528 Mr. J. W. L. Glaisher on Applications of 



The series 



x 2 r x 2 3^ 3 4^ 4 . 



00 



which occurs in Elliptic Functions, is equal to S x % r (n)x n , where 

 f r (w) denotes the excess of the sum of the rth powers of the 

 uneven divisors of n over the sum of the rth powers of the 

 even divisors of n ; and we thus see that we do not obtain by 

 Mobius's theorem formulas involving % r (n) corresponding to 

 those that have been given in the case of E r (w) and E/(w). 



It may be remarked that the functions o- r (n), A r (n), &c. do 

 not satisfy the conditions to which e n is subject, for the equa- 

 tions <7v(m) <T r (n)=G r (mn), A r (m) A r (n) =A r {mn), &c, hold 

 good only when m and n are prime to one another. 



Principle of the Method, § 14. 

 § 14. The method employed in §§ 6-11 gives a pair of 

 connected theorems relating to a function P(n), where P(w) 

 denotes the coefficient of x n in the expansion of F(#) in 

 ascending powers of x, viz. if 



F(«) =?M + «*/V) + e *A* s ) + */M + &<>• 



= ^ + P(2)^ 2 + P(3)^ 3 + P(4)^ 4 + &c. 

 and 



f(x) = x + 7] 2 x 2 + rj 5 a? + rj^x* + &C, 

 then 



+ ...=V 



nj 



.=e n . 



^-M5) + M5)"Mi) 



pw-w©tai,i(J).w(i) 



Development of a certain Symbolic Expression. The 

 Function (f>(n), §§ 15-19. 

 § 15. In § 3 the expression 



{l-eaE a )(l-e b E b ) (1-eoEc). . A x) 



was considered and found to be equal to Xe n f(% n ), n having all 

 values of the form a a b&c*. . . 



We now consider the expression 



(1-E fl )(l-E»)(l-E,)... 



(l-eaS} a )(l-e b E b )(l-e c E c )..A w) ' 



Since 



-i -p 



± =l4.(« a -l)E« + (e fl8 -f?«)E fl 2+ &c. ? 



