CHAP. IX] SUM OF FlVE OR MORE SQUARES. 315 



J. W. L. Glaisher 45 evaluated the number R w (ri) of representations of 

 n as a sum of t squares for each even integer t ^ 18. The simplest results 

 are 



the first two of which are due to Eisenstein 2 for n odd and to H. J. S. Smith 1 

 for any n. Here E r (n) [or E' r (n)~] is the excess of the sum of the rth powers 

 of the divisors of n which [or whose conjugates] are of the form 4k -f- 1 

 over the sum of the rth powers of the divisors of n which [or whose conju- 

 gates] are of the form 4k + 3; also, 



r r (n) = 2(- l)*- 1 ^, $r(n) = Z(- lY +d 'd r (dd f = n); 



while 4xi(n) is the sum of the fourth powers of all the complex numbers 

 having n as norm. In an addition to this paper, Glaisher 46 evaluated by 

 elliptic modular functions the sum of the rth powers of all primary com- 

 plex numbers of norm n and (p. 274) evaluated E (14) (n). 



W. Sierpinski 47 noted that the number of representations of n as a sum 

 of r squares is 



(2r) n I 1 1 1 



| oo(w) +-ai(ri) + - 2 a 2 (n) + - - J , 



where a,i(ri) is a polynomial of degree 2i with rational coefficients. 



G. Humbert 48 derived the formula, in which 771 = -fiTi(O), 0i = 0i(0), 

 in Jacobi's notations for elliptic functions of the variable q, 



(2) 4r?X + vie! = 4 Z (2m + 1) y +1/2 /(l + <? 2m+1 )- 



m=Q 



Let G p , g (a) be the number of decompositions of a into p + q squares of 

 which the first p are odd and the last q are even. By equating the coeffi- 

 cients of q N+llz in the two members of (2) and in the formula obtained by 

 changing q to q, we get 



4G 6 , 4 (4AT + 2) + G 2 , ,(4N + 2) = 4(- 1)"Z(- l)" l (2m + I) 4 , 

 50io. o(4# + 2) - 6G 6 , ,(4N + 2) + G 2 , 8 (4N + 2) = 42(- l) w (2m + I) 4 , 



the summations extending over the odd divisors 2m + 1 of 4N + 2. If 

 N is odd, N = 2M + 1, G w , o(4N + 2) is evidently zero. The preceding 

 equations give 



G 6 , ,(SM + 6) = G 2 , s(8M + 6) = fz(- l) m+l (2m + I) 4 . 



The total number of decompositions of SM + 6 into ten squares is 

 evidently 



. Jour. Math., 38, 1907, 1-62, 178-236, 289-351; summary in Proc. London Math. 

 Soc., (2), 5, 1907, 479-490. 



46 Quar. Jour. Math., 39, 1908, 266-300. 



47 Wiadomosci Mat., Warsaw, 11, 1907, 225-231. 



48 Comptes Rendua Paris, 144, 1907, 874-8. 



