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



Results deduced from the Theorem, §§ 6-11. 

 Formulce involving <r (n), §§ 6, 7. 

 § 6. Let 



e n —n and /(<£)=- . 



J. "~~ x 



We then have 



w . x 2 r x* &x* 4^ 



= x + ar(2)x 2 + <r r {3)x d + <T r {4:)x* + &c, 



where <tv(w) denotes the sum of the rth powers of the divisors 

 of n. 

 Thus 



f(x)=x + a? 2 + x* + ^ 4 + &c. 



<j>(a) = a!+ 2 r x 2 + 3^ 3 + 4 V + &c. 



F (a?) = x + a r (2)x 2 + o>(3) X* + CJ>(4> 4 + &c. 



e n =n r , Vn = l; 



and by substituting these values in the formulae of the pre- 

 ceding section, we find 



x + x 2 + $ + <& + &c. 



= x + a r (2)x 2 + o>(3> 3 + ov(4> 4 + &c. 



-2 r {x 2 + <r r (2)x* + <r,(3)a> 6 +cr,(4> 8 + &c.\ 



-3,{^ 3 + ov(2> 6 +<rr(3)x 9 + <r,(4> 12 +&c.} 



- 5'{ x 5 + o>(2> 10 + * r (3> 15 + o>(4> 20 + &c. } 



and 



# + 2 r x 2 + 3'^ 3 + 4V + &c., 

 = a? + o>(2> 2 + o>(3> 3 + er r (4>? 4 + &c, 

 -{a? 2 + <7 r (2> 4 +er r (3> 6 + o>(4> 8 + &c. } 



- {a? 3 + ov(2> 6 + ov(3> 9 + o>(4> 12 + &c. } 



- {^ 5 + ov(2> 10 + ov(3> 15 + ov(4> 20 + &c. I 



Equating the coefficients of x n in these two equations, we 

 obtain the formulas 



*">-*'*'© +*"* (s) -**'*(£) +• • - 



