and 



viz. 



524 Mr. J. W. L. Grlaisher on Applications of 



of n, and a,b,c,... denote, as before, all the prime factors 

 of n. Thus, if n be even, A, B, C, . . . differ from a,b,c } ... 

 only by the omission of the prime factor 2; and if n be uneven, 

 A, B, C, . . . are identical with a t b 9 c 9 ... 



Taking, as examples of the formulae, w=6 and w=9, we 

 find 



A y (6)-3*A.(2) = 1, 



A,(6)-A r (3)-A,(2) + A r (l) = 0; 

 viz. 



F + 3 r -3 r =l, 



A,(9)-3"A,(3) = 1, 

 A r (9)-A r (3) = 9"; 



l r + S r + 9»'- 3 r (l r + 3 r ) = 1, 



If w be uneven, A r (n) = <r r (n), and the two formulae are 

 included in those given in the last section. If n be even and 

 = 2 h m, where m is uneven, the first formula becomes 



^(m)-SAV r g) + 2A'Bv^)-. . .=1, 



where A, B, C, . . . are all the prime factors of m ; and it is 

 therefore included in the first formula of the last section. 



If^ = 0. A r (w) denotes the number of the uneven divisors 

 of n. The two formulae do not coalesce except in the case of 

 n uneven. 



Formula? involving A/(w), § 9. 



then 



w v so _, 2^ 2 3^ 3 4^ 4 



= # + A/ (2> 2 + A/ (3> 3 + A/ (4> 4 + &c, 



where A/(w) denotes the sum of the rth powers of those divi- 

 sors of n whose conjugates are uneven. Thus we have 

 f(x) = x + x z + x b + ^? 7 + &c, 

 £(#) = # + 2 V + 3 r x 3 + 4V= + &c, 

 F(» = a? + A/(2> 2 + A/(3> 3 + A/(4> 4 + Ac,, 



n = rc r ; *? 2 =0, 7] 2n+1 = l; 



