Formulae for snQzr. 



By 



0. Sudo. 



The folIoAving calculations of sn du were made to show the 

 advantao-e of the method of tindino; the multiplication-formula of 

 elliptic functions, given l)y Prof. Fujisawa in the second part of 

 his paper Bescarches on the Multiplication of EUiptic Functions (this 

 journal, vol. \l, pp. 151-226). For the notation adopted, as well as 

 for a full account of the method of calculation, reference is to be 

 made to that paper. 



Considering tlie nmnerator and denominator of sn du as ex- 

 pressed in terms of a ( = 1+ -- J and ^{^^s/h sn u), and putting /? = 9, 



(/ — 10 in the general formula^ for 7/^, lïf,_i .H',_2 i7,_o^ the vtdiies of 

 7^10, B^, Hg, H-, were found without much difficulty, and, thence, 

 by integrating ditferential equations (117) Qoc. cit. p. 202), H^, H^, 

 H^, H^, were successively obtained. (^n tlie other liand. H^ whicli 

 corresponds to H f)r /.•■^=— 1, was derived from the expressions of 

 sn 4 (h, i) and su 5 (u,i) by addition, and. then, l)y again making use of 

 the same ditferential equations, but this time in the reverse ordei*, 

 H^, H.J, Ho, were o-ot bv successive differentiations. The aoTeement of 

 tlie values of Hn deduced in two different ways verified the results. 



The value of E is at once obtained from that of H in virtue of 

 equation (111) (loc. cit. p. 200). 



