332 HISTORY OP THE THEORY OP NUMBERS. [CHAP, xi

of d, and f(x, 5) if x divides d. Set A(s, y) = f(x, y) - /(y, a?). Then d' - d", 5' + 5") } - S(<f -

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(#, y) = — A(t/, x). Liouville4 employed in his sixth article two simultaneous partitions

2m = m' + m", m = mi + 2a*m2 (m's odd and > 0). Set mi = d^f, etc. Let F(x) be a function for which

F(0) = 0, F(- x) = - F(x). He stated that

(L) 2{Z2(-1)^-»*[1W^^

where d, <fe, d', d" range over the divisors of m, m\, m', m", and the first summation extends over the m' and m" whose sum is 2m. For F(x) = xt

so that there are f i(m) + 4B decompositions of 8m into s + 2<r, where s is the sum of the squares of four odd positive numbers and cr 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:

cf + d" + d'") + F(df - d!' - dm) - F(d' + d" - d"1}

( j - F(d' - d"

where the two members relate to the respective modes of partitions m = mr + m" + m'", m = mi + 2**m2.

For F(x) = x8 there results the formula

Hence if £ 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 cr is a sum of 4 odd squares, then

80 + H-A{fc(w) -&(*»)}• From (M) and (F), with/(s) = xF(x) is derived

V ; = 2(2m - 1 -

each relating to the single mode of partition m = mi + 2a'm*, m» = d,-5<.

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