2 
 
 UC-NRLF 
 
 ^c ibM ?m 
 
 %l 
 
 i 
 
 "^A;*' 
 
 h'^i 
 
 
 
 ^? 
 
 '^' 
 
 ^ 
 
 wr 
 
 ^' 
 
 
 <* « 
 
 *-^<< 
 
 >• 
 
 *• 
 
 >. J 
 
 .♦fl 
 
 
 4- ' .Jf-- 
 
'/ 
 
 •if 
 
 ?/ 
 
 V 
 
 y- 
 
 LIBRARY 
 
 OF THE 
 
 University of California. 
 
 Class 
 
 A 
 
 i^ ''^ 
 
 1^^ ^' 
 
 !%. 
 
 L-. 
 
 > - 
 
 \ 
 
 K- 
 
 ^^^x.^ 
 
 w 
 
 
 
 K 
 
 N 
 
 ^ 
 
 .^ 
 
 ^ . X 
 
 
 \ 
 
 I 
 
 \ 
 
 
 
 \ 
 
 -41 
 
 - » 
 
r 
 
Researches on the Multiplication 
 
 of Elliptic Functions. 
 
 By 
 
 R. Fujisawa. 
 
 Jacobi, in one of the Suites ties notices siir lesf mictions elliptiqves, gives, 
 without denionstratioii, remarkable expreswiona of sn 2?/, sn 3?/, sii 4?/ 
 and sn 5u in terms of tlie differential coefficients of \/a;-(l— x^Xl — ^-V) 
 
 and y — ^ — ^— ! taken with respect to x^, whereby x stands for 
 
 anu,* Namely, writing- 
 
 we have 
 
 1 1 
 
 sn In. = 
 
 (IB 
 cl{x^) 
 
 fPB /^>^of THe 
 
 ,A7^' ( UNIVERSITY 
 
 sn 'Sll — — .f* ^T-: — , V ^ 
 
 (l-A 
 dix^f 
 
 1 dM 
 
 sn Au = 
 
 2.8 dix^f 
 
 1 d''B 1 dm dB 1 d^B 
 
 sn 5?< = J^° 
 
 2 d(:x^f 2 £^(a;2)2 £Z(a-=^) 2.3 d{x-'f 
 
 i_ .^!^ J_ ^-^ _i ^ _i_ ^'^ 
 
 2.8 (/(■r^)» 2.8 ^(.r^)» 2 (7(.r-V' 2.:i4 d(xh* 
 _L ^^'^ _L '^^'-^ _ J_ dU 1 r/M 
 2.8 </(a-7 2.8 £Z(a;2/ 2 d{x^f 2.8.4 f/(x='}* 
 
 * Crelle's Journal, Bd. IV, pp. 185-193, and Jacobi's gesainmelte Werke, Bd. I, pp. 266-275. 
 
2 E. rUJISAWA. 
 
 Jacobi fidds " la lor (jeneml dc la composition dc ces expressions est 
 aisee a saisir,^' iiiid lurther remarks that we sluill have aiialoo-ous 
 forinuliu by using, instead of tlie differential coefficients of A and B, 
 those of 
 
 :in(l 
 
 Vj='(l-a-2xl-AVj ' A I -J--)( 1 - /. V») • 
 
 /- 
 
 X- 
 
 The general form of tliese expressions is, however, by no mejins easy 
 to infer from the ])ai*ticular cases just given, and I have tried to tnice 
 among the writings of Jacobi, the steps which might liave led him to 
 these expressions, but without success. 
 
 The present memoir is divided into two ])art8. In the first ])art, 
 the multiplication-formulie of elliptic functions are derived from Alxil's 
 theorem for the elliptic integral of the first kind. It will be seen that 
 one of the results arrived at is the general formula in question. It 
 appears, however, highly improbable that Jacobi obtained his formula? 
 in this way. 
 
 In the }»a|)er just alluded to, Jacobi gives, also without demon- 
 stration, the partial differential equation satisfied by the numerators and 
 denominator of the multijjlication-formula?. This partial differential 
 e<iuation has since been obtained by 15etti,* Cayley,** I^riot et 
 Bou(|uet,f and others ; but the final results to be obtained by 
 applying it to the actual evaluation of the numerical constants involved 
 in the multiplication-formula?, has not, to my knowledge, hitherto 
 been develo^Ded with much completeness or success. 
 
 In the second part, Jacobi's partial differential equation is derived 
 in a manner which is most probably the one followed by Jacobi 
 
 * Betti, Annali di Mateniatica, Vol. IV, p. 32. 
 
 ** Cayley, Cambridge and Dublin Mathematical Journal, Vol. II, pp. 256-26C. 
 t Briot et Bouquet, Theorie des Fonctions Elliptiques, p. 529. 
 
MUi;i'll'lilCAI'lON OF ELLIPTIC FUXOTIOXS. o 
 
 himself, mid (lu'ii .ippliecl to tlie iiive.stigs«tioii of the inultiiHiwition- 
 forinulie. The present paper is tlius coiinx^sed of two |mrt«, eac^h of 
 them having reference to the theory of multipliciition hut otherwise 
 unconnected. 
 
 Throughout the j)aper, I have a(h)|)ted tiie notation of Fund. 
 Nova, the only exception l)eing that I write ^i, ff^, (1^, H instead of ff 
 and H, whereby 1 follow Jacobi in his lectures.* 
 
 Pari First. 
 
 §• 1- 
 
 Denote the fundamental elliptic irnitionality by .s and the corres- 
 ponding Riemann's surface by T. Let S stjind for an algebraic func- 
 tion of i, one- valued on the sm'fiice T, and of tlje order </, having for 
 its zero-jjoints 
 
 o^ I^M I -vh /^ = J, '^, 3, q, 
 
 and for its intinity -points 
 
 ^p {:v I 8p\, u = 1, 2, 3, q, 
 
 then Alxil's theorem is expressed by 
 
 (1.) y -^ - y ^ = 0. 
 
 Xow S is necessarily of the form 
 
 B 
 where P, Q, R are integral functions of i; of the degree //, q-'J and */ 
 
 * Jaoobi, Gesaumielte Werke, Bd. I, p. 50L 
 
4 n. FUJISAWA. 
 
 resjjectively. 'J'lie Miiiiienitor J' + (Js must be .so (let eniii tied tliat it 
 vjiiiishes ill q points o^ und, besides, in nnother set of cy ])oi!its 
 
 ^'v IC" I — ^""f' ^ = 1» 2, '^^ <J- 
 
 It will Ije convenient to writ^ 'V + x ii»«l<-*''>d of £„, and tben 1' + (J s 
 vanishes in 2q ])oints 
 
 '?/i {^^ i -Vl. /^ = •' '^' ^' ••• '7. 7+1. ■•• '^<7- 
 Equation (1.) tlien takes the form 
 
 (2.) 2 4^ = 0. 
 
 We now take for the fundumentui irrsitionality Uiemann's form 
 
 V-1-^. 1-A*?, 
 and write 
 
 2« = / —7- — --. , 
 
 so that * 
 
 *^/T — sii /< . 
 
 Two cjises are to be distiniiuished aecordinH" us n is odd or cvcm. 
 
 §• -'• 
 
 AVhen // is odd, put n = 2ni + 1 ^^q — l, and let one of tlie '2q jK)ints 
 in which I'+Qs vanishes, coincide with the point |C|^} ''"^^ ^^^^ 
 remaining (2y— 1) points- with the point | ^ | .s}. Ecpiation ('J.) then 
 liecoines 
 
MULTIPLICATION OF ELLIPTIC FUNCTIONS. 
 
 (3) 
 
 -^ + n = , n odd, 
 
 Mild, writins" P + Qs in full, 
 
 we must lijive, denoting differentiation with res|)ect to z by D, 
 
 «■! + a22,~ + . . . + a,„,.,{m + 1 ),^'» + b,Ds + Z)iZ)(,<j.?) + . . . + /;,„_iD(s^"'-i) = 0, 
 
 (4) 
 
 + b,,_,D^^'(f!z^-')=0, 
 
 boD^s+ 6iZ)2'»(.9^)+... 
 
 + Z>,„_iD2'»(.s^'»-i)=0. 
 
 Since a^, «,, , h^, h^ ... do not all vanish, the determinant obtained 
 
 by eliminatinof «(,, a„ , /)(„ />, ... must vanish, that is, 
 
 1 , 2^, {m + 1 ).?'", Ds, D{sz}, Disz\ D(.<f^'"-») 
 
 (5) 
 
 D^'^s, D^%<iz), D^^-iz^), D^'^isz'^-^) 
 
 Let the expansion of this determinant according to the elements of 
 the first row be written, 
 
6 
 
 R. FUJISAWA. 
 
 (G) [Po+PxC + . . . +p.+i:'"+^} + [Qo+Qi:+     +<?«-ir-^]^=o. 
 
 Then (2m + 2) roots of the etj nation 
 
 are C? and z repeated (2m +1) times. Thus we obtain the following 
 identity : 
 
 (7) {l\+P,Z+... + P^,,Z^'\'-[Q,+ Q,Z+... + Q„_,Z--'y.Z[\-Z){\-l-^-Z) 
 
 ^K,i{Z-:)iZ-zr^\ 
 
 Herein putting Z=0, /^=1, Z=-p- successively and reducing, wo 
 obtain 
 
 S5H-1 
 
 z ' = 
 
 (8) 
 
 >/nT 0-^) 
 
 2W + 1 
 
 Sm+1 
 
 Vi-A-*: (i-A^^) » = 
 
 Po+Pl+...+P^^l ^ 
 Pm + 1 
 
 P„A*"+«+PiA*-+...+P^^l 
 
 P« 
 
 + 1 
 
 In extracting square root, strictly sjX'jiking, we have to ])refix the 
 double sign ± ; but by taking some pjirticular value of n or by putting 
 fc=0 in the final results to be hereafter obtained, it comes out that we 
 
 have to take the + sign. 
 
 § 3- 
 Let us now investigate the expressions which occur on the right- 
 hand side of equation (8). For this purpose, put 
 
MUL'rrPLrOATIOX op ELLIPT[0 punttioxs. 
 
 OM 
 
 A = 
 
 
 further let the ox J KUision nf A .'locordiuir t,) tlie elemouts of the first 
 rcnv Ix' written : 
 
 (10.) A = D"'ns . Ai+ 7:)"'+^s;' . A2+ . . . +D'"+V?'"-» . A, 
 
 Xow 
 
 (11.) Po = 
 
 1,2^,... (?« + 1 ) ?'», D.s, 7).s' r, 7).'??'»-i 
 
 {m + 1 ) ! , D'" ^^s, 7)"'+'.s^, 7)"'+^«;?"-i 
 
 = A 
 
 ^, ^^ 
 
 ~TO ^m+1 
 
 \,'2z, ^^i-^-SCw+l)?'" 
 
 ?// ! , {7)1 + I )! ? 
 
 -(w + l)!Ai 
 
 -{vi + \)\A, 
 
 Z, ^^ ;?"*, /? 
 
 1,"2^, w^?"'-^ D.S 
 
 ?;/ ! , D"'.« 
 
 1, -2?, w/ ?"•"', T).*.? 
 
 7/t ! , D"*/? ? 
 
 -(w+l)!A, 
 
 1,2?, 7nz''-\ Dxz'^-^ 
 
IJ. FUJISAWA. 
 
 Writini»", i'or shortness, 
 
 we liMA'e 
 
 z, z^, 
 
 1! 2! ... m\^m\\, 
 
 yln yW + X 
 
 Z'", Z" 
 1,2,?, 7H.?'"-\(///+l)?"' 
 
 m\ , {>n+\)\ z 
 
 = m !! .?"'+^ ; 
 
 Jilso 
 
 " > -^ > •     /i , •■< 
 \,2z,...7nf" \Ds 
 
 1. I.... 1, .s 
 1, 2, ...yy/, :y).s 
 
 /// ! , 7)"'.s' nt ! , ■:"'jy"s 
 
 = (m— 1) !! [ r"'J)"'.s' — m.?'"-'D"' ^s+ . . . +(— l)"'//y ! .s] 
 
 .siiniljirly 
 
 ^ y* 
 
 . ^ , .s.r 
 1. 2.?,...m.?"* ^ 7Mr 
 
 y« ! , I)"'.s : 
 
 - (m-1)!! ,r"'+^D'",<? , 
 
 Z, Z', z^, sz' 
 
 l,2.?,...m?'" ', Dsz^ 
 
 (/y/— 1)!! r^+^D^s? , 
 
MULTIPLICATION OF ELLIPTIC FUNCTIONS. 
 
 m—l 
 
 Z, Z", ... Z^, SZ 
 
 l,2z,...mz'^-\ Disz""-^) 
 
 m ! , D"'{sz"'-'^) 
 
 = (w — 1) !! ^"•+^D"^«^"   «) 
 
 Hence 
 
 (12.) Po=/;i !! ^'"+>| A -U/i+ 1) [[dY-^\Ai+D'".s. As+ ... +D'"«^--».A J |. 
 
 Again 
 
 (13.) P,„+i=(-l)'"+^ 
 
 .SZ^ 
 
 \.yZyZ^*»% Z y Of SZ y 
 
 l,'2z,... mz'^-\ Ds, Dsz, Z)s^-»-» 
 
 mlyD'^s, D'^sz, ...D'^sz'^-^ 
 
 = (-l)"*+im!! A. 
 Substituting the values oi Pq and P«+i just found in the first of 
 equations (8.) and dividing by ^ ' , we get 
 
 (14.) 
 
 Vc~= (-i)"*+V'^ 
 
 _A-(//t+ 1)|W— \Ai + Z)'".s.A2+ ... + D'",s<j"'-2. A„| 
 
 A 
 
 The expression for A given in (9.) may be greatly simplitied. 
 By an easy reduction, we find 
 
 (15.) 
 A = 
 
 D'-'+^s, (w + 1 )D'».s-, {m + 1 )!)'» ^s {(w + l)wt. . .8 [D^'s 
 
 D'^+h, {m + 2)D"'+i6-, (wi + 2)(//i + 1 )Z)"'5, { {m + 2Xwi ■\-\)..A\Dh 
 
 D-"% 27nD^"'-\ 2fn&in - l)D^-% 1 2wi(2;/t - 1). . . (;/t + 2) } D-+»6- 
 
10 
 
 R. FUJISAWA. 
 
 that is, 
 
 (16.) A=(-l)^'<2""" 
 
 VI 
 
 (w+l)r *'(m+2)r *'"(2w)!^ * 
 
 The expression 
 
 which occurs on the righthand side of (12.), admits of being simplified 
 in a similar manner ; namely denoting, for a moment, this expression 
 by a', we have 
 
 A'=(w+1) 
 
 D'"-^, D-.S-, D-»s^"»-» 
 
 i;"'+-.v, D'"^-sz, D "+-A-^"'-i 
 
 m(m:^ (2W)!! 
 — V ^) ... 11 
 
 mw 
 
 z 
 
 — — n«< 
 
 2 ! -^ ^ ' 
 
 4 f-^*'- 
 
 ^ TJD'^'^^s, — -^D-'+^'s, 
 
 (w+1)! '(7?i+2)! 
 
 wt! -? 
 
 1 
 
 (//t + 2)! 
 1 
 
 D" 
 
 (2w)! 
 
 D2"'s 
 
 Leaving the factor ( — 1)'»+^VT. ( — 1) ' -^^ — jy— out of con.sideration, 
 tlie numerator of the righthand side of equation (S.) becomes 
 
MULTirLTCATTON OF ELLIPTIC FUNCTIONS. 
 
 u 
 
 -L-D-s — -D^<? 
 
 2\ 
 1 
 
 3! 
 
 
 (w+1)! '(w + '2)r ^' 
 
 1 
 
 1 
 
 (w + 'i)! 
 
 
 ('iWi)! 
 
 D2"'.S 
 
 «T- 
 
 a! 
 
 D".?, 
 
 2\ z ' 
 
 4! 
 
 D% 
 
 
 — T-D*"— 
 
 Vil z 
 
 1 
 
 (m + 2) 
 
 D" 
 
 (2??0! 
 
 D2»«.<; 
 
 The two determinants may lie compounded together ; thus we obtain 
 
 ^Kt> I^Kt) d:Tyi^-<f) 
 
 (17.) z 
 
 3 ! ' 4 ! ' 
 
 1 
 
 (w + 2)! 
 
 jy^^H 
 
 
 (w+1)! ''(7?? + 2;! 
 
 (2wi)! 
 
 Tr-'^8 
 
 A sliglitly different form might he given to this expression, viz 
 
 (18.) ,j" 
 
 
 4!^V^/' (w + 2;! V ? / 
 
 _J_i)™+/_l\ ^__7).«+/A\ ^LdsYA'^ 
 
 (w+1)! \z r (W/+2)! V ^ /' ••• (2w)! \ ^ y 
 
12 
 
 R. FUJISAWA. 
 
 Let this fletermi nant (omitting the factor z"") be called 3f, and the 
 determinant on the righthand side of (10.) M. Observe that M and 
 A differ from ejich other only by a muinerical factor. Equation (11.) 
 now assumes the form : 
 
 (19.) 
 
 VT =(-!)" 
 
 M 
 
 §. 4 
 
 Consider now 
 
 P, + P,-¥ ... + P« 
 
 +1 
 
 which may be written in the form of a determinant, viz. 
 
 M=^ll«+l 
 
 (20.) ^P^ = 
 
 1.1. 1. 
 
 1 z ^ 
 
 1. 
 
 ,w+l 
 
 0. 
 
 0, 
 
 sz. 
 
 sz" 
 
 I 2z (wi + l);?", Ds, Dftz, D.«j^"'-» 
 
 D'"+*9, 2)"'+*.'?^ D'"+*.«?^'"-^ 
 
 D*».<j, D*^sz, D*'"/;^--! 
 
 This determinant may be reduced in exactly the same manner ns Pq 
 and then we find 
 
MULTIPLICATION OF ELLIPTIC FUXCTTOXS. 
 
 13 
 
 2 Pm = A 
 
 M=0 
 
 1, 1, 1, 1,1 
 
 Ij/J, -4....... /4 ,/4 
 
 1, 2;?, mz'''-\ Cm + l>r"' 
 
 ?;i ! , {m + 1 ) ! ^ 
 
 Now 
 
 (w+1)! Ai 
 
 1,1, 1, 1,0 
 
 1, i2, Z^, Z"^, S 
 
 l,2z, mz"'-^ Ds 
 
 -(m+1;! As 
 
 1,1, 1, 1,0 
 
 ^, Z, Z^ Z^, RZ 
 
 1 , 1z, m?"'~\ Dsz 
 
 m ! , Jy^8z 
 
 :w + l)! A, 
 
 1,1, 1, 
 
 1, z, z}. 
 
 1,0 
 
 z^, ftz" 
 
 },2z, mz'"-\ D.fz'"-^ 
 
 VI ! , D'^sz"""^ 
 
 1,1, 1, 1,1 
 
 1,^, z^ 
 
 l,2z, w^'"-\ {m + \)z' 
 
 ^m ^"i+l 
 
 m\ , {m+h\z 
 
 = (_l)"'+i7?i!!(l-^)'»+» , 
 
u 
 
 R. PUJISAWA. 
 
 1,1, 1, 1,0 
 
 \/2z, viz'"-\ Ds 
 
 in ! , D'^sz 
 
 = -(- l)-'+i(w - 1 )!!(! -zY'+W 
 
 \-z ' 
 
 1,1, 1, 1,0 
 
 I, z, z^, z^, sz 
 
 \,2z w,?'" S Dsz 
 
 m ! , D'^sz 
 
 HZ 
 
 = - (_l)'-+i(7,j_l)!!(l-^)*+iD*^£ 
 
 1.1, 1, 
 
 \,z, z\ 
 
 1,0 
 
 ^•1 n yW* 1 
 
 1, 2.?, w?"'-\ D«?-- 
 
 7n ! , !)■•«"•"* 
 
 «^ 
 
 i»-i 
 
 = -(- 1 )"■*"'(?«- 1)!I(1 - :r"^'D"'-|3 
 
 Hence 
 
 
 
 '^ . .A, + D«-^-— .A,+ ... 
 
 1-^ 
 
 Pizr 
 
 + /)"• j^ . A 
 
 ■]l 
 
 = (-l)'"+iw!!(l -.:)"•+« 
 
 1 — ^ J — ^' ]— ^ 
 
 ir-^s, D^^^sz, D^+s,^"-! 
 
 D»",S D^»\<iz, 
 
 jy^sz* 
 
 Further, we deduce without much difficulty 
 
MULTIPLICATION OF ELLU'TIC FUNC'TIOXS. 
 
 15 
 
 11=0 
 
 »ii(»»-l) 
 
 C21.) 2! i'*'=(-i) * {-ir+\'i?n)\\{\-zr-^' 
 
 1 
 
 Z)2 
 
 D» 
 
 2! 1-^ ' 3! 1 
 
 3 ! 
 
 _1_ 
 
 4! 
 
 D% 
 
 -!)"•+'- '^ 
 
 I 
 
 (w+'2)! 
 
 £)m+2^. 
 
 nwt+l^ /•;)?n+2„ ^ 712»t., 
 
 or 
 
 J*=«l+1 t/t(OT-l ) 
 
 (22.) 2 ^^ = (-1) ' (-lr+H2wt)!!(l-^f«+i 
 
 M=0 
 
 2!-^'l-^' 8!^ 1-^' (w+l)r 1-^ 
 
 — Z)8— ^ 
 
 ^ D-+1- * 
 
 —1)9-^ 
 
 8! i-z' 
 
 4! 1- 
 
 .2) ".+2 * 
 
 (7// + 2)! 1-^ 
 
 ^ -jr)"»+i * 
 
 1 .s 
 
 -jr)"«+2. 
 
 L_2;2»__l 
 
 (w+l)! l-^'(w+2)! i-^''*"(2w)! 1- 
 
 Denoting the determinant on the righthand side of ('22.^ by M^ 
 
 P,u+i=(-irH2m)!!M, 
 
 and renieniljering 
 
 Ave get by division 
 
 
 Substituting this in the second of equations (.S.). we obtain 
 
 (23.) vr< =(1-^)=^^. 
 
16 
 
 11. FUJISAWA. 
 
 §.5. 
 
 Let u.s consider the determinant 
 
 (24.) 
 
 P»+2, A~^, A^-*-», ... 1, 0, 0. ...0 
 
 1, z, z^, ... ^"^', s, sz, ... s^"*~^ 
 
 1, 25r, ... (w+1 )-?"*, Ds, D«^, ... Ds^~-i 
 
 
 This determinant is of the Siime form as the one in (20.), and we find 
 likewise 
 
 M=0 
 
 
 =(-ir+'//i!!(i-^-*^r+' 
 
 7V"-»-1__LL_ / )m+l ^^ 7)"»+l__^f__ 
 
 D*»s, 
 
 D*»s^, ... B^'^sz'^-^ 
 
 whence 
 
 2 
 
 M=0 
 
 (25.) 2] i^A'^'"'"*"'""*=(-l)"'^\2w)!!(l-A,-*^) 
 
 1 
 
 ,w+2 
 
 1.2). ••' 
 
 Z)« 
 
 1 
 
 .£>" 
 
 nsc 7)«s _L__n"i+2e 
 
 ___L__ 71«»+lc ___r)iH+2c __L_7~)2»»c 
 
 (w+i)r *' (/;i+2)r ^' ■••{2w)r * 
 
MULTIPLICATION OF ELLIPTIC FUNCTIONS. 
 
 17 
 
 or 
 
 )t=in-t'l m(Tit— 1) 
 
 (26.) 2. ^V^-^^-^'-^^^-C-ir+X-l) '' (2w)!!(l-^^)' 
 
 Sm+l 
 
 -Ld^-A 
 
 2! l-h^z 
 1 
 
 A-D»-^ 
 
 1 
 
 8! l-^='^ 
 
 -.D 
 
 iTOfl 
 
 (w+1;! i-A;*^^ 
 
 D" 
 
 •S\ l-k^'z' 4:1 l-k^z' (w+2)! i-A-2^ 
 
 1 s 
 
 .D 
 
 m+2 
 
 1 S 
 
 {m + 1 )! 1 - k^z ' {m + 2)! 1 - A;^^ ' ' ' ' (2»0! 1 - kh 
 Call the detenniiiaiit on the righthaiid side of (::^6.) M3, we have 
 
 OT+l 
 
 If 
 
 Subatitutiiig' this in the last of equations (8.) we obtain 
 (•27.) Vi-i%' =(l-i:^^)^ '^. 
 
 §• fi. 
 
 In case // is even, say n = 2/;?, put .«? = ^y — 1, and, referring to 
 §. 1., let one of the 2r/ points in which P + Qs vanishes, coincide with 
 the point \Z \ ^], another point with |0 | Of, and the remaining 
 (2r/— 2) points with the point {z \ s\ . Then 
 
 (28.) 
 
 d: , dz 
 
 —^ + n 
 
 s 
 
 , 11 even, 
 
 and 
 
 P + gs = {a^z -\-a^i'^-...-\r a^+iZ"'^') +{b,+ biZ+ ...+ 6„,_i^"'->, 
 
18 
 
 R FUJISAWA. 
 
 rto bein<{ zero. If, as will be convenient, we write 
 
 A = yk<4^.,,^.y. 
 
 l-z. l-k^z 
 
 II, 
 
 then ^' = {a^ + a2Z+ . +a„+iO + (60+ V+ • • + K V^) f , 
 
 and 
 
 
 (29.) ( 
 
 + 6«_iD* (u^;'*-*} =0, 
 
 
 whence 
 
 (30.) 
 
 1 >« r2 
 
 r", 0, 
 
 0:, 
 
 ...0r"-» 
 
 1, ^, Z^, . 
 
 .. /.'", //, 
 
 ?*-?, 
 
 ...?*^"*-^ 
 
 1, 2^, . 
 
 .. mz^'^, Du, 
 
 Dnz, 
 
 ... D<<2"»-^ 
 
 
 W + 2,« ylH 1 
 
 2)m + 2^^^ i)- + 2^*2, ...D'^+^UZ 
 
 D-'^-^u, D-'^-hiz, ... B-'^-^asr-^ 
 
 Tlie expansion of this determinant according to the elements of 
 the first row may be written : 
 
MULTIPLICATIOX OF ELLIPTIC FUNCTIONS. 
 
 19 
 
 (31.) iPi+P2:+...+p,„+iC"i + {Q.+Q,:+-..+Q,n.-,r'} = 0. 
 
 Hence 
 
 (32.)Z|Pi + P2Z+... + P„Hi'^'''r-ia + (>i^+-.- + Qm-i^"'-^p(l-^)(l-A;^^ 
 
 = PLi(^-cx^-^r'. 
 
 Herein putting Z=0, Z = l, Z= -^3- successively, we obtain, after sligh) 
 reduction. 
 
 Vc '2'" = 
 
 Qo 
 
 (33.) 
 
 Vl^d-.")" 
 
 ^yi-kx{i-j(^^r = 
 
 ^ w+l 
 
 fi+p,+ ...+p^^l 
 
 p ' 
 
 PlP« + P2P«-2+...+P^^j 
 
 P, 
 
 »i+l 
 
 The reduction of the expressions which occur on the righthand 
 side of (33.) runs on the same line as the reduction of the corresponding 
 expressions when n is odd, discussed somewhat in detail in preceding 
 sections. We may therefore at once write down the results : 
 
 (34.) 
 
 m(m ' 1) 
 
 P,»+i=(-l) ' (2m- 1^! 
 
 D — — - D^— 
 
 2\^ z ' 'S\^ z 
 
 m\ z 
 (w+l)! z 
 
 1 .<? 1 <j 1 « 
 
 m\ z'{m+\)r ^''Cim-l)! z 
 
20 
 
 K. FUJIKAWA. 
 
 m.) 
 
 r.'iii'\)V. 
 
 
 1. ^^*^' w-^-n?^*'* 
 
 I ; 'w/+ I); 
 
 J>'x. /, />*^ 
 
 •"(w+2/^" ' 
 
 (m 
 
 m.) '£ /v»(-ir^ V«--i)!?(i-*f" 
 
 
 />)>- 
 
 
 w: 
 
 ?r=ro 
 
 
 »./r - 
 
 
 />"• 
 
 (///4-h' (I ) 
 
 a»7.) Ij iVA -"•"'- ("-ir"' 'Vim-l)!!(I A":)'- 
 
 
 ^7>• ' 
 
 *(!-*■*) 
 
 V/i- 
 
 i/r 
 
 
 l)*^ >. 
 
 Cnlliiiji( tlu* (IctniiiiiiiiiitM on tlio ri^lit-luiiKl hIiU'h of (>4|iia(ioiiH (81-87.) 
 '^' V . '^ , V lit ••nh'i', hihI, Hii)Mtitiitiiiji( in (88.), w(» ohfjiin 
 
 (HH.) 
 
 
 JV, 
 
 v\-^: - O-^^"* V 
 
Mi?ii.Tt]ruaiT«x>c ©iT cujme rv:?er»i»iss. 
 
 il 
 
 Wntitt^> »» k!^ i^wil^ 
 
 
 «>«> ibait 
 
 wbi»ig » k any e^e^a cr wid uil^i;^^ 
 
 (csy>.> 
 
 igj^ ^^ 3£ ^_ 
 
 
 
 wl*i«v 
 
 <(4^V> Jf» 
 
 
 
 Km^\)r i Hw+:3or .1 
 
 
 I 
 
 ^w-f' 
 
 1S»^*T 
 
 .^ -.JJfc^J^ 
 
99 
 
 R. FUJISAWA. 
 
 (41.) M,= 
 
 ^D2 "" 
 
 ^-D' ^ 
 
 1 D".+i. " 
 
 2! 1-^' 3! l-z' "(w+l)! l-z 
 
 Kd' ' 
 
 il)^— 
 
 1 
 
 D" 
 
 3! 1-;?' 4! 1-^' "(^ + 2)! 1-^ 
 
 J — :D'»+'— 
 
 * -£)"»+«- * 
 
 1 £)2»- « 
 
 (m+1)! l-;?'(w + 2)! l-^'"" (2wt)! 1--? 
 
 (42.) M, = 
 
 Jt-D' * 
 
 Ld^-^ 
 
 ^ 2)«*+L * 
 
 2! 1-F^' 3! 1-;^/ •••(w+1}! l-Pz 
 
 4tD^^ 
 
 -hn* ' 
 
 ^ X)«»+«_ * 
 
 3! 1-A^^' 4! 1-AV ■••(w + 2)! F^A^ 
 
 1 r)m+i ••' 
 
 J 2)"»+2 — 
 
 1 
 
 yr-" 
 
 (wi + 1 )! I - Jc'z' {in + 2)! l-h^z'"' (2w)! 1 - /r',: 
 
 (43.) M = 
 
 
 (w+1)! '(m + 2)! ' 
 
 1 
 
 (w+1)! 
 
 1 
 
 (7;i+2)! 
 
 
 (27;t)! 
 
 D*"« 
 
 In case ;/ is even, say n — 2in, we have, in virtue of (38.), 
 
 (44.) 
 
 /sn,m = (-l)"^^, 
 en mi = (1 — 5')"' -jTT^ , 
 
MULTIPLICATION OF ELLIPTIC FUNCTIONS. 
 
 23 
 
 whore 
 
 (45.) N, 
 
 —jn^s ——D*s 
 
 ■6\ 4 ! 
 
 -—D*s — -Z)*.s 
 
 4 ! 5 ! 
 
 "(w+l)! 
 
 1 
 
 "■(w + 2)! 
 
 2)m + l, 
 
 
 (46.) i^2 = 
 
 1-D- •^• 
 
 1 Z)2_ <^ 
 
 1 T^^ •'^ 
 
 .J)rr 
 
 
 tJ^D' 
 
 -i-DB^ 
 
 1 ,s 
 
 '2\ z{l-zy 3\ z(\.-zy "'(m+i)\ z{l-z) 
 
 1 T^_ .? 
 
 — r-D 
 
 1 .9 
 
 1 
 
 D^" 
 
 m\ z{i—zf i^m+{)\ z{l—zy"'{^m—l)\ z{l—z) 
 
 (47.) N, = 
 
 Id 
 
 ■D' 
 
 1 _p« 
 
 1 rz{i - Wzi 2 ! /( I - A^^)' • • • m\ z({- k^z) 
 
 l.B^ ' 
 
 1_^P s 
 
 1 s 
 
 2 ! ^(i - A;2^)' 5 ! ^(1 - F^)'   " (7?i + 1)! z{i- kh) 
 
 \d- ' 
 
 1 T)m+1 S 
 
 1 .9 
 
 mr z{Y-kh)' {m+ {)\ zi,l-k'z)' '" C2m-\)\ z{l-k^z) 
 
 (48.) N = 
 
 1 ! ;? ' 
 
 2!^ ^ ' 
 
 J_2)2A 
 2! ^ 
 
 a ! z 
 
 1 T^^ 'f 
 
 ... — T-D" — 
 
 1 s 
 
 ■■■(?;i+J)! ^ 
 
 ml z '{m+i)r z '■•■{'2m-i)\ z 
 
 It will be observed that the formula3 of nnnu for w=2, 3, 4, 5, 
 are those given by Jacobi in the Memoir referred to, which appeared 
 
24 R- I'UJISAWA. 
 
 on the opening page of the present p.iper. I was not aware of the 
 existence of Jacobi's formula? when I first found the formula for 
 snwM which was, in fact, then obtained in a slightly different form 
 as furnished by (17.), and which is perhaps in some resi)ects preferable 
 to the one here given. I have, however, given it its present shape, 
 in order that, for the particular values of w, it may exactly coincide 
 with the formuhc given by the illustrious mathematician whose 
 memory is sacred to every student of the theory of elliptic functions. 
 
 §. 9. 
 
 IJefore proceeding further with the reduction of the multiplica- 
 tion-formula3, il is necessary to give a few formulae relating to the 
 differentiation of composite functions, U) which frequent reference will 
 subsequently be made. 
 
 Let « be a function of?/ and y a function of a:. It is required to 
 find the n"' differential C(Xifficient of u with respect to x. l^y actual 
 differentiation, we find 
 
 du _ du dfi 
 dx ~ dy dx' 
 
 dx^- 
 
 d^ii 
 dx^ 
 
 d*ii 
 dx* 
 
 _ dii_£y ^ / du \- 
 ~ dy dx^"^ di/\dx)' 
 
 _ du^ cPy ^ d?u dy d^y d^ii / dy V 
 ~ dy W' "^ (tf'dx 'dx^ "^ Jy^ \llx) ' 
 
 ^ ^d*y ^{ dy_^ ../dy\'l (Pu /dyVfy 
 
 dy dx* "^ dyU dx dx^ "*" \dx'J ) "*" *" dyAdx ) dx^ 
 
 ,d*u(dy\ 
 ■^ dy*\dxj' 
 
MULTll'LlCA'1'lON OF ELLIPTIC FUNCJTIOXS. 
 
 25 
 
 dx'' ~ dy 
 
 dy dx^ dy- ( dx dx* dx^ dx^ ) 
 
 d^6 {.f./dyVd^ ^(^^i 
 
 "^ dy^l \dx) dx^ "*■ dxXdxy) 
 
 dy*^ \ dx / dx^ dy^ \ dx / 
 
 and, geiierully, 
 (49) 
 
 d"ii_ _ du ^ .d^u ,^"^'^v <d"i/, 
 
 " ~ dy^'^dy'^^^'^-'^dy^'^''^--'^ dir-'^'''''^dr^'^' 
 
 dx 
 
 where Zj, X2,...A'„ denote us yet unknown functions which contain 
 only the differential coefficients of y with respect to x. A few of the 
 initial and end coefficients may at once be written down : 
 
 ' ~ dx" ' 
 y _ ^dy d'^-hj n{n-l) dhj d'-hj 
 
 dx dx'' 
 
 2 ! dx' dx''-'' 
 ln{n-\)...{n ^ + 1)^2, ^2 
 
 + 
 
 (¥)! 
 
 
 nl,n-\)...{n-Y+'^) ld^y\ , 
 
 — I 7- J (*i even) 
 
 (50) / 
 
 2-(l)! 
 
 , fZ^' '^ 
 
 X = ?t(/?- - 1 )(^^ - 2) / (^// y-'^ rZ'> 
 
 \dx ) dx^ 
 
 8! 
 
 + 
 
 ;K/i - 1 )(yi - -iX/t - 8) / dy \ "-y r/'-z/X^ 
 
 2« 
 
 (d^Y'ViriiX 
 
 \dx) \d^'J , 
 
 _ n{n-\) /dyy-'-d^y 
 "-' 2! \dxj dx'' 
 
26 R- i'UJISAWA. 
 
 To find Xj observe that X, being a function of the differential 
 coefficients of y with respect to x only, is independent of u na a 
 function of y. Hence, putting u=y, y\...y" successively, we obtain 
 
 
 
 ^ = rr'X,-\r,ir- 1)//-%+ . . . + r ! X, . 
 
 \ ^-^ = »^r-'X,+n(n-l)y*-%+ +n! X. . 
 
 To obtain X,, X,,..., we may solve tliese e(juations as was done 
 by liertrand* ; but it is shorter to proceed as follows: — Put u=(^^, 
 then 
 
 c-*"-^^ = ;.Xi+/%+...+/'X,+ ... + /i-X., 
 
 and, on the other hand, 
 
 fix" ~ rfx" "^ 2 ! dx" '*■ 3 ! dx" ■*" " " "^ vi ! ^" "^ " ' 
 
 Multiplying the last two equations together and equating the coefficients 
 of the like powers of A,, we obtain 
 
 * Bertrand, Traite de CaJcul Differential et de Calcul Integral, T. I, p. 139. 
 
MUT/ni'LIOATION OF KT-LIITir FUNC'I'IOXR. ^^ 
 
 
 (51) 
 
 
 /•! c/.r" (;-l;!l! Jr" ^(r-2)!2! da;" 
 
 ^^ ^ l!(r--l)! (?a-"' 
 
 
 ??. ! dx" {n-])\\\ dx""^ («-'2)!2! r/a-" 
 
 ^^ ^ l!(?^-l)!(7a•"• 
 
 It may be worth while to notice that the expression on the opposite 
 
 side oi Xr is identically zero for all values of r greater than n. 
 
 From the mode of derivation it is evident that X^ contains 
 
 1 d^iv^) 
 
 differential coefficients of ?/ but not y itself. Hence, if — ; ^ be 
 
 ' r ! dx" 
 
 broken u\) into two parts such that one part contains all the terms 
 
 independent of explicit // and the other part, terms having y as a factor, 
 
 then 
 
 1 d'HiD 
 X,. =: that part of -—^ • „ which is independent of explicit y 
 
 1 dHiD 
 and that part of —-^ ^ n which contains explicit y together with 
 
 V d^{y^^) y^ dnf-^) ^(^^n ?/"' ^"V 
 
 (r-l)!l! dx"" ^ (r-2)!2! dx"" "* ' ' \\{r-\)\ dx" 
 is identically zero. Now 
 
 1^ -^t^= coefficient of If in |,y + /,^ + _ -^,+ ., j . 
 
28 K FUJI SAW A. 
 
 1 d"(i/^) 
 Hence that ])art of -^-j — ^^ which i>< indeiKindeiit of exj)Iicit // is 
 
 r ! dx 
 equal to the coefficient of h" in 
 
 that is, in 
 
 r\ V dx "*" 2! dx''^'^ ' 
 
 ^|/(-r + /0-/(^)f, 
 
 where y =/(>) . 
 
 Thus we ohtain 
 
 (52) 
 
 '^^ = 7t[(7;^)1a^-^^')-/^],..:- 
 
 ,.   
 
 This form of A'^ has been obt'iinetl in u (litfeivnt manner l)y U. Meyer. 
 Bertrand gives the following form of A,, wliich is substantially the 
 same as (51) :** 
 
 r\ dx'' 
 
 where a is to l)e regarded as constant duriniif ditierentiation and is 
 afterwards to he rej)laced by //. Airain making* use of the form of 
 A',, ^iven by (51), we get 
 
 d"u _ y J_ dru d^ _ y 1 d^u d'y-^ 
 dx''~^^r\ dif dx"" ^^ ,4iO--l)! rZ?/ dx" 
 
 ^ ^ fTi,(r-2)\ dif dx" "-^^ '^ y di/^dx*' 
 
 * Grunert's Archiv der Matheniatik, Bd. IX. 
 ** Loc. cit. p. 140. 
 
MUT/I'TPLICATTOX OF ELLIP'['I(" FUNCTIONS. -}€) 
 
 wliicli !ii>Toe8 Nvitli tlie form of -7—- <>iven by K. H()t)i)e.* 
 
 Put // = //', then -jpr --== j>rf~'', wliere />,. denotes the continued 
 
 product of /• (|UJintitie.s p, p-l, })-'2,...(p-r-\- 1). Mnltij)Iyinire(iUM- 
 tions (51) by pjf"\ K/> — 1)//'"",-.. in order :ni(] nddini--, we obtain 
 
 ^''' dr" n\y"" I p-\' dx'' "^ p-2 2! dx'' "' 
 
 ^ ^ p-r r\ dx" ^■••^^ ' p-n dx") ' 
 For/> = -^, we have 
 
 ^KA^ ^"Vy _ (2-")! 1 \n, ,d-y jH y-r^ d-y' 
 
 ^ ^ (/j-» ~ 22"0<J)2 ?/»-*} 1 -^ <7.r» 2! 3 dx^'^" 
 
 +^ ^^ r\ir~\ dx"^--^^-^^ ^^;r-\-d^\- 
 
 If// 1x3 Ji rational intes^ral function of .r of the tliird deeree, -~- 
 
 ^ ' dx" 
 
 vanishes for all values of r for which 3?- < n. In this case, denoting by 
 
 / either —^ or the integer next above -7^ according ;is // is a multiple 
 
 of 8 or not, we have 
 
 cZV/7 &n^\ 1 \ lu ?y"-' d-y^ 
 
 ^^^ dx'^ ~ '2^"{n\f y'^'H' ' Z! 2/-1 r/.r"'^"" 
 
 Again 
 
 (56) ^^ = -:^ {(2?yr^Z,-(22/)"-%+ . . 
 
 + (-lr\ 1.8. ...(2r-3X2yr%+. ..+(-!)". 1.8. ...(2«-8)X„} , 
 
 * CieUe's Joiuual, Bel. XXXIII. 
 
30 R- FUJISAWA. 
 
 sind if// be a rational inte^i^ral function of the lliird degree, -j^ and 
 all the higher differential coefficients vanish, so that 
 
 X - coef of //" in ^ i// i-'^ + -I /.^i^ . 1 //« i?l/_ (" 
 
 or 
 
 X^, = coef. Of A' ,n -^^—^ | ^,^. + ^ A ^ + -^ /r-^} ; 
 
 whence; follows that A„_r vanishes for all values of r for which 
 /• > '2)1 — 'Ir, that is, Mr > :^//. Denoting the integral ]Kirt of -^ hv /, 
 \v(! liave 
 
 + (-lr.l.:i ...C2//-2r-8X'2//)%. .+ ... + l.a...(2n-8)X„| , 
 where ' 
 
 "•■ («-2;+l)!(r-'2)! 1! V/r / V 2 r/W V C, (7x7 
 
 1 /i?v Y"'' Yl ^ijvV" Yl '?!^Y4- ( 
 
 ■*" («. - 2r + 2;! (;• - 4)! 2! V ^/^ / V -^ ^•''V \ <i ^'-^ 7 "^ ' ) ' 
 
 the last term being 
 
 1 /^?/V ^/J^A^\' 
 
MULTIPLICA'I'IOX OF ELLIPTIC FUNCTIONS. 31 
 
 or 
 
 1 ('JjiS ' (1- '!ji\ fl. 'lii\~ 
 
 according as /• is even or^^odd. 
 Similarly 
 
 + (-l)M.3....(2/i-2/--lX%)'-X„_,+ ...+ 1.3. ...(2/i-l)X,.| , 
 where A„_,. has the same signiticntion as in (57). 
 
 Let / = .s^ = z{l-z){l-k^z) = ^-(l + A;V+A;V , 
 
 
 /s 
 
 1 #/ 
 ~ ^ dz^ ~ 
 
 Now write 
 
 
 
 (59) 
 
 
 1 r— Iv'-^ 
 
 ((30) 
 
 
 
 then by (57) nnd (5<S) 
 
 (61) S„ = (-1)U.8. ...(2yi-2i-3)(2/)*Z„_i+... 
 
 + (-l)M.3....(2;i-2r-3)(2/TZ„_,+ ... + 1.3... (2w-3)Z„ , 
 
32 R- FUJISAWA. 
 
 (02) Ii„ = (-l)U.3....(2«-2i-l)(2//Z„_j+... 
 
 + (-l)M.3....(2n-2r-lX2/)''Z„_,+ ... + 1.3. .(2n-\)Z„ , 
 where 
 
 /n-2r/r fn-ir-\f'r—if f n-ir—i/Tr-ifi 
 I ya , ./I J2 Ji J J \ Jt ys , 
 
 voo; ^ „_, ^^^ _ 2^j, ^, -r ^^j _ 2r + 1 )I(r - 2)! 1 ! "^ (n - 2r + 2)! (r - 4)! 2'. " " ' 
 
 the Iiist terui beinjr 
 
 -2!! I- 8r+I r-\ 
 
 yi yg _ yi ya/B 
 
 according iis /• is even or odd. 
 
 Observe tliut botli .S„ jind li„ arc ralioiml intc^^nd f'lmctii^n.s of . of (he 
 degree ''In. 
 
 § lo- 
 
 The determinant-expressions of § N, simj^lc as they may a])[x'4ir, 
 are not convenient for introducing iS'„ and JR„ of the la.st section. It 
 is desirable to derive the multiplication-formulie in forms slightly 
 different from those given in § 8. 
 
 Consider first the c;ise n odd. Referring to § :^, we might have 
 
 p 
 taken VQ, instead of 7'+(^,s and then we shall have in place of (5) 
 
 the following determinant : 
 
MULTIPLICATION OF ELLIPTIC FUNCTIONS. 
 
 33 
 
 ((54) 
 
 I y r2 
 
 /'Vl — l 
 
 r,m-l 
 
 1 
 
 c 
 
 
 
 6 ' 
 
 6' 
 
 6' 
 
 ■• ^ 
 
 1 
 
 z 
 
 z' 
 
 ^m+l 
 
 s ' 
 
 s ' 
 
 s ' 
 
 S 
 
 1 Z ^^2 ~tn+l 
 
 I, 2z, ... (w-lK-2 i)— , jD— , D—, ... D-^ — 
 
 ,s s s ,t 
 
 1 ^ ^2 ,mi: 
 
 (/;t- 1)! , Z)"'-i— , D"'-i— , D'"-'— , ... i)"* 1-^— 
 ^ ' ' s s s s 
 
 Now put 
 
 (C5) 
 
 1 ^ z^ y'" 
 
 £)»»—, !)'»—, D'»— , ...Z)'»-l- 
 
 s .s s .s 
 
 \ y y^ z 
 
 ,m+l 
 
 /^ = 
 
 -1 •) 
 
 _D2m_L X)2m^ D^mil , ^ , J)^l. 
 
 s s s s 
 
 
 »B+1 
 
 1 z 
 
 D* 
 
 yTO+l 
 
 1 ~ ~»n+l 
 
 _D2™_i_ D^—, ... D^"*^ — 
 s s s 
 
 and let the expansion of this determinant according to the elements 
 of the first row be written : 
 
 S S 8 
 
 We find 
 
 Vc 
 
 8m+l O 
 
 r « 2 = 
 
 ''^m+2 
 
 VT-:a-z) 2 = 
 
 2^ j2,+ i^,+ ... + /4H2 
 
 i^„ 
 
 VI -^'s (l-A^^) 2 = 
 
 2-ti i;?,;.2,«+2+ /^^^2«+ _ _ , ^. /^^^^^ ^ 
 
 i^„ 
 
:u 
 
 11. FUJISAWA. 
 
 and thence, // being odd, 
 
 m) 
 
 -— M' 
 
 en nu 
 
 dn^m = 
 
 -iL M' 
 
 -'■i- M ' 
 
 where 
 
 ((57) W 
 
 8 ' 
 J ! 8 
 
 1 I s 
 '2! « 
 
 m\ s 
 
 _JL_r)««+>_ 
 
 (w+l}! .s 
 
 
 (G8) M/ = 
 
 lis' 
 
 2 ! s ' 
 
 m! « 
 
 1 2 
 
 "(w+l)! s 
 
 _L_r)mj!_ __!__r)«+ij!_ ^ ram ^ 
 
 m! «'(m4-l)I .<? ' (2m)! s 
 
 (G9) Mo' = 
 
 (i^^)--!, (1-^)-., (\-zr-\ 
 
 0, 
 
 1 
 
 1 ! ^ s ' 
 
 
 ... 1 
 
 w! « 
 
 ■■(7«+l)I S 
 
 
 ^ . ^ ,^ ^_L__r)2mJ_ 
 
 (w— 1)! ,s ' 7;?,! .s '(m+D! <s '■' ('2w)! .s 
 
MULTTPLICATION OP ELLIPTIC FUNCTIONS. 
 
 35 
 
 (70) m;= 
 
 0—¥zr+\ k\i-k^zr, jc\\-r-zy'''\ ... /.-" 
 
 0, 
 
 ]_ 
 
 s' 
 
 5 ' 1 ! ^ .s ' 
 
 7n\ s 
 
 __i__r)m+l_L 
 "(7}l+[)\ S 
 
 t Dm-1_}_ _|_2)'»_L ___L__2)'«+1— __L_2)2»«_ 
 
 {m—l)\ s ' nil s ' {fu+ 1)! s ' '" {'Ainy. s 
 
 Observe that tlie expression of sn mi ns given by (P7) niid (68) is 
 that which hsjs been indicated by Jacobi. It wiJl, however, be more 
 convenient to write il// as follows : 
 
 (71) M[ 
 
 0, 
 
 J_ 
 
 s ' 
 
 1^ 
 
 s 
 
 1 ^1 
 
 z'"-\ ...(-1)'"+^ 
 
 ±dI -Ld-1 
 
 ! s ml s 
 
 1 ^.1 
 
 ir^T' IT^V' 
 
 .])m+lj_ 
 
 •■■(w+l)! .9 
 
 1 111 
 
 im-l)l 
 
 s ml s {1)1+ i) 
 
 
 (2w)! 
 
 We may here point out that the comparison of the preceding 
 formulae with the corresponding expressions of § 8, gives some elegant 
 theorems in determinants. Take for exam[)le 3/ and M'. We fijid 
 easily 
 
 M=Cs'-"''^M', 
 
 where C denotes a constant which m;iy be a function of in. To 
 determine C we may take some particular values (jf ///. For m = 1, 
 we see at once that 
 
 _1_ 
 
 •2! 
 
 D'-s = -s' 
 
 D 
 
 1 
 
 s 21 ft 
 
36 
 
 R. FUJISAWA. 
 
 For 111 = 2, we have 
 
 o|. lo4 
 
 .s- ' 2! .<? ' 8! .s 
 2! s' 3! s' 4\^ s 
 
 4. 
 
 2 
 s 
 
 1^1 1 J_ 
 
 s .s ' .s 2! 
 
 1 ±nJ i 1 
 
 s 2! « ' « • 8 
 
 2' .s- 
 
 
 
 1 j_ 2 
 
 1 1 J^ 
 
 s' s 4! 
 
 1 1 r^ol 
 
 21 .s \ s / s .V. s s 21 .V 
 
 s 2! s 5 2: s s 1 : ,s \ 2 8 / 
 
 Multiply the first row hy 1)— and suhUact the prodiK-t from the -se- 
 cond row, arid then, niultj])ly tlic first ooluuiii hy J) :1 subtract 
 the product from the .seeoiid column ; and, observing 
 
 1 / 1 \2 1 1 
 
 d ! \ •^' / .s- 2 1 s Sis 
 
 +-4-"7-3!^T(T!^'7)l-4r^*y' 
 
 we get 
 
 
 
 
 " 
 
 
 
 
 
 i , ^'s 
 
 ! , I>'.s 
 
 
 
 2 I 
 
 8! 
 
 = .s-5 
 
 
 1 
 
 1 
 
 
 
 o.^"*% 
 
 !,D*s 
 
 
 
 8! 
 
 4! 
 
 
 1 _Lz)i _Ld^1 
 
 .s ' i ! s ' 2 ! s 
 
 — -D— -—D-— — -D'— 
 1 ! s ' 2 ! .s ' 8 ! s 
 
 JL2)2j — L^^^JL _L2)*_L 
 
 2 ! .s ' 8 ! i- ' 4 ! s 
 
MULTIPLICATION OF ELLIPTIC FUNCTIONS. 
 
 Thus we find C = (— l)"" Hnd thence 
 
 37 
 
 (72) 
 
 —D\'i 
 2 ! ' 
 
 — i r)"»+is 
 
 
 = (— ]y",<;2"» + l 
 
 .9 ' 
 
 1 ! .<? ' 
 
 1 ! .s ' 2 
 
 ! .s' 
 
 
   (w+1)! s 
 
 w! -s ' (w+1)! .<?'■■■ (2m)! .s 
 
 which identity may also be proved directly in the manner exemplified 
 by the particuhir case m=2. 
 
 § 11- 
 
 Consider next the case n even. Referring to § (i, if we take 
 
 p 
 
 \-Q instead of 7^+^,5, we shall have in place of (80) the following 
 
 determinant : 
 
38 
 
 im 
 
 
 
 
 R. FUJISAWA. 
 
 
 I.e. 
 
 
 
 r 
 
 
 rm+l 
 
 1, z, 
 
 ^,. 
 
 
 
 
 ^«.l 
 
 .S 
 
 1, 
 
 2z,. 
 
 . {m- 
 
 l)z-~\D^, 
 
 -^.   
 
 .... D ^ 
 
 
 
 {vi- 
 
 -1)!, D->-f, 
 
 2* 
 
 
 
 
 
 ^-T- 
 
 -f • 
 
 D- ' 
 
 s 
 
 
 
 
 J? 
 
 «2 
 
 .s- 
 
 
 
 
 
 2* 
 
 
 =0, 
 
 
 Now 
 
 VC^"'=7J 
 
 a 
 
 A/r-c(i--?r= 
 
 1M + 1 
 
 P« 
 
 + 1 
 
 
 m+l 
 
 and thence, ;i being even, 
 
 snmi = ( — 1)"* ^:* 
 
 
 (74) . cnnw = (I--?)"* ^, 
 I dim?. =(1-A^^)-^ ^, 
 
MULTIPLICATION OF ELLiri'lC FUNCTIONS. 
 
 39 
 
 where 
 
 [ir>) N': 
 
 1 ! .s ' 
 
 — D* — 
 '2 ! s ' 
 
 
 
 
 ! D-+IJL 
 
 {m+\)l s 
 
 1 ^ 
 
 ■('2«t-J)I s 
 
 0, 
 1 
 
 (76) iv;= 
 
 1 ! s ' 
 
 1 J)ni-ll_ 
 
 (7)1 - I ) ! s 
 
 (?/i— I) ! s ' 1)1 ! .9 
 
 ^ jyin 1 _L 
 
 ('2w-l)! s 
 
 (1-^)'", (1-^)" 
 
 (77) n:= 
 
 1 1 s 
 
 -1-D — 
 
 1 ! s ' 
 
 _1_ 2)2 — 
 
 2 ! s ' 
 
 rD" 
 
 m 1 
 1 
 
 (w+1)! 
 
 J)m+\J_ 
 
 1 
 
 (w-1)! 
 
 D" 
 
 
 1 Z 
 
 -^ 2)2'» -1 _ 
 ^27;t — i ) ! s 
 
 (1-A;2^)"', k^il-Jc'zr-K 
 
 (78) iv;: 
 
 1 ! s 
 
 — D- 
 1 ! s 
 
 — D2 — 
 
 2! s 
 
 m ! s 
 
 ^ r)m+i_L 
 (?/i+J)! s 
 
 1 Z \ 7' 
 
 1 D*»-i— — - D'" — - 
 
 (m— 1) ! s ' 7?i! ,s 
 
 1 
 
 D^" 
 
 (2m- 1)! s 
 
40 
 
 11. FUJISAWA. 
 
 iV', N'-i, N\miiy also be written as follows 
 
 (79) iV'- 
 
 z, — , 
 
 i ! s ' 
 
 
 2 ! .<? ' 
 
 2! 
 
 8! ^ s 
 
 (-ir, -^D- ^ 
 
 ^ r)m+i_ 
 
 7/t ! s 
 
 1 1 
 
 ! !>*«->— 
 
 CJwi-l)! s 
 
 ^80) ;y^;= 
 
 0, (i-^r, 
 
 (I --?)"•-', 1 
 
 s ' (m-1)! 
 
 v«-l 
 
 -Ld-L 
 
 1 ! .s- ' 
 
 (-ir+s 
 
 1 
 
 (7;i- 1) ! 
 
 D--^— , -Ld 
 
 jD*-»- 
 
 1 £)«_L 
 
 in 
 
 
 (81) n;== 
 
 0, 
 
 —z' 
 
 (i-k'zr, k\\-Pzr-\ 
 
 0, 
 
 1 ! J? 
 
 -1- D-1 
 
 (?/i— I)! .s 
 
 ?;t ! s 
 
 {-ir^\ 
 
 ^ r>m-i ^ ^ r)>i» _L ^ 7)2m-l_L 
 
 {m- 1) ! s ' w ! s ' (2w- ])! .s 
 
 § 1-^. 
 
 We are now in a position to introduce 7i„ and (S'„ into the mulfi- 
 plication-formulie. In some cases, we might have introduced <S„ 
 rather than R^ ; but, for the sake of uniformity, we have chosen those 
 
MULTIPLIOATION OF ELLIPTIC FUNCTIONS. 
 
 41 
 
 ibrin-s which arc cxj;rcssibJe by /i*„. On introducing 7/„, it will be 
 seen that there comes out ji certain power ofs as ;i factor in the numera- 
 tor as well as in the denominator, Avhich may be cancelled against 
 each other, lejiving rational integral functions of V? — sn it both in 
 the numerator and the denominator. The final results are: 
 
 n (xld, )ii = — , 
 
 (8'2) anna =(-l)"V^ 
 
 1, ±l-zA-l'z, m-^.^-l^^?, ...C2.1-^.l-^2^)'"+i 
 0, 1, Bi, ... B^ 
 
 1, Bu 
 
 B2, 
 
 B 
 
 m+l 
 
 ■"m-l> -"'>«> 
 
 B,H+1, 
 
 Bo 
 
 , ■*■ 
 
 (88) on mt ^'sJV — z 
 
 1, -%-i.\-l^z, (XzX-Vzf, ...(-2.,^.l-^2^)'»+l 
 0, 1, B^ ... B„, 
 
 \, Bi, i?2, ... B,„,i 
 
 B,n-l, B,n, 
 
 Bm+lf 
 
 Bi, 
 
 (84) dn;m ^V^ — k^^z 
 
 1 , -'2.k-z. l-z, {^2.Pz. 1-^A . . . {-2.k^z. l-zr^'^ 
 
 0, 1, Bi, ... B„t 
 
 1, Bi, B2, ... B„^^l 
 
 Bm-lJ B„ 
 
 B 
 
 »»+i» 
 
 Bo 
 
 («5) 
 
 denom. = 
 
 1, ill, ••• B^ 
 
 Bi, Bn, ...B,„^i 
 Bm^ B,n^l, ... il2„i 
 
42 
 
 fa. FUJISAVVA. 
 
 u even, m — 
 
 n-l 
 
 (86) sn?m = (- l)""2.s 
 
 0, ],...i2«_i 
 
 i , Hi, . . . M,n 
 
 , -^ 
 
 (87) cii nu — 
 
 2.1 
 
 0, 1, -2.zA-hh, ... {-2.z.\-k^zr 
 
 1, 0, 1, ... B^.i 
 
 i'ZA-zA-h'zr, K_i, H^, 
 
 ^8»i-l 
 
 (KS) till 7m — 
 
 0, 1, -±h^z.\-z, ... (-2.;fc«^.l-^)"» 
 
 1, 0, 1, ... B^_, 
 'lA-x.\~Jrz, I, n^, ... iJ^ 
 
 r2.i-^. I -/.•-'?)"•, iC-1, K. 
 
 ■'^S-. 1 
 
 (89) denom. = - 
 
 1, 1, Hi, •H'm-X 
 
 2.\-zA-r-z, Eu iVj, ... M^ 
 v2A-zA-k^zr, iC B^^u-R^i 
 
MULTrPLrr.vrroN' of HLTiii'i'K' rcvcTroNs. ^3 
 
 Pari Second. 
 
 To avoid mnfiision, we shall adopt once for all the following 
 notafiou : 
 
 Follow! iig Jacobi in his lectures, we write* 
 
 H iu) = 1 + 2*2 i- 1 )" g" cos '^, 0,{v) = 2"fgCWcos ^'^"~^?^^^ 
 
 ^i(«) = 2 2, ( - 1) ^ g^ - ^ sin ^ — 2Z ' -'^"^ = 1 + 2 2 g" cos -^; 
 
 nnd then 
 
 V /' !^ 1 w/ ^ ;i7-^, ^/ — cn u — Tj— -, —777- an u = - 
 
 
 Put (,)==— log q= ^—r- , tlien 
 
 (W ^ -^ir?^// 21 clK dti ^ 
 lid) K- rhi- K (ho rf/i ' 
 
 luit 
 
 rho 
 
 '"<^ /,ti-7,=/^Y -'■''"''■' ' 
 
 \^} 
 
 hence 
 
 (90) 
 
 Observe that the same difterential equation is s'ltisHcd by ^'1, ^2 <'ind ^, 
 
 «?r <xA; du dk 
 
 * Jacohi's gesauimelte Werke, Bd. T, pp. 501, 511, and 512. 
 •* Ditto p. :j59. 
 
 t Pitto p. 860. 
 
44 
 
 R. FUJTSAWA 
 
 Equation (90) may also ]ie wriften in the form 
 similarly ^ +(^^y+2H"'^ u ^°f + 2H-'^f* = . 
 
 ft ( \ 
 
 Subtracting the former equation from the latter and replacin<f 1 
 by c, we get 
 
 du- \ du / ( du dk ) du dl- 
 
 Now 
 
 Vk sa nu = ^/ , , J -j-, on nu = tt — r, V'"' Unuu = tt^ — t : 
 tf{nu) ^ K ff(vu) ^ tf{nii) 
 
 multiplying the numerator and denominators on the right-hand side 
 
 bv ^"'~'^^> 
 
 VA- 9>\\nu — -y-, jJ-jp cnuu -^ -y-, v/- dn ;/// = -f^. 
 
 where 
 
 (93) 
 
 V 
 
 ^«\?/) ' ^'~ th\u) 
 
 'a — 
 
 ^"» ' ' ^"'(70 
 
 Put /<« instead of?/ in equation (91), thus : — 
 <f?r \ <??/ / dk du dk 
 
MUT/rrPLTCATIOX op ELLIPTIC FUXOTIOXs. ^Ij 
 
 NOAV 
 
 loo Dytni) - log V + ;/- lojv n(„) - (^/^-l) log ^(0) . 
 Snhstitntiiig in (J>1), ^^■e \fQ\ 
 
 (04) !?5flS/ + (''i!^7+2,,'fe'5'''+7./-!5^',,i^ 
 
 al- '( au^ ah ) 
 
 . . r7log^(?^0 „^ ^ .,„- AiFF* 
 
 dK E—Jf'^K 
 (lift'creritiatino', and observini*; -yr- = — ^772 — ? we obtuiii 
 
 (In Kn 
 
 (9o) -^^^+211 —jj—--J,.nu. 
 
 Substitiifiiig- tliiis ill (5)4), 
 
 dir \ du / ( dv. dk ) -du 
 
 .dk 
 
 or 
 
 + 2n'-kk'^^+ 7?2(7?2 _ 1 )Fsn«w F- 0. 
 
 Introducing ?:=\/Fsn?< as the new indeixindcnt variable, and 
 observing (92), 
 
 * Jacobi's ^esammelte Werke, Pd. I, pp. 198 and 235. 
 
46 B. FUJISAWA. 
 
 Since 
 
 efjnation*(97) tnkes the form 
 
 r//.- 
 
 This eqiijitioii is also siitisficd In- I',. I'.,, and Is. 
 
 AVifh .I;i(()hi. wo may )»in /.■ + ,- = a and then efjuation (f).S) 
 Ix^conies 
 
 in wliicli form, tlie partial differential e<pmtion Ava« given for the first 
 time by Jacobi. 
 
 The alK)ve demonstmtion of Jacobi's partial differential equation 
 is substantially the same as the one given by l^riot and ]k)uquet* in 
 their well-known tresitise on elliptic functions, the only difference 
 being that, in their demonstration, the intermediary steps are conducted 
 by means of Weierstrass' function ,1/. 
 
 Ih-iot and l>ou«(uet give also the following forms of the pirtial 
 differential ecpiation '.- — 
 
 * JiOc. cit. p. 529. 
 
MULTIPLICATION OF liLLIFTli: FUNCTIONS. 
 
 V 
 
 47 
 
 (.00 (,-^V.C.)f-.(.-i<^V-.r)f 
 /T 1 
 
 where 3j = ^-^ cii « and ;" = ypr- dn/^. 
 
 § n- 
 
 We «li;iJI luive, by «iiital)Jy deteriiiiiiiug the coiistunts!, 
 
 / V^ = ^/Tx A{x-), A{0) - n, 
 
 V2 - J~VY^^B{x\ B{0) = 1, 
 
 V, ■■= -7j:,^/V^^U^C{x\ C(0) - J, 
 
 \ V = £>(2-^), D(0) = 1, 
 
 (102) 
 
 (108) 
 
 F2- JjpB{x\ B(0) = }, 
 
 k 
 J. 
 Vk 
 
 \ V - D{x\ 
 
 n = -^ CU% 
 
 C(0, = 1. 
 
 AO) - 1, 
 
 where J, B, C, JJ are ratioii:il ii»te<;ral f'uiictioiis oi x'. 
 
 n odd, 
 
 /I eveu, 
 
n ( 
 
 tjveii, 
 
 
 x^/\ 
 
 -xVl-^*^ 
 
 'U{x') 
 
 = 
 
 B(x') 
 D{xr 
 
 Gix') 
 
 
 48 11. FUJISAWA. 
 
 I>y division, we obtuin 
 
 n odd, 
 
 xA(x^) 
 mum = — ,. ., , snnu 
 
 ,,,,, ^/U^B{x^) 
 
 (104) en nil = — r., t, , ciwm = 
 
 £>(a*') 
 
 , 'vT^=IVC(x») , 
 
 ^1" '"* = S:^i) ' ^» »»" - ^(^- 
 
 When /t ia odd, J, />, C, 1) are all ol' llie degree m*— I in .», and 
 when n is even, .4 is of the deforce «'— 4 while A^ B, C are all of the 
 degree n'- in x. Moreover, A, llj C\ D are rational integral functions 
 of A;'- whose coefficients are integml numljers. The coefficients of tlie 
 highest power of u; in A, 7>, 6', D are res|jectively 
 
 It— 1 H* — I It*— 1 M*- 1 »t— I M*- 1 
 
 (-!)» k ^ , k ^ , k ^- , (-1) » ;t A- » , 
 or 
 
 (-1) « w A; 2 , ^>, A:», (_i)» ^s^ 
 
 accordinii" as // is odd or e\en. 
 Write 
 
 A - I'A^x*^ = 2'^'3jl-x»r =- 1'AIJ\-Px:'r, 
 (105) 
 
 = 1' G^x^- ^ 1' c:^a-xr - 1' aj-i-^-'^r, 
 
 D = I'D^x'- = 2- 1)^(1 -^r = Vi);^(l-AVr; 
 then, we have the well-known relations* 
 
 * Compare the work of Briot et Bouquet alreivtly referred to, or Baehr, .S»/' Ifx jonniiles 
 pour la tnitltiplicattOH des fonctions elUptiques de la premiere expilce, (irunert's Archiv der 
 Matheuiatik, Bd. XXXVI, pp. 125-176. 
 
MULTIPLICATION OF ELLIPTIC FUNCTIONS. 49 
 
 Observe that A.,,„, B.^„„ C\,,„, !).,„„ J',,„, h',,„, ( 1,„, V/,,,, arc integral functions 
 of k^ of the degree at most equal to m. 
 
 § 15. 
 
 Consider firat the case where n is an odd number, and put 
 n^~ \ =4p = Sq. 
 
 Introducing the new variable c — \/^a: in 
 
 l'D,„.x^' 
 
 we get 
 
 (107) VkHuim- ^-^2-' 
 
 where rt.,^, ti,„, are rational integral functions of a = ^ + ^-. 
 
 k 
 
 We may suppose .^ and B to be arranged according to the 
 
 powers of a, thus : — 
 
 (108) 
 
 D - VD^,^^^'« - yd,J'- ^ lBji:\ 
 
50 K. FUJISAVVA. 
 
 From the well known relations 
 
 (109) A{x, k)=(- 1)VV-^, ^W^ D{x, A)=(- \)'TAt-l-, A;W^ 
 which may jilso be written in the form 
 
 (110) A{a, ^)={- l)^'i)(«, i)c*^ D(«, c)=(- if^A^rx, ly*", 
 
 we see that A and i> are of the stime degree in a and tliat, moreover, 
 
 (111) ^Jc)=(-l)'^H„(l)c^ H^{^)={-\y^E„,(^jy''. 
 
 When the multiplicator // is required to be put in evidence, we 
 shall write A[_n'\ and !.>[//]. Now the terms containing the highest 
 power of a in J [8], .4 [5], J [7] are -2'c»a, 2«c"a», -2«c»««, and those 
 in D[3], D[5], -D[7] are 2»c''«, 2«c»V, 2»c«a«. By virtue of the relation 
 
 (112) J[/i+2]J[w-ii]=I>-[2]J-'[//]-(l -«c^+e*)^^'[-]^W» 
 
 which is easily deducible from the addition-equation, we conclude by 
 applying mathematical induction that, generally, the terms involving 
 the highest power of a in -4[m], D[m] are 
 
 (113) !?,«'=(- 1)'^2'^''"«", H,a' = 2'c'*"«», 
 
 w here ^« + A«„ = «* — 1 • 
 
 To find Kf I^m we deduce from (112) 
 
 ^»+s+/,. s = 2/^„-|-2 = 2w«-2^, 
 which may be written 
 
MULTIPLICATION OF ELLIPTIC FUNCTIONS. 5 { 
 
 or, jmtting for ji moment (^n— 77 + 1 ) — v''.. . 
 
 1 3 
 
 Observing ^i=-q-, V^=— 9-1 we find 
 
 </',= coef. of a;" in -^ (^pjp = ("1) ' |^ • 
 
 Hence, 
 
 n— 1 ^, ^,2 »-l 
 
 (114) ;,,^| + (_i)^|i_i. ,,,_|._(_i)-i-|., 
 
 that is, 
 
 or 
 
 1 1 
 
 '''/i = -2'"0' + 1 ) - 1 , f^n = -^n{n - 1 ) , 
 
 according as — ^-j — is odd or even. 
 
 This premised, we now substitute D=^H„/f in (99), which may 
 be written in the form 
 
 + 7^^7^2-l)?2F+8n2-^, 
 and obtain 
 
 (115) e2^-(,,2_i),^i^ + 2w^gff,=0, 
 
 (116) ,-2^?^i-(,,2_l),-^?^+ 27%-l)77,_i=(l +^f^"-2(n2_l),^^' 
 
52 R- FUJISAWA. 
 
 and, generally, 
 
 (117) ^^rL^(.n^^l)^I^^2nhnH^=U„, m = q-2, ^-8, ... 0, 
 
 where U^==a + ^*)^^-2in'-l)^^-^ + n\n'-l)^H^,, 
 
 It will be seen that equation (ll5) is satisfied by f***; indeed 
 we might have determined Un by means of this equation, for if 
 
 the operator c^ be denoted by <?, then tlic dilferential ecjuation may 
 
 be written 
 
 the genenil solution of which is 
 
 where C, (7" are arbitrary constants ; and, as H^ can contain only even 
 powers of ^, 6"=0, or C=(), according as — ^ is odd or even. 
 
 Let us now investigate eipiation (ll<>). For this purpose, it 
 will be convenient to distinguish two cases according as -^— is even 
 
 or odd. When ^-^ is even, //„=—»(» — 1) and jy,=2'c*"<"-", further 
 
 =2'-«|(7t-2)(?? - l>Kn+ 1)$*''("-^> 2+ („ _ l)w.(«+ 1X«+ 2)?*"t«-i)+J| ^ 
 so that equation (IHi) now assumes the form 
 
MULTIPLICATION OF ELLIPTIC FUNCTIONS. 53 
 
 Now, q being" equal to "7 , 
 
 The complementary function i.s thus 
 
 but, as 4'/(// — 3) is odd and //,_, contains only even powers off, C"=0. 
 Again, a ])articular integTa] is — 2p^"(?i—l )»?*"<"- 1)-2_ 2" =';?,(7t + ])f*'*("-i) ^2 
 which, together with the complementary function just found, gives 
 
 where /^j is an as-yet undetermined function of??. 
 
 n—} 
 When — ^ is odd, Ave find likewise 
 
 P>y virtue of (111), 
 
 z 
 We may put the multiplicator n in evidence by writing 
 
 and then equation (112) may be written 
 
 n+2 n— 2 T " T 2 T " " — 1 2 
 
 (118) v^^«-. Vj5;„«'»=(1-c^)2[v^^«-J _4(l-«,^+e^;|^VH,„«"'J, 
 whence, equating the coefficients of the second highest power of a, 
 
 «+2 «— 2 Ji— 2 n+2 r- n -|2 f" " ^2 
 
 (119) E E ,-{-E E .=a-$*f\ E -4(1+$*)^ 
 
 ^ ' qn+2 '?H-2-l (7».-2 gn+2-1 ^ ^^ L '^"J ^ '^ L^''J 
 
 • In <ln— I , 
 
54 R- PUJISAWA. 
 
 If ^^ be odd, so that -=^5 in even, then the term cont}iinin<( 
 
 the lowest power of f 
 
 m 
 
 n+2 n-2 
 
 is 
 
 2PH+2'/''^n--" + 6 
 
 
 »> 
 
 2P»-«/fe"'-"--, 
 
 <1 -f')' P,..]' 
 
 >» 
 
 227'»^"*-"-2^ 
 
 4(l+f')[H„J 
 
 M 
 
 _ 22p«+ 2 •n» + n 
 
 T> n 
 
 >» 
 
 2ft.+3/'f"*-« + 2 
 
 Equating the coefficients of tlie lowest power of f (that is $•'-»-') on 
 the two sides of equation (119), we get 
 
 Ti = 2^''""^''*-'=2 * , ^ odd. 
 or, writinjr " instejid of « + 2, 
 
 /j=2 « =2'^" =2 ^^" , — g— even. 
 
 Again, when —5— is even, so that - — 5 — is odd, the term con- 
 taining the highest ]X)wer off 
 
 m ^a A , 1 is 2''««r,r+«-^ 
 
 »— S w+S 11+2 > , . „ 
 
 (i-fy[i:,,,J „ 22p•'^'+"•*'^ 
 
MULTIPLICATION OF ELLIPTIC FUNCTIONS. 55 
 
 Equating- the coefficieuty of tlic Jiigliest power of-r (tluit i« ?"' + »*«) yn 
 the two sides of equation (11^), we obtain 
 
 /l=2-^"'-^^"-^='2 4 , 'l_i odd 
 
 or, writing n instead of /i + 2, 
 
 n (n-SXn+B) 
 
 ^_;)^('/»-l) "'-^ 
 
 r,=2 * =2^^^-^^^^'^--^', -1^ even. 
 
 Thus we find, whether —-r— is odd or even, 
 (120) ri=2^"~^=2^^^""^^ 
 
 and thence, when —-r— is even, 
 
 (121) i?g-i= -2''-\;i-l)Me*"<"-i)-2-2^-=*n(/i+ l)c*"^"-^^+H2^-'^*"("+8', 
 
 (122) E _i= 2*-2^*"<"-»'-i-2*-»;i(«+l)^*'''»+i'-3-2M;i-3)«^*"^"^^'"^\ 
 and, when — ^ is odd, 
 
 (123) H,,_i=2^-2c*"<''-«^-2^Mw+ l)l^*"<"+^^2_2i^-8(,;^_ 2),jfi«(u+i)+2^ 
 
 (124) E^_^=Q?^\)i- l)«|^4"("-i>-3+ 2''-37t(/i+ l)i4«(«-i)+i_2i^2^i«(/i+8)-i^ 
 
 Consider next i^^_2, whereby we suppose — ^ to be even. i/^_2 
 satisfies dilferential equation (117) 
 
 where 
 
 C/,,_2 = - 2^5(;i - 1 )/i(;i2 - M - 4)(n2 - /t - 6)l^*"<"-»^ " * - 2'^^n\)v' - 2)(yi2 - y ) Ji-f"-" 
 
 - "l^hiin + 1 )(?i2 + n^4)(7t2 + ;i-G)?^"<»-i)+* + '2J^*n{n + 3)(n2 + 8n^2)^*»<"+«-» 
 
 + 2'^%(w-3)(w'- 3?i- 2)c*"("+8>+2. 
 
56 »• FUJISAWA. 
 
 The complementary function is 
 
 «-VlTn H'-fVl7H 
 
 and, as i/y_2 can not contjiin irrational powers of ^, we must have 
 
 n— 1 
 C'=0, C"=0. Hence, — ^ being even, 
 
 (125) H,_s= 2^'(n- lMn'-?i-6)^*"<''-^>-*+2*^V-iiXM*-y)c'"*'--" 
 
 + 2^y« + l)(n»+;i-6)e*''<"^»+*-2'^*«(w+ 3)c*''<'*+*>-' 
 
 -2»-*;i(»*-3)c»"<-+»^+» 
 
 and thence, 
 
 (126) E,,_a = - 2^»n(w - 3)c*'<*-»>-»- 2^«n(n + 3)^*'«("-»)+» 
 
 + 2"-'/t(n + l)(n» + n- 0)$*"<"+>>-* + 2*^71'- 2)(w'- <j)^»-<''+«-i 
 + 2r\n - l)?i(/t«-n- 6)^»"<''+»+». 
 
 When — ^ is odd, we find likewise (or, more simply, by writing —n 
 in place of n in (125) and (!-()),) 
 
 (127) H,_ = - 2'^«M(M - B)^'"*"-"^"- 2'^*7t(« + bjc'-^'-'^-^' 
 
 + 2^V«+ 1)(m'+ ;t-G)c*'<"+^^ + 2'^ V-2)^/t'-9)c*''"+" 
 
 (128) J5;,_,= -2*-'n(M- l)(7i«- n- 6)c»"<''-'^*- 2^ V'- 2)(n«- y)^*»("-»)-i 
 
 - 2'^''n{n - l)(n»+ n - o^*"*"-!'^ »+ 2'^^n{n + 3)c»"<*+»>-» 
 
 + 2'^«7i(»-3)^*"<"^«+^ 
 
 Next, consider ii^_3 which satisfies the differential equation 
 (/?-in(n+5))(<y-^(n-5)) i/,_3= b\-,, 
 where 
 
MULTIPLICATION OF ELLIPTIC FUNCTIONS. 
 
 57 
 
 When — n— J*^ even, 
 
 (/,_, = S^-'ln - 1 )7i(n - 'S){n + 2)(«2 - n - 8)(;r -n-l 0)c """ -^>-« 
 
 + 2''-''(» - 1 )n{n - 'S){Sn + 2){n* + 2»''- 21 n'- 42«, + G())f>"("-"-2 
 + 2?-»m(71 + 1X» + 3)(3/i- 2)(n*- 2«»- 2br + 42»+ G0)c»«(''-i'+2 
 + 2'-»'/i(« + 1 ){n + 3)(n - 2)(7i2 +,i- H)(:,v' +n-l 0)c*"^"-^ '+« 
 
 - 2*^ Vn + 3)(?i + 4)(n - 1 )(n2 + Sn - G)^«'<"+8»-* 
 
 - 2'^^?i\n* - 1 9;*^ + 98)?*''< "+«^ 
 
 -2'^'/i(7i- 3)(n- 4)(;i + 1 )(«.2- 3» - ^)^Mn+»)+*, 
 
 A particular integral may easily 1)e found. The complementary 
 function is /gf*"'"-®', where f^ k ii function of//. Hence Ave get, '-^^ 
 being even, 
 
 (129) H,_,= rg^^'^-s) 
 
 - 2^i«3-^?i('i - 1 )('i' - 'i - S){n- - 11 - I0)c^"<"-"- " 
 
 - 2^M/i- 3)(w*+ 2;i«- 21n-- 42« + (JO)^*"!"-^'-^ 
 
 - 2"-i"«(m + 3)(n*- 2/t«- 21«- + 42/«. + G0)^^"<''-i'+2 
 
 - 2"~^^3-hi{?i + 1 )(7^2 + n - ^){n^ + n-l 0)^*''<«-i > +« 
 + 2*-''?Kn + 3)(n2 + Sn - G)6*"<"+»^-* 
 
 + 2^\n*-- I9;i2+ 98)?*"*"+»^ 
 
 + 2^9w(n- •d){n--Hn - C)«^i"<''+»'+S 
 
 (130) E,.,3 = 2"''n{:n - SKh' - 3n - (3)e*"'"-«>-* 
 
 + 2" \«*- h)n^+ iJ8)e*"<"-''>-^ 
 
 4- 2''-''yi(vj + 3)(;r + 3?t - 6)^i«("-3)+3 
 
 _2'^i«3-^/i(n+ l)(/i2+M_8)(;i2+;i_ iO)f5'«o.+i)-7 
 
 - 2*^i«n(/i + 3)(yi* - 2n»- 2l/r + 42« + 60)l^«"<"+^)-» 
 
 - '2r-^Mn-3){n*+ 2w»- 21 ;r- 42yi + GO)?*""*+"-i 
 _2^W3-V';t- l)(n"-n-8)(/r-;i- I0)c'"<"+^^^'^ 
 
 n— 1 
 
 When — ^ is odd, we get, by writing —n instead of /< in (ll^l'), 
 
 ^^rXsT 
 
 OF THE 
 
 UNIVERSITY 
 
 OF 
 
 LtlF.C 
 
5^ K. FUJISAWA. 
 
 (131) H,_3= 2''-»«(w-3)(/i'-3n-G)e»"<'^»>-* 
 
 + 2'^n{7i + 3)(?i^ + '6n - 6)f *»<'^«+* 
 
 _ ^p-^o^-hiin 4- 1 )(n2 + 71 - 8)(;r + /< - 1 ())et"<»+i>-« 
 
 - 2'^i»n(n + 3)(n* - 2/t» + 2bt- + 42 w + 60)c""»+*>-« 
 
 - 2r-^7jin - 'S){n* + 2yi» - 21 w«- 42/t + 60)e»"<-+"+» 
 _^-ioQ-in(n- lXM»-n-8)(n*-n- 10)^'"<-+»+« 
 
 and thence, by virtue of (1 1 1), 
 
 (132) E,_8=-r,e»"'"-^^^ 
 
 + 2^"3-^»(7i- l):n«- n - 8)(/t»- /t- lO)^'"*--"-' 
 + 2'^^»n(7i - 3)(n* + 2n»- 21««- 42/t + 60)e»"<''->'-» 
 + 2'^»Vn+ 3X«* - 27i« + 21 n'+ 42n + 0())c*"<*-*»^* 
 + 2^^"3-'7i(/t + 1 )(/t' + » - 8)(?t» + /i - 1 0)c*"''-^'+» 
 
 - 2^»»(« + 3)(;i= + 3« - 6)$»"<"+»>^ 
 -2''-Vt*- I9;i*+ 98)c»"<"+">-^ 
 -2'^»7i(n-3)(7i'-37i-6)^»»<*+»>+». -; 
 
 Equating the coefficients of a * on the two .sides of (11'^), we 
 obtain 
 
 11+2 n— 2 n+2 n— 2 h+2 «— 2 n+2 n— 2 
 
 = (l-f')'(2£,i,._, + i,._,E„_,) 
 
 When — ^ is even, the lowest power of ^ in the above equation is 
 $"*"'""-. Equating the coefficients of this jjower of f on the two sides, 
 we ^et r3 2P''-2= 2^^P'^-^K whence 
 
MULTIPLICATION OF ELLIPTIC FUNCTIONS. 59 
 
 I «^ — ^ * , — PT— even 
 
 or, writing // in place of m+2, 
 
 n 
 
 In like manner, we may shew that /^ has the same vakie in the case 
 
 where -^^ is even. Thus, whether ;/ l)e odd or even, 
 
 'ill 
 
 (133) r8 = 22<»-»). 
 
 In determining H^-^, every time 8//+1 is of the form r^, where r 
 
 n 
 
 denotes an integer, there comes in the term r^^'"^"*'\ Indeed all the 
 terms of H. and E may readily be expressed in terms of f. On the 
 other hand, t\ may be determined by means of (11<S), as has been 
 actually done in the cases fi = \, 3 ; but when /j, is a large number, 
 this becomes very laborious. N^ow in the series 
 
 r r r r r r a' 
 
 ^ 0» ' 1> ' 3> '' 6» ' 10> ^ !;■)> ' 21> > 
 
 To = 22", Ti = 22(''-i), Tg == 22<^-«5, 
 
 so that most probably l\, = 2^^''"''^; but I have not thus far succeeded 
 in proving this generally. 
 
 For a given particular value of /i, the constants r may be 
 determined in a different manner. For example, take the case w = 9. 
 
 Sin 9a; = a'(9-l20a;2+432a;*-576a;«+256a-«). 
 Again E^_^ contains the term /;g*"(«-7)-i jj^^j jj^ ^^^ <-}^g ^^^.jj^ /"^^^4.a..-9)^ 
 If we put fc=0, a becomes infinite but in the same manner as -77. 
 
 K 
 
 9 9 
 
 Thus sin 9a; contains the term t\^^^ whence follows /g = 250. 
 Obviously /io=l. 
 
60 K- FUJISAWA. 
 
 The elliptic functions of ;m for u = 2, 3, 4, 5, 0, 7, 8 have Ijeen 
 calculated by Haehr and othern* by the primitive method of .succes- 
 sively applying the addition-e<| nation. Having found the values 
 of /', sn 9u may be calculated by the above method without much 
 difficulty. 
 
 § 16. 
 
 As regards sn ???/, n being odd, the analysis contained in the 
 preceding section, whereby the variables are taken to be ^k sn v and a, 
 leaves nothing to be desired ; yet for the other functions, this is not 
 the case, and it is l^etter to have as the variables sn?/ and k. For the 
 sake of uniformity, therefore, we shall once more invostiL'^nte sn mi, but 
 this time consider it as a function of sn« and k. 
 
 Changing the variable from c to x, ecjuation (OX) takes the form 
 
 (134) |^-(l + A^)r2+^^J4|^+|((2,^«_l)F_l)J.~2(n'-lU•V^^ 
 
 ajr- ax 
 
 + 2n2A-(l -A-^-^+nV-l)A:'j'' F = 0, 
 
 and the e(|uation is satisfied by the numenitors and denominator of 
 
 ^/k sn n?/, Jrj-, en w?/, —7= dn nv. 
 
 The numerator of sn «w, that is, xA(x-), satisfies the differential 
 equation 
 
 * See the pnper of Biielir already referred to, and Proceedinsfs of the Royal Society of 
 London, Vol. XXXIII (1882) pp. 180-489. 
 
MULTIPLICATION OF ELLIPTIC FUNCTIONS. (^^ 
 
 (1:^0) 1 1 -(1 +AV+/'''-n'-T^+ {{{2n'-\)P-])x-^2{i,'- l)/rV}^ 
 
 Wo inny suppose the nuinemtor of sn tiii to l)e arrnno^ed according 
 to the ix)wers of A;- and write 
 
 0S6) xA{x') = 2p^k''". 
 
 Then, denoting the operator x-j- by '>, we have 
 
 (1 87) '^ + (;/\47H. + 1 ) - /y^Pg,,, =: ( (47« - 8)«2 _ 2>t'iJ + t^V\n-, 
 
 For iii=i), we have 
 
 and, if we put 
 
 then 
 
 ^i^"+{"'-'y-)p.-^o, 
 
 Pn = 2A-.^^^^S 
 
 ,9,(2r+l)2r = -^9,_i(»;_(2,._i)2^ 
 
 And, snice ^3, = n, i^r = (-^Y— ,,, . i\ , — ^^ — ', tlius w( 
 
 obtain the well-known series 
 
 For ?;/ = !, we have 
 
 ^'+(57i2_/^)P2=(n2-2n^+<:^Po+;rV-'?)(n*- 1 -'9)Po • 
 Since the lowest power of x in Pg is 3, we write 
 
62 
 
 R. PUJISAWA. 
 
 (139) 
 then 
 
 P2 = lrrr'^'\ 
 
 2r(2;-+ l)jv + (rm2_(2,-2- 1) ) ;'^„j 
 
 =(?i«-2n«(2r-l) + (2/--l)Vr-i-(«'-2/-+3)(n«-2r+2)/5^j 
 We find 
 
 Ti 
 
 = _!^(!^), „=?Kn»:^2(2n«-8), n= - ^^"'-^y-% (3n^-5), 
 
 3! 
 
 7! 
 
 GeneniJly, 
 
 2r(2r+ l)r,+(5««-(2;-- J )Vr-i 
 
 (4;--l0r+5>;i« 
 
 (2r— 1) ! ' 
 
 + (2r-3)(2r-l) 
 / - Km'- (2r - 1 ) ) (n«- (2r - 3)') 
 + (5n«-(2r-l)«)(r-l) 
 
 ((r-l)n«-(2r-3)) 
 
 _ _ ,„_i »(7?«-l)...(n»-(2r-5)«) 
 -^ '^ (2/-1)! 
 
 )•■ u/'-( 
 C2;-l)! 
 
 + (5n>-(2r-l)'X- lr» "^"'~ ^!;, ^^^'-;^^'-^)'> (.- 1) 
 
 ((r-lK-(2r-3)), 
 
 or. 
 
 2H2r+l)[r,-(-l)- "<'''-^y;'-f'-'^>'' Kn.'-(2r-l))] 
 
MULTIPLICATION OF ELLIPTIC FUNCTIONS. (53 
 
 whence follows 
 
 (140) r. = (-ir "'"'-'>"''-;'V,^f-'^-^>\ t.«--(2.-i)). 
 
 For m = 2, we have 
 
 Since the lowest power of a: in /^4 is 5, we write 
 
 (141) P^ = ^d,x''^'; 
 
 r-i 
 
 then, 2r(2/-+ l)^,+(9M2-(2r- l)Vr-i 
 
 =( (2r- 1)2- w2(4r- 7) ) rr-i-(/i'-2r+ 8)(?i2_2r+ 2) rr-2 , 
 whence we find 
 
 , _ n{nl-'[){fi'-9) 
 
 (142) ^ ^• 
 rl - _ Mn^Y^^ 6(?i«+85?i*-671nH945) , 
 
 9 
 ^s = ''^'''~y~'^^ 2(247n''+325w^-l3757;i'' + 28025). 
 
 Thus the general law is not obvious as in the case of r, Jmd it seems 
 to be impracticable to proceed further in this way. 
 Reverting to the formulae 
 
 »«=2p »»t=2p 
 
 since, by virtue of (109), 
 
 »-i 
 
 A — (—})Tr> I.-(2p-2»») T) —(_^\^T'J T.-(2p-2m) 
 
g4 R- FUJISAWA. 
 
 we need only determine either A or D. Let us take- D and apply 
 (134). We obtain X>o= 1, A=0, A^=(- 1)^«A:^ and, generally, 
 
 (143) (2w4- l)(2w + 2)Z)2,„+2+4w|(7^2-m)A;'-w}D»„+2w'A;(l -^2)^= 
 
 + {(n'-2m+lXw'-2m+2)}A:*Z)^...2=0 . 
 
 When m^p, 
 
 (144) D2,=I>2„..o(l + A*") + D,„.,(^'+A;*-')+... + Z)8«.v(^+A**-'^)+... , 
 
 the last term Ixiing Am-mA"* or Am,m i (A"'~^ + ^'"'"*^') according as m is 
 even or (xid. Substituting this in (113), we obt;iin 
 
 (2m+l)(2wi+2)|D^+,.o(l + A-*"-'')+£>j»+i,(^*+^'"-')+.-. 
 
 + Z),«+,,^(^-«'+A-«-«^+ )+...} 
 
 + 2n» |Dj^.o(* + 2w^-*'")+A'«.s(2^'+(2w-2)A^'' 2,^ 
 
 + !>»..*. (2^*" + (2wi - 2nA«"' ^0 +.. . 
 - 2n» |Z)»^.o(*+2w^*"+») + I>j^.2(2A-*+(2w-2)^*-)+ ... 
 
 + !>,„. 5, (2/A^'-^*+ (2;/i- 2r)A'--*'+') + . . . 
 + {n'—2m+ 1Xh=-2w+2){D,«.,.,. (A=+ A-='")+ Z>s„_s., (A*+ A-*-^)+ . . . 
 
 = 
 
 Hence 
 
 (2wi+l)(2w + 2)D„^+2.s = -4(/r-„i2)Dj^ J 
 
 (2w+l)(2m+2)Z)2^+j., = -4{{m-l)7i'-7n^\D,^,, -4(^2n'^m^)D^,^ , 
 
 -{n^-2m+ 1)(m'-27/i+2)Z)„,_2. J , 
 (2w+l)(2w + 2)^2^+0. 6 = -M{m-2),i'-m^\D^,^ -M3n'-m^D^,,, 
 
 -(,n^-2m+ l)(««-2m + 2)i)2„.-2,4 . 
 
MULTIPLICATION OP ELLIPTIC FUNCTION'S. 55 
 
 and, generally, 
 
 (145) (2m+l)(2wi+2)Z)2,„+,.2, = -4{{„i-r+l)n'-m'\D,„^,,_, 
 
 the last coefficient being given by 
 
 (2m+l)(2m+2)A„+2.„.+i = -H^J^n'-M']^D,„,,,., 
 
 -.in'-2m+l)(7i'-2m + 2)D,^_,^^., , 
 or 
 
 (2m+lX2m+2)A.+2.. = -2{im+2)7i'-2m'\D„„^_, 
 
 2»»-2, j»-2 > 
 
 according as m is odd or even. 
 By means of (145), we find 
 
 (146) 
 
 Do = 1, A = 0, A = -■^^^^2Jc^, 
 
 b ! 
 ^^_ ^^'(^^'- 1)( ^^^^- 4X/^^ - 9) 321 4(,,2_i6)(fe2 ^ ;^)^ I5(n^-4)(^H A«) } , 
 
 + 8(n2- 16X47^2- 185XA;*+ J<^)+ I5(45w*- 569^2+ l544);t«} , 
 
 + l6(/i2- l6X34/i*~ 9827^2+ 3300)(A;H ^^") 
 
 + (1549/t«-489257i*+357196/i2-8l5040X^+A-8)}. 
 
QQ R. FUJISAWA. 
 
 Again equation (143) muy be written in the form 
 
 (147) (2m+2)(2m+8)A;''Ap-2.-2+'^»'^(l-^')^^^^ 
 
 + (n»-27»)(7i«-2w+ l)Z)«^s^+, = . 
 Now, m being less than ;;, 
 
 (148) A,-2« = D,^^,,^^{Jc^'^+J^+D,^^,,^^^,(k*'^*'+J^^+... 
 the last term being 
 
 according as m is even or odd. 
 
 From (147) and (148), we obtain 
 
 (2m + 2)(2w+3)D,,_j,_5.^s,., + l7i»-(2m + l)'}D^^.,^,„ = 0, 
 (2w+2)(2m+3)D^_s^_s.s^,«+{5n»-(2m+l)'[D4^j^.,p_j«+, 
 + |(4w+ l)^i»-(2m+ 1)'[I>^»..^^ 
 
 + (n«-2wXn'-27n+l)D^^+,,,^^+, = , 
 (2w+2)(2m + 3)D^2^_j,s5^,^+, + iU;i'-(2;«+ ])'}D4p_s,.jp_2«,4 
 + |(4m-3)»'-(2m+l)«|D^5,,5sp_5„+, 
 
 + (7i«-27nXn»-2wj+ l)D4p_a«+j,sp_j^^4 = , 
 generally, 
 
 (149) (2m + 2)(2w+3)D,^2^_s,j,_s«+^, 
 
 + |(4r+l);i2-(2w+l)2}D,^j„.3^j^+s, 
 
 + |(4/;i-4r+5)««-(2w+l)^}D,p_,„,j,^j„+2,_, 
 
 + (?l' - 2w)(n'- 2m + l)D4p-2„,+2. 2^ -2m+2r = , 
 
MULT [PLICATION OF ELLIPTIC FUNCTIONS. QJ 
 
 and, lastly, 
 
 + |(2w+5)7l2-(2w+l)n Ap-2«..Sp-,„-2 
 
 + (n2-2m)(n2-2wi+l)D,^2„+2,2,_„ = , 
 or 
 
 (2m+2)(2w + 3)I),^2„_2,2p_^_i+2{(2w + 3y-(2m+l)21A;^.^,2^,„-i 
 
 according as m is even or odd. 
 
 From the above equations, we find 
 
 (150) 
 
 + 6(n2-25)(w2-49)(n2- 81)(6n-'- U){}£'+ k'") 
 
 - 3(l927n«- 33068n« - 962n* 
 
 + 1033308712-181 9125)(A;^ + A-«) 
 
 - 4(3046?i« - 38037??'' + 32799?i* 
 
 + 7086477^2- 1299375)^"} , 
 
 D,^io = - (- 1)""^ ,,(„'- W- 9) ;^2p-io |(,,2_ 25)(^2_ 49)(,,2_ 81)(1 + ¥') 
 
 + 5(7i2-25Xw2-49X5??2_9)(P+ ^ 
 - 2(2477i«+ 325?i*- 13757?^^+ 23625X/iH A") }, 
 
 A,^8 = (-1) ' g , ^ F^M0^^-9)(n^-25Xn2-49Xl + A:«) 
 
 + 4(?i2_ 9Xn2- 25X4?i2- 7XA-2+ A") 
 
 -6(?i"'+ 85n*-671w2+ 945)A^}, 
 
68 E. FUJISAWA. 
 
 ^4p-4 = (-1) ' r^ , A:^M(^^'-9)a + ^)+2(2n^-3)^}> 
 
 The first five a^eflicients were given by Jacobi himself. Tlie first 
 jind the last six coefficients of all the four functions have been found 
 by Baehr by a different method. 
 
 §17. 
 
 Let us now consider the rational integral functions of a:', J5, C 
 which enter into the numerators of en //7f and dn?iw, // Ixiinj; odd. 
 From the well-known relations 
 
 (151) B {X, Jf)==C (^-^, l^ l-'" x*p, C {X, I') = B (~^, l^ l-^ .rr 
 
 (152) B (I'x, ^) = C (x, k), C (kx, y ) = -B (x, k), 
 we deduce 
 
 (153) B (kx, x) = ^ (^ ' ^ **'' 
 whence 
 
 (154) B^^ik) = B^ (j) P", Bjjc) = B,^J^\^ k^. 
 
MULTIPLICATIOX OF ELLIPTIC FUNCTIONS. ^g 
 
 Hence, if we put 
 then 
 
 m-2p / 1 \ 
 
 so that we need only determine B^^ and this only for the initial 
 values of m in view of (154). 
 
 Now (134) may easily be modified in such a manner that the 
 resulting equation is satisfied by B. We find 
 
 (155) {l-(\ + k^x^-\-k'x*\-^+ { [{'2n'-l)k'-'S]x-2{n'-2)7c'x']^ 
 
 -{-'lim\-¥)^ + {n^-\)\l + {n'-2)k'x^\ B = 0, 
 whence follows 
 
 (156) (2m+l)(2w+2)B2„+2+[ [n'-{2m-\-m +4m{n'-m)¥}B,^ 
 
 + '2,n'k(i-k^^^ + {7i'-2m){n'^2ni+l)k'B,^_, = . 
 Further we find, for m < p, 
 
 (2w+l)(2w + 2)B2„+2,„ +[n2-(2m+l)2}B2„,,o = 0, 
 
 ^2«,o = , »i— 1 <: 27W < 2p, 
 
 {2m+l){2m+2)B^^,^,+l5n'-{2m+\f}B^^, 
 
 + 4m(?i2-w)B2„,o +{n'-2m){7i'-2m+l)B^_,^, = , 
 {2m+ 1X2^+2)^2^+2,, + {9n'-{2m+ lf}B„„, 
 
 + 4{{m-l})i'-m^}B^^, +{n'-2m)in'-2m + l)B^^,^, = 0, 
 
 generally, 
 
70 R FUJISAWA. 
 
 (175) (2m+l)(2m+2)B2„+2.,, +{(4r + lK-(2m+l)nB^ 0, 
 
 and, lastly, 
 
 . {(4w+l)n'~(2w+l)']B»..», +(n«-2mXn«-2m+l)B,^.,^ = . 
 
 By means of the above equjitions, we find 
 
 (158) 
 •Bo = 1, B, = — 
 
 2! ' 
 B, = ^ [(n«-9) + 2n»F}, 
 
 Bb = - '^^{^n«-9)(7J»-25)+6nV-9)A:'+8n«(n«-4)^}, 
 
 Bg = ^4rT^ (K-^K-25)(//»-49)+12n*(n«-9Xn'-25);fc« 
 
 + 4nV-4Xl5n'- 107)A:*+ 32nV-4Xn'-9)^} , 
 
 Bxo = - - — j^ — - {«'-25Xn«-49X«'-8l)+20nV-25X«'-49)A^ 
 
 + 1 2;i V- 4X297t«- 329)A^ + 'S2n\n*-4){Un^-S9)l(* 
 
 + 1287iV-4Xw*-16)A^}, 
 
 Bi2 = ^'''~22f~^^ {(««-25Xn'-49Xn»-8lX«*-121) 
 
 + 30wV-25Xn«-49Xn«- 81)^ 
 
 + 4?i^593n«- 1708271*+ 1 795 17n'- 482708)^* 
 
 + 871^(71* - 4X575W* - 101 1 ln» + 44276)^ 
 
 + 1927iV-4X»i*- 16Xl57i^-89)A:« 
 
 + 5l27i\n^ - 4X71* - 1 6X7** - 25)A-i°} . 
 
MULTIPLICATION' OP ELLIPTIC FUNCTIONS. 7^ 
 
 § 1«. 
 
 Consider next the case where n is even. To l)egin with, take 
 sn nu. Here D is exactly of the same form jis in the aise where n is 
 odd, so that we may restrict ourselves to the consideration of A alone. 
 
 Now A is of the degree n*— 4=4p say, and 
 
 (159) A {X, k) = (- 1)'"^' A (j^, k^ k^ x\ 
 and, therefore,' 
 
 (160) A^^ = (- 1)"^ A^ A;*^'-. 
 
 In consequence of (104) and (134), A = ^ -^i«-^^"* satisfies the 
 
 TO-O 
 
 differential equation 
 
 (161) {a:-(l + A;V+A;»x»}^2+ |2+ [(2n»-5)A;»-51a;»-2(;i5-4)A;V}^ 
 
 dx^ ' ^ dx 
 
 dA 
 dk 
 
 dA 
 + %i^k{l - k^x^ + {n^-4t){{i + k^x + (w^- 3)A;V }^ = , 
 
 whence follows, 
 
 (102) {2m + 2)(2m+'6)A^+i+ {;i»-4(w+ 1)H L(4/m+ l);r-4(;;t+ l)'^] k'lA 
 
 2m 
 
 fj4 
 + 2ii^k{i-k^)-^^+{n'-2m-2){ii^-2vi-l)k^A^_^ = 0. 
 
 By virtue of (IGO), we need only determine the first half of the 
 coefficients A^^. Again, ^2m is of the form 
 
 the last tenn l)ein<>" A^^,„k'" or ^2m.m-i ('>^"'^^ + ^"'^^), according as ?« is 
 even or odd. Substituting this in (162), we get 
 
72 K. PUJISAWA. 
 
 (2w+2)(2m+3)^2„+2.o+ |w'-4(m+l)«M^.o = , 
 
 *'"•""" ^ ^ ('2w+l)! 
 
 (2/^i+2)(2m+3)^2„+j., + |9n«-4(m+ 1)*M»,.4 
 
 + |(4w-3)n'-4(wi+ l)»f^»*., 
 
 + (ro'-2»t-2Xn»-2m-l)-4ft^_^, = , 
 
 (2m+2)(2w+3Mj«+,.e +{l3n*-4(m + l)»}^ft».« 
 
 + {(4w-7)n«-4(w+l)«[^H.,4 
 
 + (n»-2w-2Xn'-2m-l)^^_,,4 = 0, 
 
 generally, 
 
 (103) (27« + 2X2w+3Mj^+2.2r +i(-A/-+l)»'-4{m+l)'}^^.^ 
 
 4 |(4w-4r+r,^n*-4(m+l)»[^^.,.,_, 
 
 + (/i- - 2;;i — 2X/t' — 2wi — 1 )^ j^.,, ^_ 2 = , 
 and, lastly, 
 
 (2m+ 2X2w+ 3)^2^+2. «+i + 2 U2m + 3);t*-4(m+ !)-[ A.r^^ ^_, 
 
 + {n- - 2w - 2Xm- — '2^n — 1 )A 2^_2. „_x = , 
 or 
 
 (2wi+2X2m+3M2»+2.« + {(2'»+ l)n2-4(w+ ir[^2«,« 
 
 + (M^-2w-2X«^-2m- 1)^2^-2. „_2 = , 
 according as m is odd or even. 
 
MULTIPLICATION OP ELLIPTIC FUNCTIONS. 73 
 
 By means of the abo\e equiitions,'we find 
 
 (164) 
 A, = n, A,= -'^^^^{\ + n 
 
 A, = '^^^^rp^ K'*'- 16)('^--36) [(n--G4)(l + k')+M4n'-\'S){r-+ A«)l 
 
 -(5(n«+ iy6/i*-21147iH4752)A^|, 
 
 -2(247?i«-882yt«-7'2l02/i*+(36lll2/i2_l368000)(A*+A")'f, 
 
 ^"^^f"^^ |(;j2- l6)(yi2-a6)(M2-64X7i2- l00)(/i2- 144)(1 + k'^-) 
 
 + 6(yt-- lGX?r- 36X^2- 64): M^- 100X6?^'- 19XA;2+ A;i") 
 + (- 578bii°+ 170472?^«- 490 i40?i«- 22744752/1* 
 
 + I989868l6n^-383754240X^*+ A;**) 
 + 4(-30467i^'' + 75579n»-260532yi«-590l554;i* 
 
 + 48907368n- - 93493440)A;« | . 
 
 A,,= 
 
 § 19. 
 
 Lastly we consider the numerators of en nu and dn nu^ n being 
 even. In this case, put n^ = 4/y. 
 
 As in the case where n is odd, C.2„ik) = B^Jj-jk-'" by (106), so 
 
74 R- FUJISAWA. 
 
 that we may here also restrict ourselves to the consideration of B, 
 Moreover, since in this case 
 
 (165) B {X, k) = B (J^, A-) k'-" x*", 
 
 it follows, 
 
 (106) B,p-5„ = B^ A*-*», 
 
 and we need only determine the first half of the coefficients B.,^. 
 Now B satiiifies the differential equation 
 
 (167) {i-a + J^)x'+}i^x*\^,+ \[(^27i^-l)Jc'-l]x-'2{ii^-l)J^x^\^ 
 
 + 2n'A:(l-^)^+ {n\n*-l)J('x*+n^\ B = 0, 
 whence we deduce 
 
 (168) (2m+ l)(2m+2)Bj^+,+ |(w'-4m')+4wt(»'-w)A:»}Bj^ 
 
 +2n'A:(l-A:»)^^+(n»-2w+ \){7i^-2vi+2)k'B,^^^ = . 
 Substituting 
 
 B2m = B^n^o + Ssnt.S "^+ ••• + ^2»».2r k '+ ... + Bj^ jm ^"j 
 
 in (168), we get 
 
 (2m+lX'2m+2)B2«+j.o +i7i--^ni')B^,o = , 
 
 jBj^ = , w <: 'Zvi < 2p, 
 
 (2m + l)(2w + 2)B2„+2. 2 + (5n»- 4w')B2„, ^ 
 
 + 47»(«--wi)B2«,o +(n'-2m+l)(?i2_2m+2)Ba«_2.o = 0, 
 
MULTIPLICATION OF ELLIPTIC FUNCTIONS. 75 
 
 (2m + ] )(2 w + 2)B,„.+., , + (9«2 - 4m')B„„^ , 
 
 generally, 
 
 (1C9) {2m+]){2m + '2)B„„,,,,, + [{4r-\-}W-4m']B„„^^ 
 
 ^4{(,f,^r+\)n'-m']B,^^,^, 
 
 + in'-2m+ l){n^-2m+2)B^_,^^_, = , 
 
 jmd, lastly, 
 
 {2m+l){2m+2)B^^,^,„,^, + {{4m+ lW-4m'}B,^^„, 
 
 By means of the above equations, we find 
 
 (170) 
 Bo = I, -Kg = — "21' 
 
 ^8 = ^V-4) [(^2- 16X^2-36)+ I2(M^- IXn^- 16)A:^+ 
 
 + 4(w2- 1X15^2- 5 W + 32(71^- IXn^- 9)A:«} , 
 
 + 12(^2- l){7v'-9){29iv'- I04)k* 
 + 64(?v'- }){n^-9XIn^-22)k'^ 
 + 128(^2- lX;t'- 9)(w2- 16)^], 
 
76 
 
 li. FUJISAVVA. 
 "'%2^.'^^ ^^'*'~ 10XM--36)(7i2_64)(»i2_ 100) 
 
 + 4(>i*- lX593;i«- 14305m*+ 1 1 3\)72n^-25n60)k* 
 + 40(?i'-lXn2-9Xll5?^*-1283;i''+2968)A^ 
 + 2880(m2- lXn«-9Xw»- l6Xn'-8)A:« 
 + 5l2(n»- lXw'-9Xn'- l6Xn'-25)A:^']. 
 
 Im})erial University, Tokio, July 1893. 
 
 OF THE 
 
 UNIVERSITY 
 
 Of 
 
"^ 
 
 ^x 
 
 f 
 
 i 
 
 
 r^ , 
 
 ^\ 
 
 'N 
 
 N 
 
 
 Y ; 
 
 ./ 
 
 BERKELEY 
 
 THIS BOOK IS DUE ON THE LAST DATE 
 
 STAMPED BELOW 
 
 Books not returned on time are subject to a fine of 
 r i V^L ^"'"^e after the third day overdue, increasing 
 to *1.00 per volume after the sixth day. Books not in 
 Uemaud may be renewed if application is made before 
 expiration of loan period. 
 
 /^ J 
 
 U 
 
 kn 3 1926 
 
 REC'D Lip 
 
 DEC 18 1957 
 
 V 
 
 , c 
 
 4 
 
 7 
 
 V 
 
 / 
 
 
 V 
 
 r. 
 
 ■^m 
 
 ^1 
 
 ■:• 
 
 s 
 
 
 -^ 
 
■' ^ 
 
 / 
 
 l".^^ 
 
 
 } -A '7^1- 
 
 / 
 
 /^ 
 
 IT-- 
 
 \ 
 
 "4 
 
 C--' 
 
 I 
 
 A 
 
 \ 
 
 >J ^ 
 
 ^V- 
 
 / 
 
 \. 
 
 ^^ 
 
 ^ 
 
 V 
 
 t"- 
 
 % 
 
 < 
 
 > 
 
 .J 
 
 \ 
 
 \ 
 
 V 
 
 ,^ 
 
 ' \^ 
 
 7 « 
 
 r 
 
 V ^ 
 
 
 \ 
 
 >c .:^r 
 
r*^ 
 
 ^•% 
 
 tar 
 
 N^=. / 
 
 '^'