c and ^(; - i (EE) + |(HH). Now the equations (16) give eg: = [EHJ = ca'h\\ + «0 {[U (VU)] - [V (VU)]}. Now [u(vu)] =v(uu)-u(uv) [v (vu)J = v(uv)-ufw). therefore g" = a%\\ + «'0 {V [(UU) - (UV)] + U [(VV) - (UV)]}. Again (vE) = caZ»{(uv)-(vv)}, (vH) = c«a5[(uv) -(vv)}, therefore «;(vE) = (vH). Also E(vEj + H(vH)=cV(^'* (! + «'; {(uv)-(w)} {u-v}, consequently c-'g-l-E(vE) + H(vH)=cV^Yi + «')v{(uu) + (vv)-2(uv)|. On the other hand 2io = (EE) + (HH) = ceh\\ + «'0 [[(uu) (vv) - (UV)'} + c^{(uu) + (vv)-2(uv)}] and {yg) = d'h\\ + «') {(UU) (vv) - (uv)'}, therefore aTc'(l + «*) {(UU) f (vv) - 2 (uv)} = 2ti?- (vg). The relation (18) is now established. In a similar way it can be proved that the velocities m, u', v are possible velocities of the aether. It should be noticed that since the lines L and JJ are polar lines with regard to the sphere S we have the relations (uu') = c'', (uv') = c',- (vu') = c', (vv') = c'. * Proc, Roy. Soc. vol. Ixxxiii. (1909), p. 110. The Pr'mciple of Eelativiti/, ch. xv. 112 Prof. Bateman., Time and electroynagnetisin. Five years ago Mr. Cunningliam remarkefl to r letter that my vector* s, which satisfies the relations c"-'E + c [sH] - s (sE), c'H - c [sE] = s (sH) (ss) = o» J-^^^)' is a particular case of his vector V. This may be proved as follows : — We easily find from the above equations that 6' [EH] + cH (sH) - cs (HH j = [sH] ( sE ), or c% -H c'E (sE) + c'H (sH) = S |(sE)'+ c'(HHj}. Equations (19) also give c' (EE) - c' (Sg-) = (SE/, c'(HH) - c' (sg:) = (sHj', hence (sE/ + c'^HH) = c' [2i(;- (sg")), and so Cunningham's statement that S is a particular case of V is verified. There are four possible vectors of type s, and these are represented geometrically by the two pairs of points in which the two lines L and L' cut the sphere S. If we compare the two representations (16) and (17) we have the relations ah [vu] = - a'h'c (u - v'), ca5 (u - v) = ah' [v'u'J. . . (20) . Now let us suppose that the components of the 4-vectors ^, i5, A\ B' are proportional respectively to the partial derivatives of four functions a, /S, a', [S' with respect to ic, y, z, ct, then the above relations take the form ^a(a, 3)^/^ d(a,0') X d(a,0)^ d(a\ 13') d{y, z) c d{u;,t) ' c d{x,t) ^ d(i/,z) '"^ '' and we have the following representationsf of E and H, d{y,z) diy,z) ~ c d{x,t) c dy-*:, tj _ i^a;a,/3') xa(«,/3)^ dioL',0') a(a./3) ' ' " c d{x,t)^cd (X-, t) ^ a (y, z) "^d [y, z)) ' Phil. Maq. Oct. (1910). Proc. Lond. Math. Soc. ser. 2, vol. viii. (1910), p. 469; vol. X. (1911), pp. 7, 96. t These foimulje are a liitle more general than those given in a previous paper, Proc. Lond. Mdlh. Soc, ser. 2, vol. x. (1911), p. 9(». It has not jet been proved that the above repreaentatiou is possible whenever k is constant. [22! Prof.Bateman, Time and electromagnetism. 113 The vectors E and H will certainly satisfy Maxwell's equations if \ is a function of a and /3, yu/c a function of a' and j3', and K a constant. If these conditions are satisfied we may replace a and |3 by functions of these quantities in such a way as to make \ unity, and similarly we can make /u,« unity by a proper choice of the variables a', /3'. If k is not a constant there will be a volume density of electricity and convection currents in the field specified by (22). The equations (21) may also be written in the following form, . 9 {x, t) ^fid (.y, z) \ d(y,z) ^ d (x, t) 8(a',/y') ca(a,i3)' cd{a',^') '^d{<x,ld) ^" ^* If now we use the notation J ,. _ dx dx dy d,y dz dz ., ^t dt ~ do. 8a' 9a da' oa 9a' 9a 8a' ' we may deduce from the preceding equations that |aa'}={a/3'j = {f3a'} = {i3/3'} = 0, V[{a'a'} {/3'y8'} -|a'^'}'J = /.'[|aa} {/3/3} - {a,^}"^]. Hence there is a relation of type dx'^dy'^-ch'-c'de = Adx* + 2Hd<xd[6 + Bd^' + A' da" + 2H'dad^' + B'd^'\ . . (24), where A, H^ B^ A', H', B' are functions of a, /S, a', /?', satis- fying the equation \'[A'B' - H"] = fi'lAB - H'] (25). From a remark made above we may conclude that it is suflScient to put X,= /Lt=l when endeavouring to determine functions a, /S, a' and yS', which satisfy the equations (24) and (25). The case in which (EH) = 0 is of special interest, for then K = 0. The equations a = const, /S=C()nst then give a moving line of magnetic force and the equations a' = const, /3' = const a moving line of electric force. It is frequently easy to find the functions a and (3 with the aid of the scalar and vector potentials.* To determine ff.' and /3' we may endeavour * See, for instance, the formulae found by Hargreaves for the case of a moying point charge. These formulae are given on p. 117 of tlie author's Ekclrical and Optical Wave Motion. VOL. XLV. I 114 Prof. Bateinaii., Time and electrjomagnetism. to choose A^ H, and B so that the quadratic differential form dx' + df + dz' - 6'dt' - {A da:' + 2Hdoidi3 + Bd^') can be expressed in terms of the differentials of only two variables a' and /8'. The functions ot! and /3' are both solutions of the equations dx d-e dy dy dz dz c" ct dt ' dx d-e dy dy dz dz c* 9^ dt ^ for 6. This method is successful in the case of the field due to a moving point charge and the functions found for a' and (5' are derivable from the solutions of two Hiccatian equations in accordance with a known result.* Conjugate Electromagnetic Fields. § 7. Two fields (E, H), (E', H') are said to be conjugate when the rehitions (EH') + (E'H) = 0, (EE') - (HH') = 0, are satisfied. If now we represent each field by means of special 6-vectors (e, k) and (e', h') so that H=h + «:e, E=e-«h, H' = li'+ /c'e', E' = e'— /c'h', the above relations take the form ( 1 - ««'}_[(eli') + (e'h)] + (k + «') [(ee') - (hh')] = o, (1 - kk') [(ee') - (hh')] - (k + k') [(eh') + (e'h)] = o. The determinant for these two linear equations is ( 1 - kk'Y + (k + k'Y =(1 + k')(1 + k"), and this cannot vanish if the two fields are real, consequently we must have (eh') + le'h) = o, (ee') - (hh') = o. This means that the field (e, h) is conjugate to the field (e', h'). Now let the special 6-vector (e, h) be represented by a force acting along a line L as in §6, and the special 6-vector * Amer. Jouvn. of Math , March (1915). Mr. Ckaundy^ On the validity of Taylor s expansion. 1 1 5 (e', h') by a force actln<? along a line M. The condition (ell') + (e'h) = 0 then implies that the two lines L and M intersect, whilst the condition (ee') — (hh.') = 0 implies that L intersects tlie polar line of M or that M intersects the polar line of L. Denoting by (y^, u^, vj the coordinates of the first point, and by («_^, u^^ u~) the coordinates of the point Iti whicli L intersects the polar line of il/, we see that the field vectors in two conjugate fields can be expressed as follows: — • H =ah [vu] + CKob (u - v) ■ = mlS [ww'] - ca.8 (w' - w), E =cah (U — v) — «;a& [vu] =a3 [ww'] + c«a/3 (w'— W), H' = a'b' [VW] + CK'a'h\w-Y) — Ka'lS' [uu] — ca'/3' (u' - u), E' = ca'b'{W-Y)-K'a'b'[YW'\ =a'/3'[uu'] + o«'a'/3'(u'~U). The point of chief importtmce is that the velocity of the aether V can he the same for both fields. Conversely, if the velocity v of the aether is the same for two fields the two fields are not necessardy conjugate. The above equations may be made more symmetrical by taking u' = w'. This is permissible since L and its polar both meet M and its polar. A CONDITION FOR THE VALIDITY OF TAYLOR'S EXPANSION. By T. W. Chaundy, Christ Church, Oxford. The conditions for the validity of Taylor's expansion of a function of a real variable have been given by Pringsheim* and W. H. Young.t Pringsheim proves that the necessary and sufficient " cc"/" (a) condition that 2 '^" should converge \o f{a + x) over the 0 ^i ' f (a + x) v'^^ interval 0 <x<R is that '-^ n — ^ ^^ n tends to intiniiy, should tend to zero, uniformly for all values of a;, y for which Q <x <x-\-y <r, where r<R and p is some convenient integer. * Math. Annalen, vol. xliv., p. 57. t Quur. Jour, of Math., vol. xl., p. 157; "The fundamental theorems of the differential calculus" (Camb. ItilU), p. 57. 116 Mr. Ckaiuuhj^ On the validity of Taylor'' s expa?isio?i. W. H. Young gives t!ie simpler condition that for eacli fixed positive r<R the function -; rr" /„ (a + x) regarded as a function of the two variables (ic, ?i), should be bounded in the region n>^, ^<x<B,. 1 seek to establish the following results: 00 (1) If f{a -\-x) can he expanded in a power series S A^x"" 0 in the interval 0 <aj <^, and if the interval of convergence of this series is that given hy \x\<p, then the function /., (a + x) n\ regarded as a function of the two variables x, n, is bounded in the region n>0, 0 <x<k\ where k' <p and < k. (2) If /„ (« + x) - I n\ I is bounded in the region ?? > 0, Q <x< k', and if </:, (a) converges for \x\ <p, the series toill converge to f{a + x) in the interval 0 <x <k, where k < p, and < k'. Similar results of course may be obtained for negative values of x. If we are not concerned to be precise as to the end-points of these intervals, we may say briefly that the region of validity of the expansion o^ f{a + x) in powers of x is the ic"/" fa) common territory of the region of convergence of 2 — • " / ( a -\- Xi ~ and the region in which j <-^ — , — ^ " is bounded. I w! .j 1. We have that /(a-f x)=A^ + A^x+ A^x:'+... for 0 <x<k, and that the series converges for 0 <x <p. Being given any k' < p and ^ k, choose p' such that k' <p' <p. Then since lini. | J^J« =p~', we have that l-^nl" <P \ if '>i> some N. Mr. Chaundy, On the validity of Taylor'' s expansion. 117 In the Interval Q<x<h', since k' <p, we may differentiate the series n times and obtain f (a + x) , , , . (n + l)(n + 2) . „ n! -A+0» + l)A..^ + ^ ^1 -A..^+-" Thus 1 f , ^1' (« + l)(n + 2) ^'" ) if n>iV, t.e. <(p'-/c') <(p'-Ic) -n-l I \-8n Hence f«('' + ^^\» <{p'-ky\ for every ti > .Y, and every x in the interval 0 <cc < ^'. But, for a fixed s,f^(a + x) is bounded in the interval 0 <x<k' ; the same is therefore true of fja + x) ' s\ and also of the aggregate of these functions when 5 = 1, 2, 3, ..., N. It follows that we may remove the restriction 7J> iV in the preceding result, and say that n\ is bounded in the region ?? > 0 and 0 < a; < //. 2. We suppose that \frM^!^ I " < Jf, when 0 < x < U . I n! I Now /(a+x) =/(«)+<(«) +...+ ^^-^^^^\ But if 0<a:</y, then 0<^x<^', so that ' "^^,'^ < 3/". It is therefore sufficient to take I*] <M~^ to secure that x^'fiex) ^"\ — ^->0, asn->a3. In other words, /(a + x) is equal to its Taylor-expansion in the interval 0 <a;<Z' where K<k', and < J/ '. 12 118 Mr. Ghaiindt/, On the validity of Taylor s expansion. Now M is the upper bound (or at any rate a superior lunit) to 1 fJ^^J!^ In 71 \ I in the interval Q <x <h\ while p'^ is the upper Tunit of /„M|-1 i.e. of the precedinf^ funetion when a? = 0. Thus M>p'^. If p = M~^ or if k' <M~\ the restriction K <k' and <M~^ is identical with the restriction K<k' and <p. In this case we have established the validity of the expan- sion over the required interval. But if M~^<p^ and <k\ we have established the expansion over too small an interval. It will be shown that the restriction K<]c and <i/~' may be replaced by the restriction K<k' and </3 by a process of '' continuation." Choose two positive quantities x, ?/, each <l/J/ and such that x+ y <k' and < p. Thus x, y are separately < k' and < p, but x + y need not be less than 1 j M (i\i ljM<k' and <p): it need only be less than 2jM. By Taylor's formula /(x + y + a) =f(x + a) +yf (x + a) +... V"/ (x + a + By) , ^ n , ^ ^_/^ -U where 0 < 0 < 1. n ! Since \x-\- y\<k' so also l^ + %i<A;', and accordingly /;, (:e + a + %) < 2r. ;/"/„ (a; + a + %) -> 0 as ;i -> cx>. y But y<\jM. Hence The expansion /(.c+^ + a)=/(:c+a) + 2//' (.'«+«) + 'I-,/" (x + a) + is therefore established for the specified range of values of a?, y. But since 0<a:;</i;' and x<M'^ we may expand /{x + a), and therefore also f'{x + a), J"" (x + a), ..., in powers of X. We then have f(x + y + a) represented by a double series in powers of x, y. humming "diagonally," i.e. tirst collecting terms of like degree in x, y, we have f{x+y + a)=f{a) + {x + y)f{a) + ^~^^/'\a)+.... Mr. Chauncbj, On the validity of Taylor'' s expansion. 119 Now this change In the order of" summation is permissible, if the double series is absolutely convergent, i.e. if SS .L.M)\ m ! n I is convergent. This, being a series of positive terms, is convergent if is convergent, i.e. if 2 pi (x + yY fM) P- is absolutely convergent. But, being a power scries, it is absolutely convergent within its limits of convergence, i.e. if x-\-y<p, which has been stipulated. Thus the expansion of f(x + y + a) in powers of (x + y) has been effected under the restriction 0<x' + y</C where K<p, 2/ M, and <k'. We have thus replaced the conditions K<ljM and <k', where (0, A") Is the interval of validity of the expansion of f{a-\-x) in powers of x, by the condition i^<2/il/, <p and <h' . By a similar process of "continuation" we could leplace these by K <'^/M^ <p and < k\ and so on, Jt is clear then that ultimately we shall get 2'lM>p or > k' , when this condition may be removed and we are left with K<p and <k\ which is what we require. The conditions of expansibility that have been proved establish the expansions of the elementary functions without difficult}', and apply also to the discussion of the expansion of a function of a function. Moreover, they "explain" why the expansion fails for such a function as e~"'^', for then /^{x), in the neighbourhood of x=0, is of the order e~^!^^x~^"', so that /> ) ' Is of the order e~^ '"*'.»;"'' (??!)-'''". If we set u-'=l/n, this expression -> co as ?* -> co and x-^0, so that is not bounded in any region 0 <x < k, n > 0. ( 120 ) ^^OTE ON THE PRIMARY MINORS OF A CIRCULANT HAVING A VANISHING SUM OF ELEMENTS. By Si?- Thomas 3Iuir, LL.D. 1. It is known from Borchardt that any axl-symraetric deterrainsint having the sura of every row equal to zero has all its primary minors equal. Such a unique minor can be expressed in a variety of ways as a function of ^n [n - l) elements, the result being neatest and most convenient when the elements chosen are those outside the diagonal. Thus, if the given determinant be abed ^ ^ / 9 c f h I d g i j it is best that the letters removed with the help of the conditioning equations be a,e,hj', the minor in question, U^ say, then being h + c-\- d —b —c -b b+f+g -f -c -/ c+/+t and its expansion* dgi-Y dg [c +/) + di (/-f-^) + ig [b + c) ^{d + g^i)[bc + bf-Vcf). * It is worth noting that this expansion consists of three-letter combinations formed from b, c, d, f, g, i; and contains every such combination except fgi, cdi, bdff, be/, each of these four being formed of the letters left on striking out a frame-line of the Pfaffian ^ \ b c d f 9 i The case where/, g, i-d, c, b is also worth noting. Sir T. Muir, Note on the 7)iinors of a circulant. 121 Determinants of this type received at an early date tlie attention of Sylvester, wlio noted that the signs of all the terms in the tinal development are positive, and^ that the number of these terms in the case of the n"' order is {n + iy-K 2. Since a circulant is viewable as a special axi-symmetric determinant, it also must have a unique primary minor when the sum of its elements vanishes. On inquiry, however, the special is found to be not included in the general. The reason for this is that in the one case it is the diagonal of symmetry from which the elements have to be removed by substitution, and in the other it is not. Thus, the circulant being I m n r r I m n n r I 771 m n I it is impossible to get rid of one of the letters by using the equation of condition Z + ?n + ?i + »• = 0 on the elements lying in the diagonal of symmetry, as was done iu the previous instance; the substitution of — ?n — n — r for I must now be made in the other diagonal, with the result that the unique minor, F, say, is seen to be m + n + r — n — r m + n-\-r — m — n —r 711 + n + r the expansion of which is 711" + 2wi'« + n^r + In'm + Inr + 7nr'' + 2«r^ + 4m«)- + »•'. 3. In regard to the outward form of this determinant note should be taken that reading the elements of its three rows in succession we are merely repeating 7n + /I + r, — m, — n, — r, 2^ times as it were. Similarly in the case of the next order, to obtain the sixteen required elements, we" repeat w + n + r + s, — W2, — n, — r, — s, 3l times. 122 Si?' T. 3Iid}', Note on the minors of a circulant This observation involves the fact that the determinant can be viewed as persjmraetric ; thus the one of the third order is P{n, ???, —m — 7i—r, »', n). Further, either way of writing shows that the determinant is invariant to the interchange of m and r. 4. Closely related in form to the two determinants above is a third, of which a three-line example, W^ say, is u-\-v-\-io —V w U i- V ^ w — V —10 it + v-i-w Laving the expansion u^ + 3u {v + iv) + Bu {v- +VW + 10^ j, with u in every term in accordance with the fact that each row-sum is equal to ii. All three determinants, U, V, W, are unisignants, the non-diagonal elements in every case being negative and yet all the row-sums positive. They differ in that the first is symmetric witii respect to the main diagonal, the second persymmetric with respect to the secondary diagonal, and and the third circulant. 5. The third, TF, being circulant is resolvable into linear factors; and therefore if it be expanded in descending powers of M, the last term of the expansion must be so resolvable. But the said last term can be shown to be, save for an arithmetical factor, a determinant of the form V. Conse- quently we have the important proposition that the unique primary minor of a circulant having a vanishing sum of elements is resolvable into linear factors. The form of the factors will appear from the consideration of an individual case. 6. Taking W^, with S written for u-'t-v-^io + x-{-y, and expanding it in Cay ley's manner according to descending powers of a letter in the diagonal, we have S —V —10 —X —y — y S — V -10 — X — X —y S —V —IV — lo-x — y S — V — V — 10 —X — y S having a tanialnng sum of elements. 123 = u^+5(t'*(y f lo + .-c + y) +,..+ 5u S-u —V —lo -x — y S—n —V —10 — X —7/S—U —V — w —X —y S-u From this it follows that W { u, V, w, X, y) \ =oV{v,w,x,y). J u=0 But the TFliere bein;^ the circulant C [S, — V, - 10, — X, —y) is expressible in the form [S - ve — we^—xe^ - ye*) • [S - ve' — we*— ae — ye') • {S— ye'— i06 — xe*- ye') • (5'— ve*— ive^-xe'^ — ye) ' [S — V — 10 — X — y) ^ where the last of the five factors is manifestly u, and e is a prime fifth-root of unity. On dividing by ic and thereafter putting M = 0 we thus have {(1-e) V + {1 - e") 10 + [1 - e') x + {i - e*) y} . {[l-6'')v+{l-e*)io + (l -e) x+{l-e')y\ . {(l-eV + (l-e) io+{l-e*)x+{l-e')i/] • {(1-e*) y+ (1 -e^)io+ (1 — e') x + (1 - e) ?/] = 5F(v, io,x,y). Similarly, as a test of the extension in § 2, we have AV{m, n, r) = [|l -\/(- 1)} m + {l + l\n+{l + V(- l)} r] »[{l+l]m + {l-l]n + {l + l]r] . [{1 + V(- 1)} m + {1 + 1} » + {1 - V(- 1)1 r] = 2 [m + r) [{2n + m + r)"'+ (m - r]^\ = 4 (?n + )•) \{m V n)-^ [n + rf]. That m 4 7- is a factor of the determinant in § 2 is seen by adding the first and last rows. 7. Another mode of verifying the resolvability of V Is to increase the first row by all the others, arriving readily at F(y, 10, x, y) = V W X - y v + io-\-x+y — V — X —y ' v+io-\-o:^y — 10 — X y y — 10 — V V 4 io + x-[ y 124 Sir T. Muir, Note on the minors of a circulant. then to perform the operation (1 - 6) colj+ (1 - e-) col^f (1 - e^) C0I3+ (1 - e') col,, when it is found that the first column has become such that the factor (1 - e) y + (1 - e-) !<; + (1 - e'j a: + (1 - e') y, can be removed from it, and the elements 1, -e% -e\ -6* left in the column. 8. From the irrational factors of V^ rational factors must arise by grouping, save when n + 1 is prime. The formal result in regard to this is that the rational factors of V„ are in nuynher the same of those of (a;""^'— 1) 4- {x—\), and are similar in degree. Thus when n is 3, we have x* — l --=(a3 + l)(c«^+l), x—\ ^ ' and hence the linear and quadratic factor of § 6. When n is 5, x^—\ ' r = (a? + 1) [x'^x + 1) (x' - « + 1) x—\ ^ I ^ \ I and F^ is resolvable into a linear factor and two quadratics. When n is 4 the F-function, V {I, m, ??, r) say, is irresolvable: and we have for its equivalent Z* + ?«'' + n* + r* + 3 [?m + ??iV + nH + r^n) + 2 [Fn + ??iV + n'^r + r^m) + {Vr + rn^n + n^m + rU) + 4 [r + r') [m' + n') + IV + m'n^ 4- 7 {Pmn + Im^r + In^r -i mtir^) + 6 {l^mr +77i'nr + n'lm + r^?il) + 4 (py?r + m"7>- + n'mr + r7??i) + 11 ?mnr. 9. In order to compare V with Boole's and other uni- signants that have no negative terms in the elements we increase each row by the sum of all the rows in front of it, and thereafter perform on the columns the same operation. The result is V{v^w,x,7/) = v+w+x-\-7/ tv + x + y x+y y v-\-w+x v-\-2to-\-2x->ry w+2x-\-y x-i-y v-\-io v-\-2w+x v+2w+2r^y to \ x \ y V v + w v + iv^x v+w+x+y 3/r. Salmon^ The twisted cubic of constant torsion. 125 Equivalents of tlie like kind for U and W can similarly be found. 10. It is also worth noting in conclusion that the converse of the property with which we started also holds, namely, if the primary minors of a circidant he equal and non-zero^ the sum of the elements must vanish. For it is known that generally C {a, b, c, d)=^{a + bco + ca>'' + do)^) [A + Boi^ + Cw' + Da>), and therefore with the data just mentioned the second factor on the right would \iQ A [I ■{■ lo + to' +■ lo'j, which is 0 ; so that the other general identity C{a,b,c,d)=^aA + bB-{-cC+dD becomes 0= {a + b + c + d) A, whence the desired result. Capetown, S.A. 7th JVov., 1915. THE TWISTED CUBIC OF CONSTANT TOESION. By IF. H. Salinon. The following seems to be a shorter and more direct method than any yet published of arriving at the general equations of the cubic of constant torsion originally discovered by M. Lyon.* These equations, by an appropriate choice of origin, axes and parameter, are here obtained in their sim- plest form, consistent with perfect generality, and refer the curve, itself imaginary, to real axes. The Cartesian coordinates x, y^ z of the general unicursal twisted curve of the i^l^ degree can be written in the form ^-M ^_Aii ,-m n) ""'Tit)' y~F{t)' F{t) ''>' where F, /,, f.^, f^ are polynomials of degree w of a single parameter t. If the common denominator F be divided into the numerators /j, /j, J\ giving a numerical quotient, the remainder is a polynomial of degree ?i — 1, and we can, by a change of origin, write the equations of a unicursal twisted curve of degree n in the form u V 10 , . ^=j^^ y=y^ ^=7' ^^^' * Aunales de I' Enseignement Superieur de Grenoble, t. II., p. 353, 1890. 126 Mr. Salmon, The twisted cubic of constant torsion. where w, v\ to are polynomials in t of degree n — \^ and F a polynomial of degree n. The torsion t at any point of the curve will be given by ^ [y^ ~ y^) + y (^^ — zx) + z [xy — xy) •(3), 7 [y^ - y^l + (^^ - «-»)'+ {-^y - ^yY where ■ ■ , dx dy dz a., y, .,..., denote ^, -, -,.... Now • uF-Fu ■ vF-Fv ■ loF-Fw ■■ uF'- 2uFF- uFF+ 2uF' {f, = . ^5 ' with similar expressions for y and z. Therefore yz-yz= IjF'livw) F+ (viv) F-{- [viv] F] where {vw), ..., denotes viv — vw, ..., and x = llF*{uF'- 3uF'F- 3uF'F+ 6uFF' + QuFFF-uF'F-6uF']. Since ^x(yz — yz) = T'2.{yz—yzy, therefore 2 [u (vw) FU'u{ vw) F'F+ u{vw) F'F- 3u ( vw) F'FF - 3u (vw) F'FF+ Qu (vw) FF'+ 6w (vw) F'FF -u(vw) F^F- 6w (vw) FF'} = tF^ \(vw) F^ (vw) F+ (vw) F\\ But ^u(vw) = '^u(vw) = ^u(vw)^ therefore F^ 2 [u[(vw) F+ (vw) F-^ (vw) F] - u (vw) F] = T'^[(vw)F+(vw)F+(vw)F\' (4). In the case of the cubic, equations (1) are ^-d/+d/+dJVd,^ F' _ ^\f+^',^ + ^\ _ " ^ ~ d/ + d/ + d./+~d, ~ F ' Cfi-C,t + C. 10 z = d/ + df-i- dj + d^ F Mr. Salmon^ The twisted cubic of constant torsion. 127 Without loss of generality we may take ^'^=1, and, increasing the parameter ^ by a constant, we may also make d^ zero. Hence we may write F=t^ + dt + e. In the above equations the origin is on the curve at the point where t is infinite; if in addition we take the axes along the tangent, the principal normal and the binomial at this point, then ^0=0) c„ = 0, c, = 0. Therefore the equations of the general twisted cubic mav be written X = y = a^f + aj + a^ f -\-dt-^e h,t + \ f + dt-^e c. t + dt-i e u V P' ^' F (5). PI« u = af -\ra^t + a^, u = 2a^t + a^, u = 2a^, u = 0 V = b^t -\-b^, v = b^, v = 0, v = 0 IV = 0, F=2,e-\-d, F=QL IV = 0 F=Q. w = Cj, F= i" + dt + e, Therefore (viv) = 0, [ivu) = 0, {uv) =— 2c<o*^i ; [vw] = 0, [ivu] = - 2a/^, {uv) = 2ffl, [b^t + JJ ; [vw) = — b^c,, {ivu) = cj2a^t + a^), {uv)= — aj>/-2aj)^t+aj}-a^b^] and equation (4) becomes + (cv /.,-«* + 4.aJb/>/+ ia;b;e - 2a^a.J)^U' + 2a^afiJ?/ + iajcj e - ia.a^b^^J + ia^ah;t + 4a„a,c; t + a^'b{- 2a^a.pfi^ + a, V + «, V + ^ V) ^' + (- ^".X^ - ^<^^) ^^ +. [iu^b; e + 8a;A,A, t - 4a„aA* + ^a^ctfijy^i FF + (- 4.a^'b;e - \2a:b^>,e - 80,%^+ ia^aJ^;U- 4a,afiJ>^t - 8a;' c; t + ^a^ij\b^ - ia^aj^^ - 4a,a.c.;) F'F] (6) . 128 Mr. Salmon, The twisted cubic of constant torsion. Equating the coefficients of f on both sides we have ajb^b.^ = 0. But a„ = Q or J, = 0 makes the curve plane; therefore for finite constant torsion it is necessarv tliat b, = 0. Equating the constant term on both sides with the condition b^ = 0, we have '^^aj^^c/ = T{iaJb'e' + iajc^d'] ; but from the coefficients of t^ we see that 12aJ}^c^ = T.4aJb^\ Hence fl'/c/(f = 0. Rejecting a^=0 and Cj=0, either of which makes the curve plane, we have d=0. Hence equation (6) reduces to 3a,5,c, {<« + 2ee + e'\ = r [a^'b^ (f + 2ee + e') + {a;b;U' + a^c^") 9t' + {a;b;t' - 2a^nJj;U' + 4.aX t' -f ^a^a^c^t + a'^b^ + a'^c^^ + ^'/c/) 9«= - la^b^ti^U" + 3e<*j + [a'b^e - a^alj'^) {Qt* + 6et) + (- aoVe + a^aJ:>;t-2aXt-a,a^c;) m'\ .(7). Equating the other coefficients we have the following relations: — Ta;V = 3«oV, (8), HV-K«A' = o (9), T(a„Ve+9a„a,c;0=3a„V,e (10), «.V + ^V + « = 0 (11), «o«A'« = 0 (12). From equation (12), since flr„, a^, 5, must all be finite, e = 0. Therefore from equation (10), a^a^c./ = 0, and so a, = 0. Hence a/ + c/=0, 2,a^G^ = 2a^b^\ and T=3c,/a„^,. Thus c^ = ± ia^ and 6, = ± iV (f a„a,). Writing A for a,,, B for a,, and < for 1/^, we arrive at the following result : — All twisted cubics of constant torsion are imaginary, and their general equations can, by a proper clioice of origin, axes, and the parameter t, be written x = At+Bt', y = ±>^[-?^AB)t\ z = ±s/{-BY, where A, B are finite, independent arbitrartf constants. The magnitude t of this constant torsion is \/ {6Bj/A -s/A. ( 129 ) ON BRTGGS'S PROCESS FOR THE REPEATED EXTRACTION OF SQUARE ROOTS. By J. IV. L. Ghiisher. § 1. For tlie orig-inal calculation of the lof^aritliins of certain prime numbers Brip^gs used a method which required the repeated extraction of the given number a great luunber of times. For example, by extracting the square I'oot of 10 fifty-four times in succession he found tlie resulting root to be 1.00000 00000 00000 12781 91493 20032 35. This repeated extraction of square roots was very lai)orious, and in chapter viii. of his Arithinetica Logarithmica (1624) Briggs gave a method of proceeding by means of differences from one root- to the next. The object of this paper is to examine this method witii reference to the principles on which it rests and the use which Briggs made of it, and also to consider other methods to which Briggs might have been led by it. § 2. Briggs seems to have observed that the decimal part of each successive square root was approximately equal to ^ of the decimal part of its predecessor, and that if each decimal part were subtracted from ^ of its predecessor the differences so formed were such that each was approximately equal to \ of its predecessor, and that if second differences were formed by subtracting each first difference from ^ of its predecessor, these second differences were such that each was approximately equal to \ of its predecessor, and that if third differences were formed by subtracting each second difference from ^ of its predecessor, these third differences were such that each was approximately y\, of its predecessor, and so on. Briggs denoted the decimal part of the square root with which he starts by A, and the first, second, third . . . differ- ences by B, C^ D, . . . and he applied his method to the calculation of successive square roots in the following n>anner. He first extracted the square root of the given number continually, in the ordinary manner, a certain number of times, and from the square roots thus obtained he calculated the first, second, third . . . differences B, C, B, . . . until, to the number of places included, one of these differences VOL. XLV. K 130 I)}\ Glaishei\ On Briggss process for the became insensible. Su[)pose that the i^-difference is the first to disappear. 'J'lien starting with ^^ of the /li'-difference as a new i^-difFerence lie derived tVoni it new/)-, C-, i>-differences and a new A^ which was the next square root*. In the exjnnple which Brig<j;s gives he starts with the number 1.0077G96 and extracts its square root 9 times successively, tlie value of its 512'''' root being found to be 1.00001 51164 G5900 05G72 95048 Sf. and by forming the first, second, third . . . differences from this and ihe preceding square roots (viz. the 25G"', 128"\ &c.), lie obtains tlie following vahies of B, U, D, and E-. B, .00000 00001 14253 77215 03190 9, C, .00000 00000 000017271197889 3, I), . 00000 00000 00000 00004 5G894 3, A', . 00000 00000 OOOUO 00000 00020 7 ; i^ being insensible to tiiis number of phaccs. Tiien starting witii J,jE^ that is, with .00000 00000 00000 00000 00000 G5, he derives from it the vahies of Z>, 0, Z?, and A for the next root, the vahie of A being found to be .00000 75582 0443G 30121 42907 60, which therefore is the decimal part of the next I'oot. Having thus exj)lained his method by woiking out an example, Briggs concludes his chapter by giving expressions for B, C, I) . . . in terms of A as far as the term in ^'*. * I Hse Briggs's leUeis A, B, C, D . . . exactly as lie did, but my fiist, _ second, . . . differences are liis second, tliird . . . differences; for, the square root being 1 + A. Briggs calls A the first difference, viz., it is the difference between the square root and unity ; and the difference between A and hiilf the previous .4 he calls 7?, itc. : but A is merely the decimal part of the initial quantity, and it seems more natural not to include it among the differences, as it does not belong to the system formed by the others. Briggs had no i)ot;ition for distinguishing the successive .4's, iJ's, . . . such as is now afforded by suffixes, nor had lie a notation for powers which put in evidence the quantity raised to the power, e f/. A* was denoted by (4). t Although Briggs used decimal fraclions and liad a special notation for them (by uuderliniiig them), still he practically treats his numbers as integers in the course of work, e.r/. he writes this number 10000,1 5 116,46599,905fi7,29604,88. Tlie commas are used, as now, to divide into convenient groups a long succession of figures, but he starts witli the first figure, not with the first decimal. repeated extraction of square roots. 131 §3. In order to Iiivesti^-;ite tlie principle of Briggs's process It is convenient to distinguish the successive square roots and differences by suffixes. Denoting the original quantity whose root is to be continually extracted by a, it is supposed that a consists of unity followed by a decimal, viz. a~l-\-A where A is a decimal. I denote the decimal parts of the successive square roots by A^, A.^, ... (in accordance with which A is equivalent to ^J so that a*=(l + ^)^-l + J„ ai=(l + ^ji=l+^„ and in general the 2''-th root of a, that is of 1 + A, is equal to 1 + A^. If for brevity we put h — —^, then a''= (1 + Ay^'=. 1 + ^,._,, a'" = (1 + ^1)^''= 1 + .1„_^, &c., d"=[l + ^f = 1 + /!„,„ aV^ = {l 4- ^)^''= 1 f ^,,^.3, &c., Denoting the first, second, third, . . . differences corre- sponding to A^ by i>^,, C,,, i>^, ..., it follows from their definitions that these quantities are given by the equations B = ^A -A , C = ^B -B , J) — XQ _ C E =T-V^ ,-i>,&C. n 1 o 71—1 n' § 4. Let the expansion of «*, that is of (l + ^)*, in ascend- ing powers of A be 1 + VJi + K/*'+ VJi^ -'t&.c. Then A^ = VJi + VJi'+ VJi'+ VJi'+&c., whence ^„_, = V^2h + V^Ir + V/'Ii' + V^2Vi* i etc., and therefore B^ = VJi'+{2'- 1) VJi'+ [2'- 1) VJi'+ {2*- 1) VJt'+&c. Putting 2h for h, dividing by 2', and subtracting, we find C, = (2=- 1) 17.^+ (2^- 1) (2-^- 1) VJi'-^ (2^- 1) (2^-1) VJi'+ &c. 132 Dr. Glaisher, On Briggs's 2)roces8/or the and, siiiiilarl}', i>, = (2^- 1) (2^- 1) VJi'+ (2*- 1) (2'-l) (2^-1) VJi' + (2*- 1) (2*- 1) (2^- 1) VJiU- &c., E„= (2*- 1) (2'- 1) (2*- 1) VJi' + (2»-l)(2*-l)(2'-l)(2-^-l)T7i''+(2^-l)...(2^'-l)Ta'+&c.* &c. &c. §5. Now suppose tliat to the nuaiber of places included F^ is insensible. This expresses that ^V^,,., =^„. Using [E^^J, [D^^J, ... to denote the new E,'D" . . . calculated by Briggs's method, he takes (^„+,) = gV^.,- Thus (^„J = (2^-1)(2^-1)(2^-1)^5| + (2^-l)...(2'^-l)F,|[+(2<'-l)...(2'-l)T/;^'+&C. The quantity (^„+,) is obtained from the formula and theretoi'e (Z)„ j= (2^- 1) (2^- 1) f/i;+ (2^- 1) (2^- 1) (2^_ 1) 1/ i; -{(2'^-l)(2^-l)(2'-l)(2^-l)-2(2*-l)(2*-l)(2'-l)}F,|' _|(2«_l)(2^-l)(2^-l)(2^-l)-2(2«-l)(2^-l)(2^-l)}F,~ + &C. * These series are convergent; for the «"» term in the series for the r^^ difference is (2'-"-'-l)(2--^-=-l) ... (2'-])Tv„A'-^'. Now, aswill beshowninS12, Fr,,/i'*' = 3 — — -^ — , the numerator of which is (»• + «) I A approxiraately equal to A^^'^ and An is approximately equal to -- . Thus Vr^\ A""*' ^r+» 2" is approximately equal to , p-- -^ . The ratio between this term and the (?• + «) ! Iv ')" previous term is therefore approximately 2'-^'-'-l A 2'-i-l ■ (M-7) 2" ' which nearly = , — — . The largest value of r is n, so that this factor is aln'avs less than . r + n repeated extraction of square roots. 1 33 Proceeding In this manner we find (^.J = (2-i)F,f;K2^-i)(2'-i)T^^+(2^-i)(2^_i)F;^ + {(2^- 1) (2*- 1) (2=' - 1) (2^- 1) - 2 (2^- 1) (2*- 1) (2^- l) + 2=(2^-l)(2^-l)jF/i-; + &c., - {(2^- 1) (2*- 1) (2^-1) (2^- 1) - 2 (2-^ - 1) (2^- l) (2^ - 1 ) + 2-^(2^-l)(2*-l)-2^(2^-l)}F^^;-&c, + {(2^_ 1) (2*_ 1) (2^_ 1) (2-^_ 1) _ 2 (2-^-1) (2^- l) (2'- 1) + 2=(2^-l)(2*-l)-2^'(2^-l)4 2^}F,|-&c., the gener.al term being {(2'-'-l) (2'-'-l)(2'-'-l)(2'-'-l)-2(2'-'-l) (2'-'-l) (2'-'-l) + 2'(2'-^- 1) (2'--'- 1) - 2^(2'--'- 1) + 2*} vjf, . § 6. Thus, up to and including terms of the order /**, (^„^,) differs from A^ only by the substitution of ^h for k and, to this degree of accuracy, it is equal to the value of ^„^.,, the square root of A^^. Now the fact that F^^ disappears (to the number of places included) shows that (2"- l) (2'-l)(2'-l) (2'-l) VJi^ may be neglected. The numerator {(2* - 1) (2*- 1) (2'- 1) (2'-l) - 2 (2' - 1) (2' - 1) (2*- 1) + &c.} VJi^ of the corresponding term in (>^„+,) is necessarily less than this quantity, and, in addition, this numerator is divided by 32. Thus (^„+,) differs from >4^^, by a quantity which is less than g^g'" 0^ ^''^ quantity which is insensible, and therefore Briggs's rule gives the value of A^_^^ with more than sufficient accuracy. The same is also true of B^^^^^ C„^,, &c. k2 134 Dr. Glaisher, On Bi'iggs^s process for the §7. The quantities B^, C„, i>^^, . . . may be expressed in terms of ^^ and tlie previous ^'s, for 1 ^ 3 . and, similarly, 1 7 7 ■ 1 15 35 15 T. 1 ^ 31 . 155 . 155 ^ 31 ^ /^ _ ^ J ^^ A ^^^ A ^^^^ J n 9^1 n-6 o20 n-5 ' olo n-4 ^IS ?l-3 651 . 63 , + ^;tt ^. , ^A ,+A. O" "-* 9° n-1 ' n In general, if Q^ be the r"' difference, then ^„ = ^)}('?-2)(^-2-0(^-2')...(^-2-)M„ where rj is an operator such that 7]'A^= A^^ ^. § 8. Briggs, however, does not express his B, C^ D . . . in terms of A and tiie preceding A's, but in terms of A only. This, in tiie notation of the present paper, is equivalent to expressing B , C. D„ . . . in terms of A . The values given by Briggs, which include terms up to A'\ are B=hA\ J) = iA* + lA^ + j',A^ + yA^+^-,A^ E=2lA'i-7A'+10Y^A'i-12j%^gA'+nl-lA'+7\llA'\ repeated extraction of square roots. 135 and so on, the values of / and K, the eighth and ninth differences, being /= 54902 ^s^\/i' + 25584G5 j-yg^'», /i:=2805527J§|-^'"* § 9. To obtain the values of B, C, D, . . . In this form we notice that 1 + ^„_, = (1 + AJ\ 1 + A^^_^ = (1 + ^J*, &c., and therefore, from § 7, c„=^3{(i+Ar-i}-|i(i+-AJ-ii+A. ^„=^aKi+^j'-i|-^.{(i+A.r-i}+|[(i+Ay-i]-A, &c. &c. § 10. It is however more convenient to derive the expressions for 6'^, D^^ . . . in terms of ^^ directly from their definitions (7^= ^Z?^_, — 5^, &c., in §3 {i.e. to derive each difference from its predecessor) by making use of the fact that the change of the suffix n into n— 1 is equivalent to the change of A^ into {l-\-AJ'—\, that is into ^,, (2+yl^). Thus, (7„ = I [^a: (2 + Ay- a:\ = 1^; + 1-^;, Similarly ^„=i {tV^;(2 + AJ-^;n-i {tV^;(2 +^„r-^„^} +&c., or, as it may be conveniently v^'ritten +/6A^{2\wi^j-i}+M;i2'(i+i^j'-i} + -L^;[2*(i+ij„r-il, which on reduction gives Briggs's value; and so on. * Briggs's values of the terms in -4'* in 1 and K are inaccurate (in their fractional parts). These errors have been corrected in the values given above (See §13). 136 Dr. Glaisher, On Briggsh process for the These formula show tliat the complete expression for the r**" difference contains 2' — ?• terms beginning with a term in A-\ As has been mentioned the expressions given by Briggs extended only to the term in A^". § 11. If we denote the (?•— ly*" difference by P„, then n I r II '/r+1 n ' I ni n i--") and therefore the r^^ difference Q^ Is If therefore we put *n irn 71 ' i)-+2 n ' Ins n ' ' then ()•) (r+l) ?..3 = ^' Pr + —^' Pr.. + 2 (r + 2) 2^,.^, + 7^.,.^3, &c. &c., where (?•), denotes the coefficient of x* in the expansion of (l+a:/. ; § 12. By means of these forraulse the coefficients In any difference may be deduced from those of its predecessor; but the coefficients in any given difference P^, may be obtained directly as follows. From § 4 we have P=(2^-^-0(2^"-l)...(2-l)F..A'-+(2--l)...(2'-l)F,,/rV... Now the F's are defined by the equation (1 + Af= 1 f F.A + V.]i'^ F,/i'+ &c., repeated extraction of square roots. 1 37 and tlie expansion of tlie quantity on tiie left-liand side Is 1 + h log, (1 + ^) + ^'-g-, + ^ ^^^1 ^+ &c., ^^ jloff (1+.4)}'" so tb at K = - ^' , — =^ . Thus p,= ^- — '' .., ' ' ' {iog,(i + A)r. + ^^"'"1" T^ ' iiog,(i+A)r+&c, (2'--l) (2'-'-l)...(2'^-l wliere ^2'-+''!-i _ 1^ ^g*"'"'-^- 1).,.(2'""-1 «„.„ = (r 4- «ij ! Expanding the powers of log^(l + ^J we find that, if as before tben ~ 2 P,..l=a.>l- o«r' r+1 ?-(3>-+5) Pr,3= «r.,- -^ «r.l+ ^^^ "r' r -h 2 ('* + !) (3r + 8) r'+ 5/ + 6r where &c. &c., (2'"'-l) (2''-'-l)...(2'-l) a, =- 2'"-l 2-'^ - 1 «..„ = - V.s (2'^-l) (r + 2) '■^" (2'-"-l) , 138 Dr. Glaishe7\ On Briggs's jn-ocessfor the § 13. By means of the formulge in tlie two preceding sections 1 have calcuhited the vahies of ^^, 0^, ... in terras of A^, thus verit"yiu|2^ Brif^<^s's vakies except in the case of five of the coefficients of y1'" as mentioned below. Suppressing tiie suffix throughout, the values of Z?^, (7„ . . . in terms of A^ are found to be B = hA\ C = \A' + IA\ D = lA^^lA^+^^A^+lA:' + -i^A^, E=^-iA'+lA'+ Vg'^'+ W8^^'+ W^'+ V2'¥^"» -^~16-^T^8^T^ 128 ^T 128~ ^ "1 256' "^ ' /7 _ 19 5 3 /17 , 19J5 347 /J8 , 1 4C 8 8 7 3 ^» i 43 75 8 0 5 JlO TT — 2JL8 0 3_1 J84.GO3 5421 /19 I 9047 6197 410 -" ~ 128 -^ T 128 ^ T 128 -'^ » 7"_7027 545/|9i3 2748 3 5 974lu ± 12 8~'^^ ^ 128 -^ » jr— 718215 099 /1 10* Briggs expresses the coefficients as mixed fractions, and his coefficients of ^'" in F, G, li, /, K are 1953|f|, 683722^18, 706845if||, 25584652|f|^-, 2805527, the true values in this form being 1953i||, 68371§1, lOQUb^^^, 2558465/2^, 2805527|§f § 14. Briggs's formulae enabled him to calculate 5^, C^ . . . from A^^ alone, but it is not clear why he should have desired to do so. He mentions that i?^, C„ . . . can be so expressed, * Although the coefficient of the leading term increases, the term itself decreases. For the first term of the i-^^ difference is (S""— 1) (2'"'— Ij... (2-1) Tr+i /i*"^', and substituting approximate values, IV+i/i*"^' approximately (l0g,(l+^„)'-H ^„'-+l ^rM (r + l)! {r + l)I (r+ljl 2C-*^';»" Also (2'"-l)(2''"'-l) ... (22-1) is less than, and may for the present purpose be replaced by, 2^''""'^''^*^ Making these substitutions, the term becomes 1 ^4''^' . ., ■ ,r-n),n-iryi ' which diminishes with r. For the last (n"') difEerence this Talue is 1 A"^^ (n + lj ! ihrnnnyl- repeated extraction of square roots. 1 39 and gives their values up to terms in ^"', without any indi- cation of the manner In wliich they were obtained. He then applies the formulge to calculate i?^, C^, D^, E^ from A^ to 30 places of decimals, A^ being the decimal portion of the num- ber 1 .00001 511G4 ... quoted in §2. Jt seems to me possible that Briggs liaving observed the curious result that B^ was equal to i^,/, was so led to express i> . C , . . . in terms of A . § 15. The fundamental relations in § 3 show that A =^A -IB , + i(7 ,-nVZ> ,+..., n 2 71-1 4> n-l ' S n-1 lb ti-l '•••} and therefore, if F^ may be neglected, If therefore Briggs had only required ^„,,, he could have derived its value from those of B^^ ^n • • • without previously calculating i?„^,, 6'„^, . . . , but these differences would have been required for the derivation of A^^^^. § 16. The values which Briggs obtained for B^, (7^, . . . in terms of A^^ would have enabled him to derive any A from its predecessor without calculating any differences. For, sub- stituting: in the htst-written formula the values of i? , (7 ... in terms of A^, we have if terms beyond ^ are neglected ; and Briggs's values of the higher differences would have enabled him to extend this expression up to the term in -4'". Briggs does not give this formula, and so presumably he did not obtain it. Jt may have escaped his notice, or it may be that as a calculator he preferred to work by differences, a method which he continually employed and of which he may almost be said to be the inventor. Series were unknown in Briggs's time, but if he had noticed that A could be expressed in terms of the preceding -<4, By C ... it would seem that he might have given the formula for A^_^_^ in terms of A^ in a finite form, rejecting powers of A which were insensible to the number of places included. There is however the important difference between 140 D7\ Glaishe7\ On Briggs^s process for the the expressions for ^,_, C„, ... in terms of A^ (which he does give) and of A^^^ in terms of A^^^ that the complete expressions for the former are finite while the latter cannot be expressed in finite terms. § 17. The preceding expansion of A , in terms of A of course follows at once from the formula and similarly we have §18. As shown in §12, the quantities F,, F,, F3, ... are log^a,^),i!^f4±^\i!^I^^ ...,thatis,the, aie-^y— , — ^j— , .... Since Ji = 2~", It follows from the formulae of §4 that the limit, when n is very large, of 2"A is log a, and that the limits of 2'"^„, 2"'(7„, 2'"Z>„, ... are (log^ay (2'-l) (2^-1) (2^-1) — ^j — J — §1 — (loge^). ^j (log^a),.... The first of these results, viz. that the limit of 2"A^ Is log^a, is involved in the formula by means of which Briggs obtained the logarithms of the early primes, and for which he calculated the value of -4^ ; for his actual process was equiva- lent to log,„a=2M„x. 43429..., this multiplier .43429... being derived from the repeated extraction of the square root of 10, the logarithm of which was known. § 19. The differences B , C . . . .do not seem at the present day to possess mathematical interest of their own. They are derived from a system of successive square roots ■which were constructed in order to obtain as the final result a very high root, but which do not form a mathematical table that would be calculated for its own sake. I am afraid that the interest in the differences is almost entirely historic, and consists in the fact that their existence was discovered by Briggs, that he used them for calculating square roots, and that he obtained formulae for them in terms of the decimal part of the final square root to which they were attached. repeated extraction of square roots. 141 A^'itli reference to these formulge it is curious that any matliematical work of so fine a calcuhvtor as Briggs should not have been quite free from error; but it is likely that he never used these formulae as far as the term in ^", and that when lie wrote the Introduction to the Arithmetica Logarithmica, he had partly forgotten the details of his work coniTected with the calculation of these early primes, and printed the formulje as he found them among his papers. I may mention that Hutton has given a very good account of chapter viii. of the Arithmetica Logarithmica, which forms the subject of the present paper, in the Introduction to the numerous editions of his Logarithms. Also Delambre in vol. i. of his Histoire de V Astronomie Moderne (pp. 538—541) has given a full account of Briggs's process and has quoted his formula for B, C, D, E in terms of A (§13). He makes no examination of Briggs's results and process, but after quoting the formulje he remarks: " Le procede precedent [i.e. the process] parait bien preferable a ces formules. Briggs ne demontre rien, il parait avoir trouve le tout par le fait et d'apr^s ses calculs; cependant, pour donner ces formules si longues, il a dil se faire une esp^ce de th^orie empirique, dont il ne parle pas." § 20. If Briggs had repeatedly extracted cube roots instead of the square roots he would have found that similar differences existed and were capable of expression in a similar manner. For, proceeding as in §§3 and 9, if a be a decimal, and if (H-a/ = l + a„ (l + a)* = l + ot„ (l + a)"" = 1 + a3, &c., and if then we find 7„ = i«; + v«: + ¥«,; + ¥a,; + 1< + 1^: + 2W. &c. • &c. ( 142 DETERMINANTS OF CYCLICALLY REPEATED ARRAYS. By Sir I'homas 3Iuir, LL.D. 1. Writers like Puchta, Noetlier, W. Buniside,* who have dealt with determinants of the type here specified, have restricted thenjselves to cases where the circulating arrays were also themselves circulant, and where as a consequence the determinant is expressible as a product of linear factors. It will be found interesting to withdraw for a moment this restriction, and to see how it conduces to the discovery of additional properties of even the less general functions. 2. The circulant of two n-line arrays ts expressible as a product of two n-line detenninauts. Eor, taking ii equal to 3 and the arrays of \a,hcA, \ah,cA as the two circulating arrays, the determinant in question is a, «. "s a^ «^5 « K K h h^ K ^ <'i ^2 ^z c^ ^5 C "* ^'s «0 a "^2 a h. Ik Ik l> h h ^4 ^'5 ^'« C. C, C3 and this, when we reverse the order of the last three columns and thereafter the order of the last three rows, becomes «l ^^2 «3 «6 «S «. \ K h K h K ^1 c, C3 c^ c, c, ^4 ^* C, ^3 ^3 ^, K h h h K h «. «5 "e "3 «j «i which being ceutrosymmetric is resolvable into a, + a^ ttj + a. a^ + a^ b, + b^ b^ + b^ b^ + b^ c,+c^ c^ + c^ c, + c^ o, -a^ «.-^5 «3-^6 ^.-^4 ^-^5 '^3-^6 c. — c, c. — c^ c, — c. * Dtnkschr. . . ., y4/<;ad. c?. Wist. (Wien), vol. xxxviii., pp. 215—221 : vol. xliv., pp. 277—282; Math. Annalen, vol. xvi , pp. 322—325, 551-555 ; Messenger of Math., vol. xxiii, pp. 112-114. Si)' T. Mui7\ Determinants of cyclically repeated arrays. 143 3. Taklnj^ the special case of the foregoing-," wliere the circulating arrays are the arrays of the circulants C{a^,a^^a^, C {a^, «j, a^), we find tlie result of the resolution to be C {a^ •+ a^, a, + a., a^ + a^) .C (a, — a^, a.^ — a^, a^ — aj , whence there come six linear factors, in agreement with the result obtained by the writers above mentioned. 4. We may note in passing that this circulant of two three-line circulant arrays is not altered in substance by changing o„ a^, a^, a^, a., a^ into a„ aj, a^, -«^, -a^, -a„ as is readily seen on changing rows into columns and' multi- plying by (—1)^. (—1)^- This being the case, if we take the product of the two forms, it must be possible to extract the square root of botii sides; and, doiiig so, we find that the circulant of two three-line circulant arrays is itself expr'essihle as an ordinary three-line circulant^ namely, C[U, F, IF), where U = a,' + 2a^a^ - «/ — 2a^a^, V = a^^ + 2a, a.^ - a/ - 2a^a^, W = a,"' + 2a fl^ — a^' - 2a^a^. 5. The circulant of n two-line general arrays is divisible by a, + c<, + o,+... a^ + a^ + a^+.. ]f, merely for shortness in writing, we take 7i equal to 3, the circulant in question is a, a., a, a, a. a, 1 2 d 4 a 6 K b, K K K K «5 «6 "l "3 ^'3 ^<4 h h ^ K K \ a^ a^ a^ a^ a, a.^ h. h^ K K Ik h.. 144 Sir T. JIuir, Determinants of cyclically repeated arrays. and tliis is seen to be equal to a 4 «3 + <7, «, + «, + «. a^ a, a a h,^-h, + h^ h, + h^-tb^ b, b^ b^ b^ h + ^.^K K + KaK ^, K ^ ^4 />3 4- ^>, + i. />^ + Z>^ + /;.^ ^,^ h^ h^ b^_ wlilch on perfoiiniiig the operations roWg — row^, row^ — roWg, row^ - row^, row^ — row^, becomes resolvable into ^ + ^3 + ^ ^^.+ ^4+^6 «.-«3 «'.2-«4 «3-«S "4-«'6 a^ — a^ ttg — a, a, - a^ a^ — a^ 6. Taking the special case of the foregoing where the given two-line arrays are themselves circuiant, that is to say, where />„ b^, b^, b^. b^, b^, = ^2' «P «4' ^35 ^61 "o-> we see that the first factor in the result is C{a^ + a^ + a^, a^ + a^ + a^), and the second is the axisymmetric determinant «, - a^ a^-a^ a^- a. a, - o^ «2-«4 «i-^ ^4-^6 «3-^5 ^^6-«2 «5-^^ «2- «4 "l-^3 7. TAe circuiant of n two-line general arrays is equal to the determinant of four u-line circuiant arrays. This follf)\vs at once from advancing the odd-numbered columns to occupy the tirst n places, and thereafter treating the rows in the same manner. Sir T. ifuir, Veterininants of cyclically repeated arrays. 145 For example, the circulant of the four aiTcays is equal to the determinant of the arrays of tlie four circulants 8. If the four given two-line arrays in the preceding paragraph be made circulants the arrays of G [b^, b^^ b^, b^), G {b^, b^, b^, b^ become the arrays of G [a,, a^, a^, a^), G{a^, a.^, a^, a^), and the resulting determinant becomes the circulant of two four-line circulant arrays, and as such resolvable into linear factors. Note must be taken, however, that it is not the eight-line determinant considered by Puchta in his first memoir. Both are eight-line deteruiinants which are cir- culants of circuhmt arrays; but, while the one here appearing is the circuhmt of four two-Hue circulant arrays, Puchta's is the circulant of two four-line arrays each of which is a cir- culant of two two-line circulant arrays. The distinction between them is perhaps more simply brought out by viewing them as eliminants, the one being the eliminant of {l,x, y, xy, y\ xy\ y\ xtf^i,, a„ «„ ..., a,) = 0 x- = \, y = l and the other, Puchta's, the elinjinant of (1, X, y, xy, z, xz, yz, xyzjti^, a,, a^, ...,«,) = 0 a?" =!,/ = !, z'=l Further, if in writing the eight-termed equations here, we use (1, X] (1, 2/, /, y) to stand for the first set of facients, so that {l,x){l,y)[l,z) stands for the second set, we have an additional aid to clearness. 9. The circulant of u three-line general arrays a^ a^ a^ a^ a. a^ a^ a^ a^ ^hh KhK KKK c, c, ^3, c, c^ Cg, c, r^ fg, ..., VOL. XLV. L 146 Sir T. Midr, Determinants of cyclically repeated arrays. is divisible by a^ + a^ + a^+... a, + a^ + a^+... a^ + a^-{- ag+... b^ + h^ + b^+... h, + b^ + b^+... K+K+h+... . c^ + c^ + c,+... c^+c^ + c^+... C3 + c, + c,+... 'J'lie proof of this is quite similar to tliat in §5, and the two toj^ellier at onci suggest the forniuhiting of a wide generali- zation. Wlien 7i is 3 the cofactor is «.-^4 «2-«5 ^3-^6 «4-^ «5-«8 ^6-^9 c, -c, c^-c, c^-c^ c^-c^ C^-C^ Cg-C, «7-«. ",-". «9-^3 «l-«4 «2-«5 «3-«6 ^7 - <^. ^8 - ^3 Cg -C, C, - C^ C, - C, C3 - C, which like that at the close of §5 is conveniently viewahle as the determinant of four arrays, two of which are identical. 10. The circidnnt of n three-line general arrays is equal to the determinant of nine {i.e. 3^) n-line circulant arrays. For example, when n is 3, the circulant of the arrays of kA^sK I^A^gI' kA^sI' is eqnnl to the determinant of the arrays of the nine cir- culants C (a„ a^, flrj, C (a^, o„ a^), C[a^, a^, a J, C{b,,h,h), 0 A, *„?>«), 0{b,,b^,b:), This is established exactly in the manner of §7, and the general pro[)osition which includes the two is readily grasped. 11. By making the given arrays in §§9, 10 circulant ari'ays we learn that the circulant of the arrays oj is divisble by C [a^ 4- «4 + «„ a^ + a, + a^, a^ + a^ + aj , (a), and that the circulant of the arrays of C (a„ a^, a^), C {a^, a^, aj, C (a,, a,, aj, Si?' T. Mid?-, Determinants of cyclically repeated arrays. 14' is equal to the circulant of the arrays of G[a^^a^,a^\ C{o^,a^,a^), G)a^, a^, a^), (/?). From the latter result it follows that the said circulant, Ca.S say, is invariant to the interchange of o^ with a^ \ a^ with (7, I (7), «g ivith f/g J and from this and the former result that (73.3 "is divisible hy C{a^-\-a, + a^, a^ + a^ + a^, a^ + a^ + o\, (6). JSJaturally eacli of these four results can be established independently. 12. It is important, however, now to note that the theorem of § 9 is susceptible of extension in a quite different direction when the number ot given arrays is the same as the number of elements in each array. For the case where this common number is 3 the wider theorem h:—the circulant of the three arrays a, a^ a^ a^ a, a^ a^ a^ a, K K K h h h K K K Cj c, C3, c, c^ C3, c, c^ Og, is divisible by rt, + 0^7 + a,7'-' a^ + a^'y + a^r ^3 + «g7 + ^7^ l\ + ^7 + /^7" ^, + ^57 + ^\1 K + ^c7 ^ ^97' c, + c,7 + cy 0^+^57 + ^87' ^3 + ^6^ + ^97' where 7 is any third root of 1. On the given determinant a, a^ a^ a^ a, «g a, a^ a^ b. b. K b, b.^ h, K ^ l> a, a^ a^ a, a,^ a^ a^ a^ a^ K \ K K K K K h h ^7 ^8 ^9 «. ^2 ^3 ^4 ^5 ^6 «4 «5 «6 ^'l-% ^'9 «1 "2 ^'3 K K K K K ''. ^ ^ ^ c. c. c, c, c, c„ c. c.^ c. 148 Sir T. Maii\ Determinants of ct/clicalhj repeated a^n^ays. we first perform tlie operations col, + 7001^+7'" col^, col,+ 7Col. + 7"colg, C0I3 + 7Colg + 7'col5, and then on the resnltlng determinant the operations ro vVg — 7 r o w g, ro w^ — 7 ro w^, ro w^ — 7 ro w^, followed by ro vVg — 7 roWj, row. — 7 row^, row^ — 7 row,. The resnit of this is that 0 appears in every place of the first three colnmns except those sitnated in the first three rows; and the determinant oi' the non-zero elements being I "^t, + «47 + ^77' ^>2 + ^'37 + ^h 7' ^3 + ^57 + ^"97' 1 -, is, as expected, a factor of the original determinant. The CO factor is «.-7«'4 «2-7a5 «3-7«6 «4-7«7 «5-7«s ^^6-7^9 differing from that of §9 merely in having 7 prefixed to the second term of every element. 13. As we shall presently see that the three determinants, got from the three-line factor by giving 7 its three values, do not have a common factor, there follows the important theorem that the circidant of the three arrays of\ah^c^\, \a])^c^\, 1^7^8*^9 1 is equal to the product of the three determinants I <^i + ^i + "7 ^'2 + ^3 + ^s *^3 + ^6 + ^9 1 ' 1 «, + «47 + ^',Y ^\ + ^'57 + ^7' C3 f Cg7 + c^i' I , where 7 is a privie third root of 1. 14. Taking the second of these determinant factors, namely, «, + ^'47 + ^',7' «2 + «57 + «s7' «3 + "eV + «97' K + b,y + b^y- h^ + Ki + Ky" K + Ky + hy' c, + c^7 + c,7' c^ + c,y + cj' c^ + c^y+c^y' and seeking to express it as a sum of determinants with monomial elements we find that there are nine of the deter- minants free of 7, nine with 7 as a factor, and nine with 7' as Sir 2\ jUuir, Determinants of cyclically repeated arrays. 1 49 a factor. If we denote each of the twenty-seven by the suffixes occurring in it, for example, I^A'^sl ^y 459, tiie lesult of the development is (123 + 456 + 789 + 159 + 267 + 348-168 -249-357) + (126 - 135 + 234 + 189 - 279 + 378 + 459 - 468 + 567) 7 + (129-138 + 237 + 156 -246 + 345 + 489- 579 + 678)7"': or, say P+ Qj + Ry\ With this notation it follows that the product at the end of § 13 must be (P+ Q-^E){P+Qy+ Rj') (P+ Qy' + Ry\ and this we know to be equal to the circulant P Q R R P Q Q R P AYe thus have the theorem that the circulant of the arrays of I^.Val) l«A^6 kvVsl' is expressible as an ordinary three-line circulant C\P, Q, R), where P, Q, R are aggregates of nine determinants whose columns are taken from the arrays a. a. a„ a„ a, a„ a, a„ a„ one f-om each. ^K^ KKK KhK 15. Another notation for the determinants in P, Q^ R would be that in which Imn would denote the determinant whose columns are the Z"" of \af^c^\, the ?u"' of \a^b^c^\ and the ?/"' of |t'3^gCg We should then have P= 111 + 123 + 132 + 213 + 222 1 231 + 312 + 321 + 333, ^=112 + 121 + 133 + 211 + 223 + 232 + 313 + 322 + 331, P = 113 + 122 ■\ 131 + 212 + 221 + 233 4 311 + 323 + 332, l2 150 Sir T. ^fuir, Determinajits of cyclically repeated arrays. wliere the sum of tlie intej^ers specify inp^ a determinant is in P of the form 3;-, in Q of the form 3/-+ 1, and in R of the form 3r + 2. 16. Continning now the specialization interrnpted at the end of § 11 we learn from § 13 that the circulant C3.3, of the arrays of G(a^, a^, a^), G (a^, a^, a^) C (a^, a^, a^) is equal to the yrodact of three circalants C(a+«, + a„ a, + a,+ «3, ^3 + ^6 + ^ • C (a, + fl,7 + a^f\ a^ + a.7 + «y , a^ + a^y + a^f) . C (a, + a^i' + «_7, a., + a,7' + a^7, a, -^a^'y' + 0^7), and therefore, by the allowable interchange, also equal to the product of other three circulants C (a, + a^ + a,, «, + o^ + «'«, «, + a^ + c/g) • C (a, + a^7 + a37', a^ + a^y + ^^7=, «. + 0^7 + 0^7') 'C(a^ + a^y' + aj, a^ + a.Y + a^y, a^ + a^y' + a^y). It should be noted, however, that independently of the said interchange, the identity of these two products of three circulants can be readily shown. As a matter of fact each circulant is resolvable into three linear factors, and the nine linear factors obtained from the one group are the same as those obtained from the other. 17. Similarly from § 14 we learn that the same circulant of circulant arrays, C3.3, is equal to where (^=123 + 456 + 789 + 3(159-168), X = S(12Q f 459 + 378), ^ = 3(129 + 345 + 678), and the column-numbers refer to the array «1 «3 «3 «4 «S «6 «7 ^8 %i «S «1 ^2 «6 «4 «i ^9 ^7 «8' «2 «3 «I ^'5 «6 «4 «8 ^9 «7- the minors of which are no longer all different — for example, 126 = - 135 = 234, 459 = - 468 = 567, .... Sir T. Muir^ Determinants of cyclically repeated arrays. 151 To this there is also a companion form, namel}', where 0', x, 6' ai'e outwardly identical with (^, ;y, 6, but the cohimii-nuinbers now refer to the array «l «4 «7 «2 ^^3 «g «3 «6 «9' «7 «. «4 «8 «» «5 «9 ^3 ^6' «4 «7 «1 «S «8 «2 «G «9 ^3- 18. From the preceding paragraph we obtain two rational cubic factors of C3.3, namely, ^■^x-^^ ^"*^ ^' + x' + ^'- As such, however, these are not new, being essentially the same as those obtained in § 11 ; so that the resulting equalities ^ +x+^ = C(a, + fl^+a3, a^ + a^ + a^, o^-^ a^ + a^), 0' + X' + ^' = ^ («i + ^^4 + ^'7' «2 + «5 + ^9' ^'3 + «G + ^9)? are nothing more than the expression of the change of a three-line determinant with trinomial elements into a sum of 27 determinants with monomial elements. The tirst-obtained forms, too, have the advantage of showing that the two cubics have a common linear factor, namely, the sum of the a's: so that, up to this point, three rational prime factors of Cs.s have been found, one linear and two quadratics. It remains to ascertain the character of tlie others. 19. To do this it suffices to arrange the nine linear factors of § 16, not in one uninterrupted series, but so as to form a square array, say the array F F F s F F F , •^21 22 23' F F F , ■^31 32 33' the positions being chosen so that the Fh of the first row are those of ^K + «3+«3' «4 + P5 + ^6' ^j + ^a + ^O) the i^'s of the first column those of C{a^-{a^ + a^, aj + «5+^8' ^a+^e + ^s^' 152 Sir T. Muir, Determinants of cyclically repeated arrays. and therefore F^^ identical with la. The arrangement brings at once to light the existence of two other circulant factors simihu' to those just mentioned, namely, C (a, + a. + ffg, a^ + ttg + a^, «3 + «< + o^, 0(a, + ag + fl^, a^ + a^ + a^, a^ + a. + a,), these being equal to F F F F .F .F . II 22 33 We thus learn that C^^ is resohahh into five rational factors, one linear and four quadratic^ the latter being of the form {x^ + %/ + z^ - ?>xijz) -^{x+y + z), i.e. x'^ + y^ + z^ — xy —yz — zx. 20. After this, one is not surprised to find tiiat there are two other results like (/3) of §11, and therefore also two additional ways of expressing C'3.3 as the product of three oidinary circulauts, namely C (a, + rtj + a^, a^ + «6 + ^'7' <^'3 + «4 "I- ^s) . C (a, + a^7 + 0^7', a^ + a^ry + a^ry\ a^ + a^y + a^f') 'C{a^ + a^f + a^Y, a^ + ^^7' + ^^7, a^ + 0.^7' + 0^7) , and G (rtj + Og + Og, a^ + a^ + (Tg, ^3 + «5 + «J • 0 (a, + ag7 + a37'-', a^ + a^y + ^1^7', «^ + a.7 + «,7') • G (a, + flg7'-' + 0^7, a^ + a^7' + a^-^, a^ + a^7' + 0,7) , Of the two former expressions of this kind (§ IG) one com- bined the i^'s of our square array by rows, and the other by columns. In the first of the two just written the sets of tliree i^'s forming a circulant are taken from the secondary diagonal and its parallels, and in the second from the main diagonal and its parallels. There are thus twelve three-line circulants that are factors of C3.3, and each linear factor has a set of four in each of which it occurs ; for example, F-i^ occurs as iS\ + «27 + ^'37') + («< + ffjT + %!') 7' + («7 + «87 + «97') 7, («. + a,7'+ ^',7) + {% + a^f" + ^^7) 7 + (^3 + ag7' + ^^7) 7', (a, + a, + flg) + {a^ + a^ + «,) 7 + {a^ + a,, + a^ y\ an d (a^ + a^y + aj'') + (a^ + aj + a^r) 7 + (^3 + a^y + a^Y) 7*. Sir T. Mid)', Determinants of cyclically repeated arrays. 153 21. It will have been observed that the formulae occurring in these statements of tiie properties of G3.3 are characterized by different groupings of the suffixes of the a's, the four leading groups being 1, 2, 3 >^ 1,4,7 1,5,9 1, 6, 8 ; with associates The first of the four is the originator, and the three others are derivatives of equal status which evolve their associates in one and the same manner, namely, by use of the cyclical changes 1, 2, 3 into 2, 3, 1, 4, 5, 6 into 5, 6, 4, 7, 8, 9 into 8, 9, 7. Now it is very interesting to note that if we return to the general determinant of § 12, which includes C3.3 as a special case, a further segregation lakes place among the four groups, 1, 4, 7 being removed from its fellovvship with 1, 5, 9 and 1, 6, 8. Thus, while it will be found that in regard to the two latter we have the theorem that the circidant of the arrays of l^l^^s!' l«4^Cfil' l«7^8C9l> is equal to the circidant of the arrays «I«S«9 «7«2^6 «4«8«3 h,Kb, bj,K hKK and to the circidant of the arrays «7 «3 ^; C. C„ 0. 2' there is no corresponding theorem in regard to 1, 4, 7, the nearest approach being the theorem of § 10. Again, while we have the theorem of §12 in regard to 1, 4, 7, there is no such theorem in regard ta 1, 5, 9 or 1, 6, 8. Capetown, S.A. 27id Dec, 1915. ( 154 ) SOLUTION OF A PROBLEM IN LINEAR DIOPHANTI^E APPROXIMATION. By TV. U. H. £ertvick. The theory of continued fractions supplies a solution to the i'ollowing arithmetical problem : given a positive number a, it is required to find the least value of [a^ -h\ for integral values of {h, Ic) with 0<k<N. In fact if a be expanded in a continued fraction with integral partial quotients, and piq, p' Iq are the inferior and superior convergents (principal or intermediate) whose denominators are next less than N, qoL—p takes a smaller positive value and q'a — p' a greater negative value than k'a — h' for any integral values of h', k' within the above limits for k' . This theory, iiowever, is Insufficient to determine what pair of integers x, y, 0<7/<N, gives its least value to the linear expression f= ax + 1)1/ 4- c when a, J, c are real numbers all different from zero. In the following note I have developed a modification of the method of continued fractions, which enables a solution of this latter problem to be given. In giving a solution of the arithmetical problem it is con- venient to make use of geometrical notation and ideas. The geometry is only introduced formally, however, for the sake of visualising the arithmetical processes involved. The solution is a strictly arithmetical one and could be presented logically without mention of any geometrical entity. On a Cartesian plane, cut up into unit squares by the straight lines x = m, y = n, for integral values of jh, ?«, the distance of a point (?», n) from the line /, ax -\-hy + c = 0, is proportional to am-\-bn + c. So con6ning attention to the strip of plane lying between y = 0, y = jSI (including the boundary line 3/ = 0, but excluding y = N when N is integral), if {m, n) [tn\ n) are the two integer-points within the strip which are nearest to I, on its positive and negative sides respectively, «m + 5n 4- c takes a smaller positive value and am' + bn' + c a greater negative value than afx + hv ^c where (/u., v) is any other pair of integers with 0<v<N. It will be convenient to call a point with integral coordi- Mr. Berivick^ On linear diophantine approxi7nation. 155 nates a node, and the straight line joining two nodes a node- line. For our purpose then it will be sufficient to show how to find the nearest node to the line I on each side of it within the strip. The equation of any node-line PQ can be put in the form Ax-\-By-\- C-0, where A, B, C are integers and A, B have no common factor. Every node B then lies on a line Ax + By+C=C' for some integral value of C, and the distance of R from PQ is ± C (^' + ByK Further, each of the lines Ax + By+ C = ±l (1) passes through nodes, and nodes lying on these two lines are equidistant from Ax-\- By+ C=0, and nearer to it than any other nodes. From this property the two lines (1) are called the node-lines nearest to Ax + By+ (7=0. If (^, 7}) is a solution of Ax + B>/+\ = 0 (obtained by expanding Aj B in a continued fraction) all the nodes on the lines Ax + By +0=1, Ax + By+ 0=0, Ax + By+ 0=-l, have coordinates of the form {Bt+{0-l)l -At+{G-l)vl \ [Bt+Ol, -At+Ov), \ (2), [Bt+[0+l)l, -At + [0+\)ri] ) respectively for integral values of t. A plane area is defined by Minkowski to be convex when (1) it is limited in all directions, i.e. lies entirely within the rectangle bounded by a; = ^„ x = A^, y = B^, y = B^, for finite values of ^„ A^, B^^ B^, and (2) no part of the segment of the straight line joining two arbitrary points on the boundary, and lying between them, falls outside the area. If Q, R, S are three non-coUinear nodes which lie within 156 Mr. Berwick, A solution of a problem in a convex area It can be shown that there is at least one node within the area and lying on one of the two node-lines nearest to QR. For let P be that node on the segment QR which is neai'er to Q than any otlier node on it. (If there is no node on QR between Q and i?, P coincides with R). The triangle PQS then lies entirely within the convex area, Sliould this triangle not contain any node other than its vertices, neither can TSQ, T being the fourth vertex of the parallelogram PQS7\ and therefore a node. The entire pai'allelogram thus contains no node other than its four vertices. Now the whole strip of the plane between PQ and TS can be cut up into parallelograms homothetic to PQST and contiguous to each other, the four vertices of each parallelogram being nodes, and no one of these parallelograms can contain any node except its fi)ur corners. There can thus be no node at all between PQ and TS. Hence either S lies on one of the node-lines nearest to PQ, or else there is at least one node P' within PQS or on PS or QS. In the latter case P' lies in the convex area, so repeating the argument for the triangle PQP' it either follows that P' lies on a node-line nearest to PQ or else that there is a node distinct from P' within the triangle or on one of its sides. And since the area PQS, lying within a convex area, can only contain a finite number of nodes, a node in this area and lying on one of the node-lines nearest to PQ must be discovered after a finite number of such steps. We are thus able to state the following theorem : If P, Q aie two nodes within a convex area, and if the area contains any other node at all, then it must contain at least one node on PQ or PQ produced or on one of the two node-lines nearest to PQ. When the boundary of the area passes through nodes, these nodes, or a part of them, can be included amongst the points belonging to the area if desired, subject only to the restriction that when two points P, Q on the boundary are included amongst points in the ai'ea, then every point between P and ^,on the straight line PQ is also included amongst them. A solution of the problem enunciated at the beginning of this note now follows immediatelj'. Fii-st, when it is required to find the nearest node to ?, given by , , ^ ^ f = ax + hy + c = 0, between y = M a.nd y = N, any two nodes ^.(a.,/3,), P,Kft,), M<0<N, one on each side of the line, are to be chosen. linear diophantine approximation. 157 The parallelogram bounded by y = il/, ax^-hy -\- c = aa^ + Z^/S, + c y = iV, aaj + hy + c = aa, + 6^?, + c (Including P, and P,) is then a convex area, and this area contains no point more distant from I than P, on one side or Pj on the other. So if there is any node nearer to I than P^ or P. there must be one such node on P, P., or on one of the node-lines nearest to it. rutting down the coordinates ot nodes on these three lines in the form (2) above, that value of t is to be found \vi)ich corresponds to a node Pj within the area (3) and gives \J'\ its least value. Should the only possible values of t belong to P,, P, these are the two nodes nearest to I, one on each side of it, between the limits M <y <N^ and there is no need for further calculation. But when Pj is on the same side as Pp say, the convex area (3) is made narrower on replacing the side ax + % + c = aa, + h^^ + c by ax-\-hy + c = aa^ + h^^ + c, and this narrower area either contains a node P, on P,Pg, or on a node-line nearest to P^P^, or else it contains no node at all. In the latter case P^, P^ are the nodes nearest to I on its respective sides within the limits considered, and no further calculation is needed. But in the former case P^ replaces P, or P3 (according as P, is on the same side of I as P, or PJ, and a further narrowing of the convex area ensues. Since the original area (3) is limited in all directions it only contains a finite number of nodes, so after a finite number of such steps a stage must be reached when there is no node on P^P, or on a node-line nearest to it within the area y = M, ax + by -i- c = aa^ + 5/3^ + c, y = xV, ax-\-hy-\-c = aa, + h^, + c. When such a stage is attained P^ is the nearest node to I on one side of it, and P, on the other, within the limits M<y<N. Of course when I is parallel to a node-line it may happen (and will liappen if M—N is great enough) that there are several nodes equidistant from I and nearer to it than any other node on the same side. When, secondly, it is required to find the node nearest to I on one side of it, the positive side say, two nodes P„ P, are to be chosen on the positive side of I in the strip 158 Mr, Berwick, A solution of a jjroblein in M<y<N, P, being more distant from I than P,. The first convex urea is bounded by y ^N, ax + by + c = aa^ -{- hjS^ + c j and Pj Is that node in tliis area which lies on P^P^ or on one of the node-lines nearest to it, and is nearer to I than any other such node. P^ may or may not be nearer to I than P,, but in either case the convex area (4) is narrowed when the side through P, is replaced by a line parallel to it through P^ or P3 according as P^ or P, is nearer to I. The next node required in the approximation lies on P.jP„ or on one of the node-lines nearest to it, and is nearer to I than the more distant of these two points. ►Since the area (4) only contains a finite immber of nodes a stage must be attained, after a finite number of such steps, when no node on P^P„ or on a node-line nearest to it within the limits M<y < ^ is nearer to I than P„, the nearer of the two points P^^, P„. When this stage is reached P„ is the nearest node to / within the limits considered and P,,^ the second nearest. If it is required to find the third nearest node to I on the positive side within the same limits, the nearest node to ax+by + c — «a,„ - 6^,„ -c = 0 on its positive side is determined. But it is unnecessary to approximate again from the beginning in finding this third nearest node, only from P^. the nearest of the nodes to I, except P„ and P^, already discovered. A further approxi- mation enables the fourth nearest node to be found, and the process can be carried to any desired number of steps. It may happen of course that a set of ^ nodes equidistant from I is found as the «*^ nearest instead of a single node. As a numerical example we will find the four nearest nodes to the line f='7rx-€y-l=0, 7r = 3.141592 653 590... , 6 = 2.718281 828460 ... , on its positive side within the limits 0<?/ < 10 000. The first two nodes being taken to be P,(2, 0), /, = 27r-l = 5.283185, PAhO),/,= Tr- 1=2.141 593, linear diophantine approximation. 159 P3 lies on y = 0 or y= 1, tlie other nearest line y = —\ being entirely outside the area considered. Now fit, 0)=7r«- 1 = 3.141 593^-1, f[t, l) = 7r«-e-l = 3.141593^-3.718 282, and the least positive values of these tvvo expressions for intejrral values of t is taken bv the second when i = 2. So P^ is (2, 1), /; = 2.504 903. The line F^P^ is x — y—\ = Q, and P, must lie on this or on x-y=0 or on ic — y — 2 = 0. Its coordinates are therefore [t, t) or [t + 1, t) or [t + 2, t). Now /[t, <) = (7r-e)<-l = .423 31U- 1, f{t+l, t) =(7r-e)^ + 7r-l = .423311« + 2.141 593, /(«+2, t) = {7r-e) < + 27r-l = .423 31U + 5.283 185. The least positive value of these three expressions is taken by the second when t = — 5, but this gives a node (—4,-5) outside the limits 0 < 3/ < 10 000. Within these limits the least value is taken by the tirst expression when < = 3. ISo F. is (3,3), /,= .269 932. Continuing the approximation we find i'. (4, 4), y;=.693 243, i'elQ, 10), /, = .091516, P, (22,25), /, = .157 993, P, (41,47), y;=.046 053, P, (73,84), y;, = .000 590, P.J118, 136), /;„=.021604, P,, (240, 277), /„ = .018 170, Pj, (362, 418), /„=.014 736, P,3(484,559), /,,= .011302, P,^ (606, 700), /,^=.007 868, P,, (728, 841), /,,= .004 434, p,^(85o, 982), . y;^=.ooiooo, P,^(5634, 6511), /,,= . 000 023, P,J6411, 7409), Z.^^. 000435. 160 Mr. Berwick^ On linear diophantine approximation. There is no node on P^^P^^, or on a node-line nearest to it, within the parallelogram ^ = 0, y=10 000, TTX- €2/-l=0, TTj; - e_y - 1 =/,,, so the nearest node to ttx— ey —1 =0 within our limits is (5634, 6511) and the second nearest is (6411, 7409). To find the next one, we tind the nearest node to iTx-ey- 1-/^ = 0, on its positive side. Among the points already noticed the nearest is P^ (73, 84). Tiie equation of P^P^^ is F= 7325a; - 6338^ - 2333 = 0, and the coordinates of all nodes on i^+l = 0, F=0, F-1 = Q, are of the form (6338^+5634, 7325^ + 6511), (6338^ + 73, 7325^+84), (6338^ + 850, 7325^+982), respectively, for integral values of t. Now writing jr= 63387r - 7325e = - . 000 155, ■we have /(6338<+5634, 7325^ + 6511) =7r/+y;^ = -. 000 155< + . 000 023, /(6338< + 73, 7325^ + 84) =ia+/3 -- .000 155<+.000 590, /(6338i + 850, 7325f+982) =Zf +y;g = -.000 155<+ . 001 000, and within the limits concerned there is no node nearer than that given by i = 0 in the second of these expressions. Since this point lies on the further boundary of the parallelogram 3^ = 0, 2/=10000, ■rrx-ey-l-f^^ = Q, ;r.c - 6?/ - 1 -/, = 0, it must be the nearest node, except P,, and P|g, to TTX — €y —1 = Q. Carrying out one more approximation, the nearest node to the line , 77-cc-e?/- 1-/^ = 0 on its positive side is found to be (7188, 8307), giving 71887r-83076-l = .000 845. The required nearest node is therefore (5634, 6511), and the next three, in order of distance, are (6411, 7409), (73, 84) and (7188, 8307). ( 161 ) ON EQUIPOTENTIAL CURVES AS POSSIBLE FREE PATHS. By aS*. Brodetsky, 3f.A., S.Sc, Ph.D., Lecturer in the Uniyersity of Bristol. 1. The motion of a particle in a plane is governed by two equations : v' = 2V+C (1), where v is the velocity, p is the radius of curvature of the path, V is the potential of the field in wliich the motion is taking place, and dn is an element of normal to the path measured in the same direction as the radius of curvature. If an equipoteiitial line is to be a free path, V is the same at all points of the path, so that the velocity is a constant. Equation (2) then gives 1 dV p en Let 8n represent the normal distance, at any point, from the equipotential in question to a consecutive one. Then 7^— on = const. on Hence p a Bn (3). This Is the property that must be satisfied If the curves F= const, are to be describable as free paths. An obvious possible case is when the equipotentials are concentric circles, provided the constant V in (1) is properly adjusted. 2. I am not aware that the general solution of (3) has been obtained. To do so in the simplest manner it is convenient to use tangential polar equations. Let p be the perpendicular from the origin on the tangent at any point of an equipotential, and -ip be the inclination of the tangent to the X axis. Then the condition can be written p + Tp cc 8n (4). If the equipotentials are given- by the family VOL. XLV. M 162 Dr\ Brodets/oj, On equi potential curves \ beliif^ a parameter varying from equipotential to equl- potential, ihea hn — hp. In going a very short distance along a normal from one equipotential to a consecutive one, ;// remains constant to the first order of small quantities. Hence and the condition (4) becomes ^+B? = ^W| f^)' L being a function of \ only, and \ being invariable in finding the radius of curvature. and the equation is p +-7^ = ^ (6), dip cfj' and we can suppose fi to be the parameter defining the various equipotentials. The most general solution is p = ^e^'+'''>^{A/^ + B^e-''^) (7), summed for any number of values of h, which can assume any value without restriction. The equation (7) gives us the most general form fx)r the tangential polar equation of a family of equipotential curves which are describable as free patiis. The equation (7) gives for special values of h concentric circles, families of equiangular spirals, cycloids, epi- and hypocycloids, etc. To plot a curve given by (7) one would first plot as if p and xp were ordinary polar coordinates, and then construct the first negative pedal. 3. The problem can also be solved by means of pedal coordinates. Let the pedal equation of a family of equi- potentials describable as free patiis be r' = /{\p) (8), where A, is a parameter defining the individual members of the family. Then ^ dp 'dp ^^' as possible free paths. 163 the differentiation being partial because X Is constant for any patli. Let ?•', 2^' refer to a consecutive curve A, + t\. Then r"=/(\ + SX,, p). When h\ is very small we can put Bn = p — p = Bp', and r'' — r" =p'^ —p' = 2p Bp. We get 2p Ip =f [X + S\, ^ + hp) -f {\ p) Thus {'P-%)'P = %'^- But p oc S«, t'.e. oc 8/?. XT d^ iience p = -,,, where Z is a function of \ i.e. is constant for any path. Substituting in (9) we get for f the partial differential equation ^^f-.,) + .W§(=o ,0). where h is an arbitrary constant, and the solution is f=h' -k^\^,+ \{v±^[f+k^)]dp (11), where k' is another arbitrary constant. Thus we get the paths r' — j\p ± ^J [p' Jr F) ) dp = a constant varying from path to path and the radius of curvature at any point is \[V±sl{p'^k')\ (13). This is the complete integral. To obtain the general integral, we put k' = c^[]c% 164 Dr. Brodetshj^ On e.quijJOtential curves ^vllel•e ^ is an arbitrary functional form. If we elimlnale k* between the two equations ...(14), dk' ]L{X)-^ \k) we shall get the general solution for r in terms of X and p. There is no singular solution. 4. In (14) make -^, = const.; then p/h is a function of \, and we obtain r' = A + M[\)f.,... [15), A being a constant, and M a function of \. J/(X) depends upon L (X) in (10), so that it can assume any arbitrary form. M (X.) = 1 and J. = 0 give concentric circles. A = Q and M[\) constant give equiangular spirals. If A does not vanish we get epi- and hypocycloids. The value of p in the case of a solution given by the equations (14) is found as follows. In virtue of the equations (14), we have ^- =0 die' ^' Now P = where the brackets denote that we must first substitute for /c* from tiie second equation in (14). Hence P~^dp '^^Gk' dp 'dp ;i6). The result (16) is identical with (13) obtained for the complete integral. But we must remember that in (16), k' is not a constant, but is a function of ^; and \ defined by the second equation in (14). Thus if d<f) _ die' const. as possible free paths. 165 we g&ipjh as a function of X, so that P = ^\^)-P (17), v/liere iV(X,) Is a function of A,, and is tlierefore constant for any given path. We thus again get concentric circles, equi- angular spirals, cycloids, and epi- and hypocycloids. 5. In the two-dimensional motion we have not considered the forces outside the plane of the path. If the motion is not constrained to be in one plane and there are forces out of the plane, we proceed as follows: Consider the forces acting upon the particle at any position in its path. Their resultant is along the normal to the equi- potential through the position occupied by the particle, 'jlius the motion of the particle Is equivalent to that of a particle on a surface under no forces. The path must therefore be a geodesic on the equipotentlal surface. For the normal force we have p dn as In Art. 1. Thus, as In the two-dimensional problem, we get p oc Sn (18). p depends upon the Inclination of the geodesic to the lines of curvature of the equipotentlal surface. Let p,, p^ be the principal radii of curvature at any point on the surface, and let (j) be the Inclination of the geodesic path to the line of curvature corresponding to />,. Then 1 cos'^ sin'^ PR, P, - and our condition (18) becomes ^ + ^«^ 19. Pi P, 6« If a line of curvature is also a geodesic, we can put ^ zero. h>uch a line of curvature is a plane curve. The problem reduces to the two-dimensional one already in- vestigated. It is in this way that free plane paths along equipotentlal curves arise. 6. First suppose that the equipotentlal surfaces are cylinders. One of the principal ladii of curvature at any point is Infinite, say p,. Then (19) reduces to Pj sec''^ a Bn (20). m2 166 Dr. Brodetslvj, On e qui potential curves If now the cyliiulilcal equipotentials are such that their section hy a plane pel peiidicuhir to the generators sutisties the condition for free paths, we have But a geodesic on a cylinder Is given by ^ = const. PTcnce the condition for three-diniensioinil free paths is also satisfied. We get the result that if a faniilj' of cylindrical equipotential surfaces is such that the })lane curves obtained by a perpendicular section are describable as free paths, then any loxodronie on any of the cylinders is also freely describable, the velocity being chosen appropriately. 7. We now take the equipotentials to be surfaces of revolution. The meridian ciu'ves are geodesies, whilst the parallels of latitude are not geodesies except at points where the tangent plane is parallel to the axis of symmetry. By Clairaut's theorem, the geodesies are given by r%'\\\<^ — k (21), where r is the distance of any point from the axis, ^ is the inclination of the geodt-sic to the meridian curve, and k is a constant defining a particular geodesic. In the equation (19) we may use for p, the radius of curvature p of the meridian curve, and for p^ we may put rls'iiiip, \p being the inclination of the normal to the axis uf symmetry. Thus our condition becomes sin*0sin;i 1 + — a -.- , r on -^ r ^ Y- (22). pr on If we can find a solution of (22) corresponding to any value of k, then the geodesic >-sin0 = A' will be a possible free path. It is of course obvious tiiat for given k there are an infinite number of equal geodesies on the surface, obtained by imagining one of them to rotate through any angle about the axis of symmetry. For ^ = 0 the problem reduces to tlie two-dimensional one already considered, the meridians being the corresponding geodesies. as jjossible free paf//s. 1 67 8. Suppose that we can ^et two sets of geodesies on llie same surface, botli bein^ possiltle i'vee pallis, tlie cori-espondiiig constants being k^ and k^. We get tlie two conditions 1 + ^•, (P H\nxp — r) a 1 f> Pr' hi' 1 + K (p sim// — r) a 1 p P>' ^a' Ignoring tlie trivial case p sini^ = ?•, wliicli gives us concentric spheres, on whlcli all great cirtdes are clearly geodesies and possible free pallis, we get tlie two conditions p a hn ; The latter becomes r — p sin;// 1 ^^ ^ ^"5 pr on r — p sin^ or r' (23), the geometric meaning of v^^liich is that the distance from the axis of symmetry of the centre of curvature at any point varies as the cube of the distance of this point from the axis of symmetry. 'i'lius we conclude that for surfaces of revolution, if more than one set of geodesies are described as free paths, the meridians must satisfy the condition for possible free paths in two dimensions, and must also have tlie property indicated by the condition (23). If this is the case, then it follows that all geodesies are freely describable. It is of coui'se clear that concentric spheres are a family of such surfaces. It is also not difficult to show that on a surface on which all the geodesies are free paths, the cori"esponding velocity either continually increases or continually decreases, as the geodesies get more and more inclined to the meridians. 9. If the conditions of the last article are not satisfied, there is still the possibility of one set of geodesies being free paths, if Jc can be found so that the condition (22) is satisfied. ( 168 ) CRITERIA FOR EXACT DERIVATIVES. By T. W. Chaundy, Christ Church, Oxford. Prof. Elliott lias recently* exhibited a set of criteria of exact derivatives, wliicli differ from the classical system of criteria investigated by Euler, Bertrand, and others. The present paper aims to show how these criteria of Prof. Elliott ina)' be connected vvith the older criteria, and how certain other sets of criteria may be established and similarly con- nected. Reference is made to Prof. Elliott's paper, named above, under the letter E2 and to an earlier paper on the subject by the same authort under the letter Ei. It should be added that I have not contemplated the presence of more than one dependent variable _y, although the results, 1 believe, admit, in general, of extension to the case of many dependent variables. § 1. A convenient notation is the following: Define 0"^=^ pD-^ V- — ^r-i — Dr. ... to uifimty, the numerical coefficients being those of (l +a;j~^. Jn particular 0 " = 77 pD TT- + '-^^. — ' D — ... to mfinity, and (9/ = ^ — . Such operators obey the two fundamental identities and o;i)=o;:; | '''• In this notation the classical criterion that F be an y"" derivative is its annihilation by all of the set 0,°, 0^^ ..., OP*. In addition, I write 0 for yO^~\ and 9, 3^ for — , ^ — . ^y oy,, * Mesaeiif/er of Mathematics, vol. xlv. (1915). f Luc. cii., vol. iliii. (1913). .(2). Ml'. Chaiuuhj, Criteria for exact derivatives. 169 § 2. Introduce the set of operators o), = ?/a, + 2^,a,+ 3^,83+... I &c. &c. / the numerical coefficients in w^^ being those in the expansion of (1 — x)"'"'*''^ The operator w, is that called co by Prof. Elliott in E2 : co^ is of course that due to Euler for homo- geneous functions. These operators are, in point of fact, the operators a a a , , d^'z a^'a^'a^'-' ^i'"'e - ^ lo^i/, -. ^ a^. • Thev are commutable and obey the identities w^D — Do}^= (w^_, In addition we have the results 0, = ft»^-2Z)a,, + 3Z>'co3-... O3 = a>, - %Dw^ + GD'oj^ -. . . &c. &c. / (These latter identities represent, of coarse, merely the trans- formation of the operators 0]^ ' from variables y^ 3/,, y,, ..., to variables z, 2:,, 2;,, ...). Now the annihilator of r^^ derivatives given by Prof. Elliott in his recent paper [E2) is the operator (r, w] = [DoD — ri] [D(0—[r-\- 1) i] ... {Day — wi)^ where iv is the greatest weight of the function In ?/,, y^, y^, .... From the identities (3) we have the set 0= co^-Dco^ + D^i 30^-^2DO, + n'0, = 3(o,-noy^ + D'{ &c. . &c. Hence (2, 10) 0, = (2, w) (w, - Bco^), since (2, iv) 1)' = 0, (3). 170 Mr. Chaumhj^ Criteria for exact derivatives. [4), and since the operators on the rij^Iit are of zpro weight. But on a function homogeneous of degree i, w^ = i. Thus (2, to) 0, =-(2, w) {Day- 0 = " (^ H SO (3, iv) (2(9. + i>0,) =-(3, w) [Du>-21) =- (2, w), &c. &c. And SO on: finally {10,10} {[ic-1) 0^+ [to -2 j D0.^-{-...\ = - {w - 1, w) wO^+ {to -1)1)0,+... =wi-D(o = - (Z(7, W). This gives {-Y{\,w) = [wO, + {w-l)DO^_+...] > \{w-l)0,+ {xo-2)DO,^...]...0^ {-Y-\2^io)=[wO^ + [w-\)DO,-V...]...[20^>cDO,) &c. &c. expressing the operators (?•, ic) in terms of the operators 0^, and showing that a function ainiihilated by all ot (9,, 0.^, ..-, 0^ is annihilated by all of (1, iv), (2, lo), ... (r, %o). §8. We miy farther deduce from the equations (3) the identities 0,= i- D (a>, - Dcv,-\-D'(c- ...) 0^ + DO=l-I)'(a},-2D(o^+3D'oy^-...) [ (5) &c. &c. The function is supposed homogeneous of degree i. From these identities we see that anniliilation bv the one operator P^= 0^ + DO^^- D'0^+...D-' 0,.\^ (if the function be homogeneous of degree other than zero) sufficient to prove it an /'' derivative D'<p, and that /^ can be expressed as The fact that annihilation by P. necessitates annihilation by P , P P can be exhibited by means of the identities i-P = P,P,...P l--^P=P,...P^ &c. &c. :6). Mr. Chaundy, Criteria for exact derivatives. 171 §4. Now we have expressed the aniiiliilators (1, xo)^ (2, i6] ... (r, w) in terms of the annihihitors 0,, 0,, ... 0,, and shown that the latter necessitate the former. Conversely, althouf^h it does not seem possible to express 0^ in terms of (1, 1(7), (2, w) ... (?•, ?<;), we can show that annihilation by (r, ?y) necessitates annihilation by 0^, ..., 0 . For this we employ the identity of El (§§5, 6), namely, that 1 =^„ll, ?.t') + ^,Z>a)(2, i:;)+J^i)V(3, w) ^ (7), where A^^ A^., A^ ... are numerical quantities: precisely ^ =(-l)-T«'-i — ^ -, . '^ ^ ^ r\ [w-r)\ It may be first mentioned that the set of identities 0^(0 — (1)0^= r i 0^. J holds, whence o>0^,= 0^, , [Dco — ri) ; in addition we see that (9., o.co, oy, ..., oy form a set of annihilators equivalent to 0^, 0.^, ..., 0^. Operating on both sides of the identity (7) with 0^ we have, since O^D = 0, 0=A„0,{l,io). Operating with 0.^ we have, since 0^D= 0^ and O^D^=0^ 0=A^0^{l,iv) + A^0^co[2, iv) = {A^OJ,Bco-l)+A,OMi^,^o) =={A^<oO^ + A^O,co){2,w). Similarly 0,- [A^ 0^ + A^ooO^co + A^Oy) (3, iv), and so forth. It is clear then that annihilation by (r, w) necessitates annihilation by 0^, 0^, ..., 0^. §5. I pass now to the case in which the functions dealt with are isobaric in l/^^ y^, ..., y^. Since, when x occurs explicitly, d_ dx ^ = l; + 3/iS+i/A+.-., I.e. is not isobaric, a function and its derivative will not, witli rare exceptions, be isobaric, and we therefore stipulate, for 172 Mr. Chaundy, Criteria for exact deiHdatims. . isobaric functions, tliat x be not present explicitly, and there- fore accurately In this case we can introduce a new set of annihilators E^, defined as follows : — 1 + ^^ = D-^ (y, o: + y.fi:^ + 3/3 or +•••)• To prove them to be annihilators we proceed thus Thus ED = DE^^. r r— I In particular so that Ep==0 and E^D'' = D'-' E^D = 0. Thus E^ is an annihilator of r"" derivatives. § 6. To prove the converse we observe in the first place that a function E annihilated by E^ satisfies tiie equation and thus is certainly an (r— 1)"* derivative. It is therefore annihilated by all of J5',._,, E^ ^, ..., E^. We may exhibit this fact, that annihilation by E^ necessitates annihilation by all the E^s of lower suffix, by means of identities of the type E . E = E . r r+t r It is to be noted that we have also proved that, for a function annihilated by E^, (>• — !) integrations can be performed by direct differential operation, and this without knowledge that the function is either isobaric or homogeneous. Now, employing the identities (1) of § 1, we have ^i!hO: + y.^O:' + y^O:^+...) il/r, ChauacJy, Criteria for exact deriva,tives. 173 Hence E^-Vl=E^_^ + l- D'-'y^ 0';\ i.e. E^_^-E^ = D-'yfi';' and in particular DE^ = — y^0° Thus the ^'s are connected with the O's, and if E^F=0, since we know that E^._^F must also be zero, it follows that If X is not to occur explicitly, this must lead in all but a few exceptional cases to 0^' F=b, which proves ^an r'*" derivative. We see then that functions annihilated by E^ are r'^ derivatives. The properties of tlie particular operator E^ iiave been previously described by Prof. Elliott {El, part I). § 7. There Is finally a svstem of annihilators analogous to the system discussed by Prof. Elliott in E2, and mentioned above under the symbol (r, ?<;). We introduce these as follows: Define the set of operators (analogous to the set wj V, = y.d,^Sy,d,+ &y,d^ + . &c. &c. / Writing r] for t/j, we have rjD — Drj = t]^', but rj^ operating on a function isobaric of weight w multiplies that function by w. By the symbol [O, w] mean the operator where the r'^ factor from tlie beginning Is Bt] — (^r — 1) {w — ^r) and there are iv factors In all. Mean by [l, tv'] tlie operator obtained from the foregoing by removing the first factor, i.e. it starts with Dr] — [io- 1): mean by [2, lo] the operator obtained by removing the first two factors of [0, w], and so forth. Then we shall prove that [0, iv] annihilates all Integral algebraic functions isobaric of weight w] [l, to] annihilates all integral algebraic first derivatives of weight w] and generally [?•, iv\ annihilates all Integral algebraic /*" deriva- tives of weiirht w. .{8). 174 ^F^•. Chaunchj^ Criteria for exact derivatives. To prove this observe that [O, xo\ may be written 10 [\o— 1)] jdUd- 2 j [vD- [210-3]} {r)D-{lv-l)]V' But if F is of weight w, tjF is of weight (?« — l) : the factors r}D — [io — \)^ T]D—[27v—3), &c., are of zero weight. Thus in the above operator we may write rjD = Drj + [to— l). This reduces it to j{j,-"''-'y"'-^'}...lJ,-C.>-2)|J,.,. In other words since vF'^ is of weight lo— 1, [0,iv-l]r]F^=B[0,io-2]'n''K- Proceeding in this way we see that [0,tv]F=D'-'[0,l]7)-^F^,. But, the weight of rf"~'^ F^ being unity, it can involve only y and ?/,, and is thus annihilated by [O, l], which is Dq. Again [l, to'] D may be written Now if F(^=D<f) is of weight lo, ^ is of weight w — V so that [1, ro-]D^ = D[Dn] \Dn-{io-2)]...^DrjJ''-^\^'"-"^]^ ^ = D[0,w-\](p = 0, by the preceding result. In like manner we may prove that and proceed similarly to show that [>•, lo] annihilates r"" derivatives. §8. To prove the converse, that functions annihilated by [r, to] are 7-"' derivatives, we establish the identity \=A^[\,io]^A^Dr^[2,xo]^A^Dr^[Dn-{w-\)][3,w] + ...{<^l Mr. Chauiuhj, Criteria for exact derivatices. 175 where ji^, A^^ A^, ... are the numerical quantities which secure the partial-traction identity A -4, A^ X x—^w—l) x—{2io — ^) 1 f w{io-\)} x\x- {lO-l)]...^^ ^ 1 . Writing the identity (9) in tiie t'orni we see that a function F annihihvted by [l, lo] is a first derivative and that its integral is \A^r^[2, W-] ^ A^D^i'l?,, w-]+...\F. Since [2, iv] is a factor of [l, w\ annihihition by [2, to] of a function F shows that i^ is a second derivative Z>'^, and allows us to write down ^ by direct differential operation only. So generally for [?*, loj. §9. It remains to connect this set of annihilators with the foregoing systems. To do this we need the operators 17^ defined above and the following identities that may be obtained, expressing the operators E^ in terms of these : E= - 1 + Z>7;, - ID'h, + ^D\ - - E= -I + Z>'77,-3Z)X+- ^■•(lO)• E= -\ -H D\-.. &c. &c. From these we obtain the set E = {v-i)-D[rj^- Dv,+ D\-...]^ E^+E = (v,-2)-n'{v,-2Drj,+ 3D-'v,-.-A E^ + E,+ E = {v-^)-D'\v-^Dv,+ QD\--] &c. &c. Since a function annihilated by E^ is annihilated by all^ of E^, ^,, ..., E^_^, we have here an additional proof of the fact that a function annihilated by E^ (if it is isobaric of weight 17G Mr. Chaundy, Criteria for exact derivatives. other than r) is an r"' derivative 2)''^ : moreover we see that we have an expression for (lo — r) ^, namely „ rir+l) _, 'r Ir-H ' o f 'r+i § 10. Another set of identities that can be deduced from the set (10) is sE^ + 2E^-^E^ = {3v,-&)-nn + n\...) &c. &c. Since [r, ro] D'' = 0, Tve have [2, to] E^ = [2, to] [{lo — l)- Dr)\ = - [l, to\ so [3, io](2^, + i^;) = -[2, t4 and finally [tf-1, tv] [{lo -2) E^ + {to - ^) E^+...] =^-[w -2, w] \{xo-l)E^ + {w-2)E^+...]=-[to-\,to]. Hence [Uto] = {-r-^[{tv-l)E^+...]\{to-2)E^^...]...\2E^ + E:^E, [2,t.]^(-rM(to-l)^.+...!((t.-2)i^.+...|...{2i^, + ^,} &c. &c. Thus the operators [r, w] are expressed in terms of the operators E^^ and it is clear that anniiiilation by E^ necessitates armihilation by [r, to]. To pi'ove the converse we employ the identity (9) and proceed exactly as in the case of the annihilators 0^ and o)^. §11. It may be remarked that it follows from the fore- going results, in conjunction with previously known facts, that a function (separable into isobaric and homogeneous parts), not involving x explicitly and known to be an ?•"' derivative, can always have its r integrations performed by direct differential operation, except in the case when, with one differentiation left to be performed, the function is of unit weight and zero degree — -that is, of course, the case covering the possibility of the original function being logarithmic. ( 1- ) AN ARITHMETICAL PROOF OF A CLASS RELATION FORMULA. By L. J. 3Iordell, Birkbeck College, London. Let F [m) be the numbei" of uneven classes of negative determinant — m, with the convention that the class (1, 0, 1) and its derived classes are each reckoned as ^, and that F[0) = 0. It is well known that F{in) - 2F{m - \') + 2F {^m - 2')-... =-S(-l)^("^'V...(^), where the left-hand side is continued so long as the argument of the function F is not negative; and the right-hand sum- mation refers to all the divisors d of ih, which are <\J[vi) and of the same parity as their conjugate divisors a, but when d = \/(7n), the coefficient d in the sum is replaced by ^d. Kronecker* proved this formula and sinnlar ones by considering in the theory of elliptic functions the modules which admit of complex multiplication. Hermite* shewed that formulae of this kind could be proved by expanding in different ways functions represented by products and cpiotients of theta functions, although when a formula is given it is no easy matter to see a priori what is the function to be ex- panded. This method was not unknown to Kionecker.* Liouville* showed that the general formulae introduced by him in the Theory of Numbers could be applied to give an arithmetical proof of some fornudte of this kind. Ki'oneckert also gave an arithmetical proof depending upon the general theory of bilinear forms with four variables. By considering the subject from rather a different point of view, I was enabled to find directly various formulae of this kind, some of which are given in my "Note on Class Re- lation Foi mulse.:]: But some of niy analytical methods suggest a very easy arithmetical transformation, and as an illustration, I prove the above formula. Let m be any given positive integer (all the letters used denote Integers), and consider the representations of m by the two forms * An account of these methods will be found in H. J. S. Smith, Report on Theory of Numbers, § 6, Collected Works, vol. i., pp. 022-350. t Collected Works, vol. ii., p. 427. Ueber Bilinenre Formen mil vier variaheln. X Messenger of Mathtmatics, 1915, vol. xlv., pp. 7(i-80. Mr. G. Humbert writes to me that the formula [A) in this paper has been given by him in his paper, " Nombre de classes dea formes quadratiques/' in Liou villa, 1 1)07; and that it is due to K. Petr, Acad, des Sciences de Boheme, 1900-11)01. Both of these authors have found this formula and various others, some involving the representation of numbers by simple indefinite forms, by means of Hermite's classical method. VOL. XLV. N 178 Mr. ^fordell, An arithyietical })roof s* + ?i' + H (2i+ 1) -r'' =-m (1), d[d+2h)=m (2), in wlilcli s takes all values, positive, negative, and zero; n all positive values, zero excluded; r all positive, negative, and zero values from — (" — Ij to », l)otii included; and t all positive values, zero included, d and S are positive, but S may also take the value zero. Let /(x) be any even function (either an analytic function or an arithmetic function) of x, so th at /(x) =/(-«)). Then ^[-lYf[r + s)=-2^{-\ydf[d) (Z?), where the summation on the left extends to all solutions of equation (1), and the summation on the right to all solutions of equation (2j ; but when S = 0, the coefficient 2 in the sum is replaced by unit}'. For putting r + s = ±h, and supposing h a given positive integer, the coefficient of/ (7^) on the left-hand side is S(— l)"" extended to all solutions of F±2/.T + n'+«(2^+l) = m (3), where t, n and r are limited as in equation (1). But this coefficient is equal to 22 (— 1)"'", extended to all solutions of P + n- -{- n^ -]- 2ka = m (4), where n is as before, | takes all positive and negative odd values from — (2/;— 1) to 2k— 1, both included, <t all positive values, zero included, but when o- = 0, 2 (- 1)"^" is replaced The proof of this depends upon the theorem'* that it n and k are given positive integers, 2(— l)*" extended to all so- lutions of nl2t+l)±2kr = N (5), in which ?• and t are limited as above, is equal to 2S (- 1)"*^', extended to all solutions of n^-\-2ka = N (6), in which ^, a are limited as above, and with the above con- vention when cr = 0. Putting ^ = 2?; + 1 and iV— ?« = 2P, where we niay suppose P is an integer, otherwise both equa- tions have no solution, and the theorem is certainly true, this is the same as 2 (— 1 / extended to all solutions of nt±kr = P (7), -(n-l) oo * It was suggested by evaluating 1. E 7"(-'*')i-*^ of a class relation formula. 179 IS equal to 2S (—1)"^'' extended to all solutions of 7}r] + ka = P. (8), where 7; = 0, ±1, ..., ± (Z; - 1), - /.-, and ct, t, r are limited as above. Putting -r for r in tlie equation with the negative sign in^ (7), this is the same as 2(-l)'' extended to all solutions of nt + kr = F where r takes the values —n to « — 1 and then again the values n — 1 to —n plus (—1)'' for the solution r = n [if r = n does not give a positive integer value for <, (—1)*^ must be replaced by zero, and similarly in other cases] minus (-Ij*" for the solution r = -n equals 22 (- 1)"" extended to all solutions of ^jt; + /ca= P where rj=0, ±1, ..., ± (/c — 1), — /; minus (-1)"^" for the solution o- = 0, where now the con- vention for cr = 0 is removed. But (—1)'' for r = n minus (-l)"" for r = -n is equal to — (-l)"'" for o-=0. For we may consider P/n to be an Integer, as otherwise none of these solutions exist. If k> Pin > — k, there is a solution a- = 0, but the case r = n does not arise while the case r = — n does and the equality is clear. If Pjn lies outside these limits, the cases r = n and r = — n both arise and (- l)" cancels (—1) ", while the solution a = 0 does not arise, so that again the equality is evident. Putting now r — ti for r and i] — n for ■»/, we have to show 2 (- l)"" for all solutions of nt^ lcr=Q (9), r = 0, 1, 2, ..,, 2n - 1 equals 2 (- l)" for the solutions of 7ir]^k<T=Q (10), 1^ = 0, 1, 2, ..., 2A,-1. But we can establish a unique correspondence between the solutions of equations (9) and (10). Thus if (10) admits of a solution [rj, o), we may suppose T)<k, for otherwise ■»; — i', a + n is such a solution. Hence we can arrange its solutions as follows [t], a] with 0 <a<n, and in pairs such as [t), a) and [rj + k, (T — n) with cr>n. If a <n we can take r = a and t = r,. If o->?j, we write r = a + e)i, t = r) — ek, and there are two consecutive values of e for which we can make 0<r<2n and these make ^>0. ISince for the pairs of solution of (10) for which (if a> l) {0<r]<k, an<(r<{a + -i) nj-or [k <v <-2k, [a-l] n <<t <an] we find {0<r<n, ak<t< (a + 1) /cj or [n<r<2n^ {a-l) k<t<alc]y 180 Dr. Wilton, On the zeros of Eiemami's ^-function. the correspondence is obviously unique. But (_ 1)<^ + (_ 1 )-"= (_ 1)-- + (- 1 )-(^^^)" which proves the statement for equations (9) and (10); so that we can replace equation (3) by equation (4). Hence S (—1)'' extended to all solutions of equation (3) is equal to 2S [— If^" extended to all solutions of for which n takes all positive and nej^ative values, zero ex- cluded ; I takes the values 1, 3, ..., (2Z;-l), (7 = 0, 1, 2, ... Avitli the convention for cr = 0. Noticing now that we can group the solutions in pairs, such as ?j, ^, a and ^ - n, ^, o", if % is not equal to ?«, then since ^ is odd the sum 22 (- l)'"^" is zero for this pair of solutions, and we need only consider those solutions for which n _ sr But then k [k -f 2a) = in, and the sum reduces to 2 (— l)^"^, which being summed for |=1, 3, ..., [2k - I) gives — 2^(— Ij"^. This proves the result {B). Taking now /(«)=(- lj% we have 2 (— 1)', extended to all solutions of equation (1) is equal to —2I,{- l/^'^el extended to all solutions of equation (2). But when s is given it can be shown that the number of solutions of (1) is 2F {in —s'). This 1 have done in my forthcoming paper, " On Class Relation Formula." It can also be proved very simply by means of the modular division of the plane, a method due to Humbert [l. c.) Hence S (- 1 )• F{m -s') = -^{- 1)*"'W, or F{m)-2F{m - l') + 2F{m-2:')...^-^ ( - l^t^'^W. NOTE ON THE ZEROS OF RIEMANN'S ^FUNCTION. By J. R. Wilton, M.A., D.Sc. The following slight extension of Mr. Hardy's result that ^ (s) has an infinite number of roots on the line <t = ^ may not be without interest. It is here shown that both the real and the imaginary parts of T (|s) tt"'' ^ (s) have an infinite number of roots on any line a = a^, such that 0<o-g<l.^ 'J'he method followed, except in the actual determination of the value of the definite integral which leads to the result, is the same as that adopted by Mr. Hardy.* * Comptes Rendiis, April, 1914, pp. 1012-4. Dr. Wilton, On the zeros of Riemamis ^-function. 181 Kieinann's integral for ^ (s) is ' s[s-l) J, „=1 /* In this put and fji = e' ; we obtain* = i— 2 COS 2; A,, e'^ ^ e a A. 1+^" J, «=1 = cos z\ .f {X) dX (1), J 0 «=i provided that — !<?/<! (2), ■which is equivalent to 0 < cr < 1. Further, from Jacobi's relation 1 + 22 e-^»'> = /i* (1 + 2 2 e-^»V), »i=l w=l we obtain, on putting /j, = e*^^ the relation y"(\) =/(— '^)5 so that of tlie two functions 6'(A,) = cosi);/?i./(Vi, ;j^(A.) = sinhj/A-./(X) (3), 6 is even and ^ is odd, and on account of (2) both vanish together witii all tlieir differential coefficients at infinity. Also d^'"-'\Oj = 0 and ;^'"'^0] =0. From (1) and (3) we have, by successive integration by parts, 2x'"^ (x, y) = a;'" cosxX . 6 (X) dX, = (-)"[ cosxX.6^''"\X)dX, 0=f smxX.6<'"^(X)dX; * In Riemann's notation £ W = HI + 4t=) Fi2t) = (" cos ^\t .( j5J,) - 1 {^"J^^"'"' '*"} ^'^ by integration by parts. And as in (1) the subject of integration is an even function of X. N2 182 Dr. Wilton, On the zeros of lUeynann s ^-function. Hence J_^ a, v / 2a;"'".^ (x, tj) cos (/3 - i(x) X = (-)" f cos [x -0 + ia) X.e^'"\\) dX = (-)" f COS {x-/3)X. a"") (X. - ia) dX, J -co by an evident contour integration, provided that <a <-. 8 8 .(4). ..(5). Similarly, under the same restriction as regards a, 2x'"xP (a;, 7/) sin (/3-ia) .-c^ -)"'• j" cos {x-/3)X . x^'"^ (X-ia.) dX. And we have immediately, by Fourier's theorem, - f x"'<p(x,7j)cos{[5-ia)xdx=:(-ye^'"^([5-ia) ' - I x"\p {x, y) sin (/3 - ta) x dx = (-)'"' x^'"'\^ - ta) Mr. Hardy's equation (3), p. 1013, is obtained by putting y = (),i3 = 0. It is easy to verify that, on account of (2), 6 (X) steadily decreases as X increases from 0 to cc . And it readily follows that (-)"^("')(0) is positive, while ^f""(/3) vanishes for n values of IS between 0 and oo . Further the radius of con- vergence of the power series for 6 (X) is clearly tt/S; hence on account of (4) we may expand (— )"0''"^(fa) in powers qf a and every term will be positive, and therefore x'" (p (x, y) cosh (XX dx J n Is essentially positive so long as the inequality (4) is satisfied. In the particular case when /3 = 0 and yj = 0, equations (5) become I ^ (S^i y) ^'^^^ ^^ ^^ —f{^^) cosya J 0 = 2 cosa cos^a - e'" cos_j/a [l + 2 E e-^«'(cos-ia+»sin4a)j^ Prof. Burnside, Determinants of repeated arrays. 183 2 r" . — yp (x, y) sinli ax dx = —f(ia) sin ya. = — 2cosasIn?/a 4 e'"sin?/a(l + 2 S e-^«'(co3 4a+i8in4a)j. Since and arc both of constant sign we see, on making a->7r/8, and following Mr. Hardy's argument precisely, that tor any value oi' y in the strip (2) of the 2;-plane both (j>(x,y) = 0 and ;/. (.r, 3/) = 0 have an infinite number of real roots when regarded as equations to determine x. The UnirerBity. Slieffield. DETERMINANTS OF CYCLLCALLY REPEATED ARRAYS. By Prof. W. Burnside. In a recent paper* Sir Thomas Muir has shown that when the number, », of arrays is equal to the number of lines (or columns) in each array, the determinant can be expressed as the product of n determinants, in whose elements the oiiginal elements enter linearly. The residt is, in fact, true without limitation; and may be proved by an obvious extension of the method which exhibits a circulant as the product of its linear factors. Let D denote the circulant of the n ?n-line arrays a;„ a;^, ..., o;„, <, a;, ..., a;^, r=l, 2, ..., n. Messenger of Mathematics, vol, xIt., p. 142. 184 Frof, Buniside, Determinants of repeated arrays. Take lo an assigned primitive n^^ root of unity, and put SO that nal-^io^'-'^^'-'^A'... Tlie elements of tlie [(p — 1) m + ^]"' row of D are a'',, a'',, .... a^ , a^^', .... a''^', c/.^", .... (f'\ where tlie a.'s are to be replaced by the A.'s by the preceding formula. The terms containing the upper suffix s that occur are - wj<'-'K'-^> X n (^' ^' .... ^' , lo'A', iv'A' , .... w'A' , tf;M'„ ..., i(;"M' ) V ^l" q2' ? gm^ 9P 52" ' gm^ q\' ' gni' The terms that contain the upper suffix s occurring in the [(p'— 1) m 4-^]"' row differ from the above only by the out- side factor. Hence it follows that D can be expressed as the sum of a number of determinants of nm rows and columns each of which is a numerical multiple of the determinant wiiose [fx - 1) w + ?/]"' row is A\, A', ..., A' , vfA\, .... vfA' , w'''A\, ..., xtT'^A' ; y\< 1/2' ' yrn^ yV 7 ym> yV ' ym ' 30 that D itself is a numerical multiple of this determinant. E-eplace the [(n - 1) w -f ^]''' column of this determinant by t"* column + to''' (m + lY^ column + lo'*'' [2m + /)"" colunm +...-\-io''[[n — 1) m + ^P column for each i from 1 to ?7i ; thereby affecting the determinant only by a numerical factor. The elements of the last m columns will then all be zeros, except those that belong to the rows from the [(a; — l) m + l]'** to the xm^^\ and these will be «^" «4" .... nA' , nA"",. nA' , .... nAl , 21' -2' ' 2ml jiJ'',, nA'„, .... i^A"" . ml/ mZi 1 mm Lt.-Col. Cunningham, Factorisation of N = (a.-^ ^P /). 185 Hence, apart from numerical factors, D is divisible by A' A' A' A' A" for each x. No two of these determinants can have common factors for arbitrary values of the rt.'s, since when each of the n «i-line arrays is itself a circulant the nm linear factors of the determinants are known to be all distinct. Hence a: i) = N.n x=l A" A"" ^11' ^12? •••) Al, Al, ..., A where N is numericah It is easy to show by comparing coefficients that N= ± 1 according to the order in which the rows of D are written. FACTORISATION OF N=(x'rf). By Lt.-Col. Allan Cunningham, R.E., Fellow of King's College, London. [The Author is indebted to Mr. H. J. Woodall for help in reading the proof sheets.] 1. Introduction. The numbers, whose factorisation is considered in this Paper, are of form X=zxv-y'', N' = x''+y' (1). It will be supposed throughout that a; and y>] ; and x prime to^* (la). These numbers rise so rapidly as x, y increase that complete factorisation (into prime factors) is possible only for a very small range of x, y ; in fact-=— 2":p2o-, 4":f13*, 8»?:9», 16»:p3'«, 5":?:1P, lO'rf:?'", &c. are beyond the powers of the present large Factor-Tables. 186 Lt.-Col. Cunningham, Factorisation of N={x'" ^ y"). 2. Algebraic Factors. When x, y liave certain forms, then N or N^ is algebraically resolvable into two or more factors. 'J'liese cases are — i. Difference of Squares. ii. Binomial Factors. iii. Aurifeuillians. 3. Difference of Squarea. N is algebraically resolvable at sight into a continued product of (a + 1) factors, when one of a;, y is of form (2Aj^, where e = 2". Ex. A- = 2» gives ]S^^\y -y*={'ly -y'')\\(2y Jry*) (2a), x^l*' gives N=l&y -y^' = ('iy -y*)\(2y +y^)\\{2"-y ^yi) (26), These are the only cases worth recording: as, when either h>\, or a>2, the numbers iV ai-e too high* to admit of complete factorisation. A number of examples of these two cases completely factorised are given in the Table at the end of this Paper. The first case iV"=(4'' ~ ?/*) has the peculiarity that its two algebraic co-factors (2"-^), (2^+v/') are themselves of the form iV, N^ of (1). This is the only case possessing this property. 4. Binomial Factors. Each of N., N^ contains an obvious algebraic binomial factor when one of x, y is of form (»//)" with n odd and > 1. jEx. Take ;c=2", j=3», [n = 3, A = 1J; then N or i\r' = (2«)«'q:272" = (29")' + (32")3, which contain the obvious factor (2^"+3^") (3). This is the only form worth record: as when either A> 1, or n> 3, the numbers N, N^ are too highf to admit of complete factorisation. Examples of these forms with x = '2, 4, 16, y = 21 will be found in the Table at the end. 5. Aurifeuillians. These may be of three different orders (?i), which must be separately considered. i. H = 2. ii, 7j = w (odd). iii, n = 2u) (twice an odd number). ..(4). 6. Bin- Aurifeuillians. These ai'e numbers of form N' = 4X«+r« (5). which are algebraically resolvable into two co-factors (say L, M), viz. N' = L.M=(Y*-2XY+2X'')(Y^ + 2XY+2X-) (5a), * The smallest of these is N- (36' - 6'«). t The smallest of these is (S^'rp 27*). Lt.-Col. Cunningham, Factorisation of N ^ [x^ T y'). 187 The conditions foi- cc, y that N^ — x^ + y'' may be expressible in tlie above form (5j are x = 4A*, ^ = 2i) + l (an of^fiJ number) (6). whereby = 4(2"/(!')H(yM*; [X = 2''/(y, r=/ ] (6a). Ex. a; = 4, >'=2r,+ l (odd), A=l. N'=4!'+y=4.2<"+y = /..i)/=(j--2''+'j/ + 22''+»).(j/' + 2'''-'j' + 2^"+') (66). This is the only Bin-Aurifeuillian form of iV^ worth detailing: as when /^>l, the numbers N^ are too high* for complete factorisation. A number of examples (with a; = 4, y = 3 to 27) completely factorised are given in the Table at the end. Qa. Bin-AiiriJ'eniUians as factors of N. When x = 2*h\ ?/ = 2j7 + 1 (odd) then JV"=A-2'-j'*=(2*As)!'-yl6/i» = (2yh"'J-y*''^)\{2yli-y + y*''^)\\{2''yh''y + y^'*) (7). Here the largest factor (say Z) is Z=22J'A<J'+j.8a' = 2«"+2A<J'+_>'8a' = 4X*+ r< (7a). which is a Bin-Aurifeuillian (see 5), and therefore resolvable as in (5a). Ex. A-=16, >' = 2)) + l (odd), /j = l. then Z = 2*i'+y = 4.2*'' + (_>--)♦ = i.M = (/-2''^'/ + 2'^"^>)(y + 2''+y + 2'''+') (76). This is the only case of N possessing a Bin-Aurifeuillian as an algebraic factor, which is worth detailing here: as when h> 1, tiie numbers .Z" are too bighf for complete factorisation. A number of examples (with a;=16, .y = 3 to 27) completely factorised are given in the Table at the end. 7. Aurifeuillians of odd order [n). These are numbers of form N =(X"-F")-^(A'-r), with w=4i + l (8a), N' = (X''+r'')-f-(X+F), with « = 4t + 3 (86), along with the condition nXY = n (9). * The smallest of these is iV' = (645+B") given bj li = 2. t The smallest of these is Z = 2" + 3'*". 188 Lt.-Col. Cunningham^ Factorisation of N= {x^ ^ if). Eacli of these is alj^ebraically expressible as a difference of squares, and therefore resolvable into two co-factors (say X, M). The simplest forms of X, F satisfying the condition (9) are X = H\ Y=nK' (9rt]. By taking a;, y of forms x={2h+\f, odd; y = n»k"-», [;i odd] (96). the numbers N, N^ = x^'^y^ can be expressed in the forms X"^Y" along with the condition nXY=n, and will then be divisible by [X^Y). The co-factors will be the Auri- feuillians N, N\ It is not worth while developing this further, as the smallest numbers of this kind are too high for complete factorisation. The smallest example of each kind is shown below. 1°. i\" = 25" + 27^^=5" + 3'S which contains (5'8 + 3"), and the co-factor N' is seen to be a Trin-Auiifeuillian ^''=Sf^=(5"-3.3'-.o«+3")(5'« + 3.3'2.59 + 3==). O'^ + ii" 2°. i\r= 93125 -31 259= S^^'^-S^s^ which contains (S'-^o-S^), and the co-factor N is seen to be a Quint- Aiir if extillian, Q62I0 _ C4( N=-^,— -^=(3-"° + 3.3'«».5» + 5'•)=-(5^36»)^(3'«• + 5»i^ 3 ■* — 0 8. Aurifeuillians of even order [n = '2(a). These are numbers of form W = {X'^''' + Y"^"') -{x- ^y"-), with n' odd (10), and with the condition 1n'XY= D ^11). These are algebraically expressible as a difference of squares, and are therefore resolvable into two co-factors (say L, M). The simplest forms of X, Y satisfying the condition (11) are X=H\ Y=2ii'K'' (11a). By taking x, y of forms X^ylh+Xf odd; y = (2«')-»'.>&*'" [ix' odd^ (116), the numbers W=x}'-\-y'' can be expressed in the form (Z'"'+r'"') along with the condition InXY^-u, and will then be divisible by {X'^-\-Y'^). The co-factor will be the Aurifeuillian N\ It is not worth while developing this further as the smallest numbers of this kind are too high for complete factorisation. Lt.-Col. Cunningham^ Factorisation of N={x^ ^ if). 189 The smallest is iV' = 25""« + 46656« = 5«-"'« + G«'«, which contains (5-'"*+ 6--") and the co-factor N"_is seen to be a Sext- Aurtfeuillinn. 56.-T76 , (56 25 N' = „ ■■,, „ „ = (5""^° + 3.5""'.6'^ + 6---')^ - 6.6".o"'«(o""'' + 6")'. 9. fse of Numerical Canons. 'Y\\q factorisation of large nuuibers N. N^> 10', wherein the elements x, y are powers of 2, 3, 5, 7, 10, 11, has been rendered possible by the help of certain Numerical Canons (Binary, Ternary, &c.) which have been compiled* by tlie author. These give the Residues, both -f- and — of the powers (n), of the above bases (2, 3, &c.) up to the limits named below, Residues of 2"; 3", 5'', 7", 10», 11", Limit of n 100 ; 30, after division by every prime (p) and prime-power p*":^ 10000. 10. Perfect Squares and Poioers. No perfect squares or powers have as yet been found among these numbers N, N^: 80 that it would seem probable that none exist. 11. Dimorphism. No case is known of any number being expressible in two ways in the same form N or N: and only one case is as yet known of a number being expressible in both forms N, N\ viz. 17 = 3«-43 = 32 + 25. If, however, the value ?/ = 1 be admitted — hitherto ex- cluded, see (la) — every number Nov N^ would be expressible in two ways in the same form, and every number whatever (say Z) would be expressible in both forms, for iV=(7V+l)'-l^'+l and i\^' = (iV'-l)' + l-^~^ z=(z+l)'-l^+^=(z-l)'+l^~^ 12. Factorisation Tables. Here follow two Tables giving the factorisation of the numbers iV, iV' in separate Tables. 1°. Arrangement of Facton. Each number N, N' is shown resolved first into its Algebraic Prime Factors (A.P.F.) and Aurifeuillian Factor* (L, M) : these are arranged in order of magnitude, the smallest on the left, the highest on the right, and are separated by special symbols. Each A.P.F., and each L, M is sliown resolved as far as possible into its numerical prime and prime-power factors {p and ju") : these are arranged in order of magnitude of the primes, .the lowest on the left, the highest on the right. * At present only in MS. Tliose for bases '2, 10 were compiled by Mr. H. J. Woodall and the author jointly (not iiidepeudently). The rest are due to the author. 190 Lt.-Col. Cunningham, Factorisation of N= (^^ T 'if). The powers of the small primes ^ H are printed in a condensed form, thus : — 4, 8, 16, 32, &c.; 9, 27, 81, 729, &c, ; 5, 25, 125, &e. ; 49, &c. ; 121, &c. 2°. Special muUiplication-symbols (. | || ; :). These are used to separate rarious kinds of factors in such a way as to indicate the nature of the factors. Use of dot (.). This is used between arithmetical factors in the same A.P.F., L, or M (but not between the A.P.F. or L, M themselves). A dot on the rijjjht of an arithmetical factor, followed by a blank, indicates the existence of an other arithmetical factor of unknown con- 5<titution. Use of bars (| and ||). These are used between the A.P.F. of (X = — F^), {X*-Y% &c.; thus X2- r2==(X- Y) II (X+ Y) ; X*- F* = (X- Y) \ {X+Y)\\ (X-+ 1--=) ; the double bar (||) being placed just before the highest A.P.F. U'se of semi-colon (;). This is used between the A.P.F. of (X"4. 1'") where n is odd, thus x«+r"=(x+r); (x»-'+X"-«r+ &c.). A semi-colon on the extreme right indicates the complete factorisation of the highest A.P.F. Use of colon (:). This is used between the twin " Aurifeuillian Factors" {L, M) of an Aurifeuillian. These Aurifeuillians occur as complete A.P.F., so that their ends are marked by either bars (|) or semi-colons (;) — [see above] . 3°. Symbols (f J). These symbols are used (in incomplete factorisations) to show the limit to which the search for factors has been carried, thus t to 1000, X to 10000 [or a little further]. 4°. Use of qtieries {}). A query (.'') on the right of a large arithmetical factor (>10') indicates that this factor is beyond the power of the Tables to resolve. 13. Tahh o/a;*'*^'+ (a;+ 1)*. The case of 3/ — a; = 1 seein.s of some special interest. Accordingly the short Table below (extracted from the larger general Tables) gives tlie factori- sation of N= .r^+' - {X + 1 )% A' ' = a-^+' + {x-\-\ )'. X, y N^x'J-y'. N' = xy^y''. 1, 2 -I ; 3; 2, 3 — I ; '7; 3, 4 i||i7; 5:29; 4, 5 79113-59; 17:97; 6, 6 47-167; 7-3343 ; 6, 7 162287; 5. 131. 607; ,7, 8 23-159463; 3.II. 19.12539; 8, 9 257-354751; + 9, 10 X II. + 10. 11 3-37-53-12589^ ^53; 253>-4975797i; 15, 16 7.2551 1 89.93; II 25793:277.509; Lt.-Col. Cunningham, Factorisation of N = [x-' ^ y""). 191 TahJe A. X, y iV. X, y N. 2, 3 — i; 7, 9 2.103.172673; 6 7; 7, 11 4.13.19.1981619; 7 79; 9 431; 8, 3 -23.263; 11 41-47; 5 -23-15559; 13 71. 113; 7 -23-159463; 15 74649: 9 257-354751; 17 130783; 8, 11 X 19 523927; 21 b4i.327i; 9, 11 X 23 7.1198297; 25 7.167.28703; 10, 3 -58049; 27 503; 7-38119; 7 -3.90825083; 29 23.97. 240O41; 9 : 2, 31 7-I7-47-599-64I; 10, 11 3-37-53 12589253; 3, 5 2-59; 12, 5 23- 7 4461; il 8.21977; 14, 3 25.107.1787; 3, 13 2.796053; 16, 3 -731891153:125; 4, 3 -ill'7; 5 -593l9-73li457:857; 5 7II3-19; 7 -2273[9.28i||5.349:33i3; I 79113-59; 9 -23.263111.643116481:5.1933; 9 43'll593; 11 -49-25713-556311 11 41.47II9.241; 115.1789:53.461; • 13 71.113II9.929; 13 -20369I3.1251II 15 7.46491132993; 1115121:11677; 17 130783II3-43787; 15 -7.2551:89.93711 19 523927113-1:9-977; 1125793:277-509; 21 641.3271II 2097593; 17 7-67Q3l3-233 307ll 23 7. 1198297II3. 67.41737; II '25 533:362561; 25 7.167.28703II3.U185019; 19 7.23.2447I3.113.1931II •n 503;7-38ii9;i52i;73 3529; 115.56989:109.9397; 29 23.97. 240641II27.11.41.44089; 21 190267 1 1 1097. 2089II 4, 31 7-i7-47-599-64'l|27-79536467?t Il5'3 41-521:3194801; 23 73-I13-983I9-227.4243II 5, 7 2-23.31-43; 115. 1300333: io835233.?t 9 4-473519; 25 33163807I9.3771073II 5, 11 2.3-23-36373; 1153.113.4813:39065057?^ 16, 27 43i;7-73-6o7l593;2272233l| 6, 5 -47.167; ||8Si;i39393:5. 61:13.36997; 7 162287; 6, 11 5- t i 192 Lt.-Col. Ciuiningham, Factorisation of N= {x^ ^f i/). Table B. X, y N\ X, y N\ 2, 3 I"; 7, 9 64-705259; 5 3-19; 7, 11 2.3. 17. 19576607; 7 3-59; 9 593; 8, 3 11.643; 11 9.241; 5 3-141131; 13 9.929; 7 3.11. 19 12539; 15 32993; 9 17 343787; 11 3.19.43.11933011; 19 3- '79-977; 21 2097593; 9, 11 4. 25. 367.919319; 23 3-6741737; 25 3.11185019; 10, 3 11-53-103; 27 521:73-3529; 7 11.4397.6047; 29 27. II. 41. 44089; 9 II. 2, 31 27.79536467?! 10, 11 2531-49757971; 3, 5 16.23; 12, 5 13.19.463 2137; 7 2. 5. II. 23; 11 2233.383; 14, 3 677.7069; 3, 13 8.5.167.239; 16, 3 17.2532401; 4, 3 5:29; 5 '7- : 5 17:97; 16, 7 17- X 7 5-13:17-17; 9 5.61:881; 11 5.293:13.13.17; 13 37.181:25.401; 15 29153:36833; 17 173-709:5-109257; 19 13.38861:5.108821; 21 5-13-73-433:2140601; 23 17.409.1193:5.13.130513; 25 33350257?t:29-373-3i2i; 4, 27 25-i7:6i7;i57-'38i:i3-24373; 5, 7 4.81.293; 9 2.1006087; 5, 11 8.31. 251. 787; 6, 5 7-3343; 7 5.131.607; 6, 11 7. t END OF VOL. XLV. METCALFE .VND CO. LTD. TKINITY STREET, C.MIBRIDGE. 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