310 HISTORY OF THE THEORY OF NUMBERS. Glaisher110 considered the set Gn{\f/(d\ xM, . - . } of the values of x(d), . . . when d ranges over all the divisors of n, and wrote — G(\f/t x> • • .) for G(-& -x, • • •)• By use of the f -function (12), he proved (p. 377) that the numbers given by all cancel if n is not a triangular number, but reduce to one 1, two 2's, three 3's, • . ., g 0's, each taken with the sign ( — )c"~1, if n is the 0th triangular number 0(0+l)/2. For example, if n = 6, whence 0 = 3, {1, 2, 3, 6} - {1, 5; 2, 6; 0, 4} + {1, 3; 2, 4; 0, 2; 3, 5; -1, 1} = {1,2,2,3,3,3}. Let \l/(d) be an odd function, so that $( — d)=—\l/(d}, and let Sr/(d) denote the sum of the values of /(d) when d ranges over the divisors of r. Then the above theorem implies that 2) } 3) } + . . . where 5 = 0 or 1 according as n is not or is of the form g(g+l)/2, and where $(d±i) is to be replaced by $(d+i) +\l/(d—i). Taking \l/(d)=dm, where m is odd, we obtain Glaisher Js108 recursion formula for crm(n), other forms of which are derived. For the function74 f 3, we derive + d2-22+32)f(n-6)-...} . . . +(-irV) or 0, according as n is of the form g(g+l)/2 or not. Next he proved a companion theorem to the first: ( 2d+5 \ r ( 2d+7 \ ^ all cancel if n is not a triangular number, but reduce to 1, 3, 5, . . ., 20 — 1, each taken with the sign (— )°, together with ( — l)0+l(2g+l) taken g times, if n is the 0th triangular number 0(0+l)/2. For example, if n = 6, 3, 5, 7, 131 r 5, is r 7, ni Hence if x(^) be any even function, so that x(~"^)=: Taking xW = km+1, where k and w are odd, we get Glaisher's109 formula. 110Proc. London Math. Soc., 22, 1890-1, 359-410. Results stated in London, Edinb., Dublin Phil. Mag., (5), 33, 1892, 54-61.