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



and, operating on x by 



(1-2E 2 )(1-3E 3 )(1-5E 5 )...> 

 we obtain as result 



x + 2x 2 + 3x 3 + 4:X* + &c, 

 which 



x 



§ 18. We have thus found for the expression 



(1-E 2 )(1-E 3 )(1-E 5 )... x 

 (1-2E 2 )(1-3E 3 )(1-5E 5 )... l-x 

 the three results : — 



^ <t>(n)x n 



(ii.) S^W^-S" o-W^-S" o-(n)^- &C, 



(iii.) 



the first being obtained by operating with the operator as a 

 whole, the second by operating first with the denominator, 

 and the third by operating first with the numerator. 



By equating the coefficients of x n in these three expressions, 

 we find that 



*(!)+*(/) +*(?) + *(*)... +*(n), 

 and 



■w-*<9+*e)-HK) + " 



are each equal to n, where l 7 /,#, h, n denote all the 



divisors of n. 



The first result, viz. that 



<Ml)+<M/)+<M#)..- + *(*) = *, 

 was given by Gauss in § 39 of the Disquisitiones Arithme- 

 tical. The second result is the particular case ^ = 1 of the 

 second of the two formulas proved in § 6. 



It is curious that two theorems so distinct in character 

 should be derivable by means of different developments of 

 the same operator. The symbolic expression for the series 

 2 X <f>{n)f(x n ) seems also deserving of notice. 



§ 19. It will be observed that the processes of the three 

 preceding sections are equally applicable if we put e n = n r , 



