296 History of the Theory of Numbers. [Chap, x 
where A and B denote the number of positive and negative terms respec- 
tively, not counting cr{0) =n as a term; 
n<T(n)+2{(n-2)(r(n-2) + (n-4)(r(n-4)l 
+3{(n-6)(r(n-6) + (n-8)(7(n-8) + (n-10)tr(n-10)} + ... 
= a{n) + {V-+3')\a{n-2)+a{n-4:)] 
+ (l-+3- + 5') {(r(n-6)+(r(n-8)+<r(n-10)l + ... (n odd). 
He reproduced his^ formulas for d{n) and E{n). He announced {ibid., p. 86) 
the completion of tables of the values of ^(n), T{n), (T{n) up to n = 3000, and 
inverse tables. 
Mobius^^ obtained certain results on the reversion of series which were 
combined by J. W. L. Glaisher^^ into the general theorem: Let a,h, ... 
be distinct primes; in terms of the undefined quantities e^, %,..., let e„ 
= e^V/ ... if n = a'^lP . . . , and let ei = 1. Then, if 
F(a:)=SeJ(a:"), 
where n ranges over all products of powers of a, 6, . . . , we have 
/(x)=S(-l)^6^(a;0, 
where v ranges over the numbers wdthout square factors and divisible by 
no prime other than a, 6, ... , while r is the number of the prime factors of v. 
Taking 
Glaisher obtained the formula of H. J. S. Smith^^ and 
aM -2aV,(^) +2a'fcV.(^^^ - . . . = 1. 
Using the same/, but taking €2 = 0, Cp = p\ when p is an odd prime, he proved 
that, if Ar{n) is the sum of the rth powers of the odd divisors of n, 
A,(„)-.A,©-f.A,(|) 
= or rf, 
according as n is even or odd. In the latter case, it reduces to Smith's. 
If A'r(^) is the sum of the rth powers of those di\'isors of n whose com- 
plementary divisors are odd, while Er{n) [or E'r{n)] is the excess of the sum 
of the rth powers of those divisors of n which [whose complementary divisors] 
are of the form 47?i + 1 over the sum of the rth powers of those divisors which 
[whose complementary divisors] are of the form 4772+3, 
A',(n)-2a'A',(^)+2a'6'A',(^^) - , . , =;, = ijl-(-l)-), 
A',(„)-2A',Q+2A-,(^)-...=n-, 
"Jour, fiir Math., 9, 1832, 105-123; Werke, 4, 591. 
"London, Ed. Dublin Phil. Mag., (5), 18, 1884, 518-540. 
