332 HISTORY OF THE THEORY OF NUMBERS. [CHAP. XI 



of d, and f(x, 5) if x divides d. Set A(x, y) = f(x, y} - f(y, x). Then 

 S{SA(d' - d", b' + 5") } = 2(d - 1)A(0, d) 



', 2) + +A(,d- 1)}, 



where A (5, ?/) is to be suppressed from the final sum if y divides d. The 

 last formula is valid for any function A for which A(z, y) = A(y, x). 

 Liouville 4 employed in his sixth article two simultaneous partitions 



2m = ra' + m", m = m x + 2 a2 m 2 (m's odd and > 0). 

 Set mi = difii, etc. Let F(x) be a function for which 



f (0) = 0, F(- x) = - F(x). 

 He stated that 



(L) S{S2(-l)tf'- 1 >' 2 [^(d / +d'0+^^^ 



where d, d\, d', d" range over the divisors of m, mi, m r , m", and the first 

 summation extends over the m' and m" whose sum is 2m. For F(x) = x, 



2ri(m')p(w") = fiM + 4Sri(w!)p(m 2 ), 



so that there are $\(m) + 45 decompositions of 8m into s + 2<r, where s is 

 the sum of the squares of four odd positive numbers and a is the sum of the 

 squares of two such, while B is the number of decompositions of 4m into 

 s + 2V. 



For a like function F(x), another formula was stated: 



f + d" + d'"} + F(d' - d" - d'") - F(d' + d" - d'") 



( > - F(d' - d" + d'"}'}} = 



where the two members relate to the respective modes of partitions 

 m = m' + m" + m'" } m = mi + 2 a *m z . 



For F(x) = x 3 there results the formula 



Hence h (r is the number of decompositions of 4m into a sum of 12 odd 

 squares, and H that of 8m into s + 2V, where s is a sum of 8 odd squares 

 with s/8 odd, and a is a sum of 4 odd squares, then 



SG + H = 

 From (M) and (F), with/(x) = xF(x) is derived 



- l)F(d) 

 - F(d l + 2* 



= 2(2m - 1 - d*}F(d} + 82S(2 a!l - 



each relating to the single mode of partition m = mi + 2 a 'm 2 , m,- = 



4 Jour, de Math., (2), 3, 1858, 325-336. Sixth article. 



