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



mode of proof is different, as Mobius obtained bis results by 

 equating coefficients, and without tbe use of symbolic methods. 

 In his paper Mobius first proves several particular cases of 

 the theorem, but his results are practically of the same 

 generality as the theorem given above. 



If the number of primes a, b, c, . . . be finite, the number 

 of values to be assigned to n is infinite in the first equation, 

 but finite in the second equation. If the number of primes 

 a,b,c,... be infinite both series extend to infinity. 



Particular Cases of the Theorem, § 4. 



§ 4. The most important cases of the theorem are (i.) when 

 a,b,c, . . . denote the complete system of prime numbers, and 

 (ii.) when they denote the complete system of uneven prime 

 numbers, 

 (i.) Putting 



a = 2, Z> = 3, c = 5, . . . 

 and 



e 2 = 2% e B = S% e 5 = b r ,... 



so that e M = n r (§2), we obtain the result: — 

 If 

 F(«) =/(*) + 2 W) + 3y(* 3 ) + 47(^) + 57(* 5 ) + &c, 

 then 



/(^) = F(^)-2^F(^)-3-F(^)-5-F(^) + 6 >, F( l r 6 ) 

 -7 r F(0 7 ) + WF(x 10 ) -1MV 1 ) + &c. 



In the first series all the natural numbers occur ; in the 

 second only those numbers which are divisible by no squared 

 factor. 



(ii.) Putting 



a=3, 6 = 5, c = 7, . . . 



and e n =n r as before, we find: — 

 If 

 F (0) ==/ (*) + 3'/ (* 3 ) + 5 -/ (* 5 ) + Vf (^ 7 ) + 9"/ {*») + &*, 



then 



/ (0) = FO) - 3'F (^ 3 ) - 5 r F (^ 5 ) - 7-F(«* 7 ) - lM> n ) 



-13''F(^ 3 ) + 15^F(^ 15 )-17^F(^ 17 )- &c ; 



the first series involving all the uneven numbers, and the 

 second the uneven numbers which contain no squared factor. 



In these theorems the value of r is unrestricted. 



Putting r equal to zero, we see that, if 



F(*) = /(«) +/(**) + /(*•) + /(*') + /(*•)+ 4c., 



