CHAP. XI] LIOUVILLE'S SERIES OF EIGHTEEN ARTICLES. 333 



Liouville 5 remarked that if we multiply the members of (a) by x p , 

 where p = 2 a m, and sum for p = 2, 4, 6, , we get 



M 



which includes various formulas of the theory of elliptic functions. He 

 stated that it is easy to prove (a) and then deduce (a), and that he had 

 in his lectures at the College de France given a direct, elementary proof of 

 (a), based on Dirichlet 42 of Ch. VIII, the method applying to (b) and with 

 slight changes to the other formulas. 

 For any integer m, let 



(2) m = m' 2 + m", m" = 2*"d"b" > (d", 6" odd and > 0), 

 while m' may be negative. Then for F(- x) = - F(x), F(0) = 0, 



V) Z2(- 1K->F(2>V' + m') = 



10 if m 4= square. 



A discussion of the case F(x) = x shows that, if we set 



<r = fiW - 2fj(m - 1) + 2f l (w - 4) - 2^ - 9) + 2ri(w - 16) 



continued as long as the argument of f i is positive, then for m even, 



(m\ ( m 4\ ( m if m = square, 



<r ~ fl \2/ 2ri \~"/~ 1 if m * square, 

 while for m odd, 



/w-l\ f-^ I m if w = square, 



V~2~; + 2ri \~^~J~ 10 if m * square. 

 Using the same partitions of m and a function such that 



Liouville stated in his eighth article that 

 S2(- W-iffQfd" + m', b" - 2m') 



= or <T(Vw", 1) + *"( VS, 3) + - - - + ^(V^, 2V^ - 1), 

 according as m is not or is a square. As a special case, 

 P(TTI) - 2 P (m - 4) + 2p(w - 16) - = or (- l) (v " 1)/2 j/, v = 



For ^"(x, y) a function of a; only, (7) reduces to (0). 



Set &(x, y) = ( l) y / 2 F(x, y), so that F is an odd function with respect 

 to x and to y. Then (7) gives 



ZS(- l) (5 "- 1)/2 F(2 a "<r + m', 5" - 2m') 

 () = or (- 1)"+MF(^, 1) - F( Vro, 3) _ 



+ F(4m, 5) - ..- F(4m, 24m- 1)}, 

 according as m is not or is a square. 



8 Jour, de Math., (2), 4, 1859, 1-8, 72-80. Seventh and eighth articles. 



