308 HISTORY OF THE THEORY OF NUMBERS. [CHAP. IX 



Liouville 12 stated that the number of representations of n = 2 a m (m odd) 

 as a sum of 10 squares is 



where X is the positive value of Z(dJ dl), where di ranges over the divisors 

 41 + 1 of n and d 3 over the divisors 4Z + 3 (X being the same for m as for n), 

 while ju is the number of integral solutions, positive, negative or zero, of 

 n = s 2 + s' 2 , and v is the sum of the products sV 2 for all the solutions. 



When m = 3 (mod 4), n = v = and the formula becomes that of 

 Eisenstein 2 if = 0, and that of Liouville 10 for a = 1. In the notation of 

 that paper, X = p 4 (ra). Thus 



N(2 a m, 10) = 



The last sum is multiplied by 4 when a is replaced by a + 1. Hence 



N(2* +l m, 10) + 4N(2 a m, 10) = {16 a+2 + 4(- I) (m - 1)/2 }p 4 (w). 

 The values of N, = N(2 a+2 m, 10, 4) and N 8 = N(2 a+2 m, 10, 8) follow from 

 2 4a p 4 (ra) = # 4 + 4tf 8 , 4(- I) (m - 1)/2 p 4 (w) = 5N(2*m, 10) - 96AT 4 + 256JV 8 , 



N 4 - 16^8 = |2(s 4 - 3s 2 s' 2 ) (s 2 + s' 2 = 2 a m). 



H. J. S. Smith 13 stated that the principles indicated in his paper enable 

 one to deduce by a uniform method the theorems of Jacobi, Eisenstein 

 and the numerous recent ones by Liouville on the representation of numbers 

 by a sum of four squares and other simple quadratic forms; also the 

 theorems of Jacobi 1 on six and eight squares. In view of Eisenstein's 

 remark that there is a single class of quadratic forms of discriminant unity 

 in n =i 8 variables, but always more than one class if n > 8, the series of 

 theorems relating to representation by sums of n squares ceases when 

 n > 8. There remain the cases n = 5, 7. Smith gave a description of the 

 general theory on which are based the formulas for the numbers A 7 ^ and N 7 

 of primitive representations of 4"co 2 6 as a sum of 5 and 7 squares, respec- 

 tively, where co is odd and 5 has no square factor : 



where, as in N 7 , the product extends over every prime dividing co but not 5, 

 while F 5 is defined as follows: For 5 = 1 (mod 4), 



(?)(-, 



= 1 \ / 



and 77 = 12 if 5 = 1 (mod 8); 77 = 28 or 20, if 5 = 5 (mod 8), according 

 as a. = or a > 0; while,* if 5 = 1, 77!! is to be replaced by 2. But, if 

 6 ^ 1 (mod 4), 



"Comptes Rendus Paris, 60, 1865, 1257; Jour, de Math., (2), 11, 1866, 1-8. 

 "Proc. Roy. Soc. London, 16, 1867, 207; Coll. Math. Papers, 1, 1894, 521. 

 * The ijS here used was replaced by 77!! in his 31 later paper giving proofs. 



