MULTIPLICATION OF ELLIPTIC FUNCTIONS. 209 



»+2 iJ^±^_, n-l 



' 3 = ^ * , — T— even 



or, writing n in place of // + 2, 



n»-l 



n — 1 



/; = 2^-" = 2^^^-«), ^ odd. 



n 



In like manner, we may shew that /^ has the same value in the ease 

 where \^ is even. Thus, whether // he odd or even, 



(133) r^ = 22(' -^J. 



In determining Hq^ix, every time 8/^+1 is of the form /', where r 



n 



denotes an integer, there comes in the term r^ç-"^"*'\ Indeed all the 

 terms of H imd E may readily he expressed in terms of /'. On the 

 other hand, E^ may be determined hy means of (11<S), as has been 

 actually done in the cases /z = l, o ; but when p, is a large number, 

 this becomes very laborious. Now in the series 



r r r r r r t' 



r, = 2^'^ /; = 2'^''-^\ ri =^ 22(''-='>, 



so that most probably /;, = Q^n^i-i^). ]^^^^ j i^^^^g j-,q^ ^j-j^,g f.^^^ succeeded 

 in proving this generally. 



For a given particular value of //, the constants E may be 

 determined in a diiferent manner. For example, take the ca^e n = d. 



Sin 9x = œ{9-l20x^ + 4:32x*-51Qx^+2rDQx^). 

 Again E,^_e contains tlie term /;ç*"<" "-i j^^d iî,,_i„ the term Al""^"'^'. 

 If we put A:=0, a becomes infinite but in the same manner as -j- . 

 Thus sin 9a; contains the term f\x^^ whence follows /g = 256. 

 Obviously /w=l. 



