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.