Formulae for sn 10?/, on 10?^, dn lOii 

 in terms of sn u. 



By 



E. Sakai, Student, 



College of Seionee. Iniperiul Univovsity. 



Formiilfo lor sn 10?/, r-n 10?/, dn 10?/ in terms of ;r = sn 7/ nre 

 ofiven in the followinfr fonr tables, in which the mode of arrangement 

 will he apparent on inspection. Tliey were cah-nlated hv two entirely 

 di Heren t methods. 



I'^irstly. they wei-c dcdiieed from the well-known equations : 



S'-'2S-P- + JrP' 

 S'-lrP* ' 



S*-'2V-S'^P- + l-P' 

 Dix-) " S'--Jrr' 



where A. B, C, ]> denote alo-ebraic rational intearal functions of .r- and 

 P=.r {r, - (20 + 20 Z"-^) .r-'+ (10 + 94 7.2+ 16 V) x'- (SO A- + 80 V) x'- 105 7.' x' 

 + (8(-,0 Z-^+ :\m ]/') .7-1"- (240 7.-^+ 780 7.''+ 240 V) x^-+ ((U k*+ 560 7/' 

 + 5(i0 7.' + 64 7.h x^' - (100 Ä-« + 445 k' + 1 60 Z^") ./"' +(140 k' + 1 40 7,-^") x^' 

 ^:,Ok"'x-'+ l^ :>-'), 

 Ç:^^r^' -;i-12 ./■-+ (16 + 50 Z-) X*- (80 Z- + 140 7.^1 .7«+ (885 ZH 160 7/') x^ 

 - (264 k' + 464 Z'' + 6)4 Z'^) x^" + (208 k* + 508 Z'' + 208 Z-^) x^- - (64 k' 

 + 464 7.'ß+264 k^) .'ri^ + (l60 ZH335 Z-«) x^'-{\àO k' + HO k'"} x''+{r,0 k'" 

 + 16 k'^) x^' - 12 Z'^- .r-- + 7.-12 x'-*] , 



