ox (J E. SAKAL 



.R = V'i-/r.r--' |1 - 12/;^/-'+ (507.2+ Ul')x'- (140 /c-+ 80/.'').r«+ (100 /r 

 + 335 ^.4) 3,H_ ((34 7,->4G4 A^ + 264 7.") x'"+ (208 Â'*+50R 7^«+ 208 Ji") x'- 

 - (264 Ä;'''+404 7.'"+G4 1'') j''^ + (335 7cH 160 Z.-i«) x''-{80 Ä-^+140 Ä-^") ^r^« 

 + (16 ^'^+50 7.'i").T-^-12 7.-i».T-- + 7c:^-^-*}, 



5f=l - 50 A- .7'' + (140 7r + 140 I') x''' - (160 Jc'+ 445 7.^+ 160 ¥} .t«+ (64 7.- 

 + 500 7,4+ 5(-o 7,G + 64 JA x''- (240 A'H 780 V'+ 240 7.;«) .r^-+ (360 !/■ 

 + 360 A-«) .T^'- 105 Z-" •r^'''- (80 7,''+80 ¥') .t^^ + (16 7i-' + 04 7.-i"+ 16 A'-; 2-' 

 -(20 Ä-^'' + 20 7/^-) ,r"+ 5 A'^- •t--'. 



To obtain P Q R S, P S ^vas first formed and the result multiplied by 

 Q and P. successively. P' was ,2:0t by multiplying P' by P twice in 

 succession. S^ was found bv squaring -S^ and. as a verification, they 

 were substituted in tlie well-known relation — 



,<?C.r,A) - P(-JJ' ^•)^-'^"'' ^''^^ 



where 4/)= ;r — 1, «being odd. P^S'- was obtained by multiplying 

 P^ and N^ and also l)v squaring PS. In every case, the result was 

 further verified by putting /.:-=:l and /.:' = /. 



Second I V, A, B, C, P) Avere calculated by a totally independent 

 method given bv Professor Fujisawa in his paper entitled " Researches 

 on the Multiplication of Elliptic Functions" (this Journal, vol. VI, 

 ]>p. 151-22(3). 



In the course of calculation, these functions were considered to be 

 arranged aceording to the powers of .r" as well as aecording to the 

 ]i()wers of /,'.- 'I'he i'ormula^ to he used in the former case are given in 

 the paper just alluded t<\ and the corresponding formulae in the latter 

 case are as follows : 



wlicre 



7 - o y 'm ■= r 



