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LVIIT. Applications of Mobius's Theorem on the Reversion of 

 certain Series. By J. W. L. Glaisher, M.A., F.R.S* 



Definitions Sfc, §§ 1, 2. 



§ 1. rpHROUGHOUT this paper the letters a,b,c, . . . are 

 -I used to denote prime numbers only. 



I. The operator E n is defined by the equation 



E,/(*)=/<y), 



viz. the effect of operating with E n upon any function of x is 

 to convert it into the same function of x n . This definition 

 is supposed to hold good for all values of n, positive or 

 negative, integral or fractional. 

 Since 



E„/(<)=/(*-)=E„»,/(tf), 

 we see that 



E n xE TO = E nTO =E OT xE„ 



universally, so that, if n = a a b 13 c y . . . , then 



E n =E:xE?xEj... 



II. The quantities e a , eb, e c , . . . (in which the suffixes are 

 primes) are supposed to be absolutely independent constants ; 

 and by e ab we denote e a x ei>, by e a 2 we denote e a x e a = e a 2, and 

 in general, if n=a a b& c y . . . , then e n is defined to denote 



It is perhaps most convenient to regard, as in this defini- 

 tion, e a , e^ e e , ... as arbitrary quantities merely distinguished 

 from each other by the suffixes a, b, c, . . . ; but if we regard 

 them as functions of a, b, c, . . . respectively, it is to be 

 noticed that they are independent and arbitrary functions of 

 a, b, c, . . .; viz. we have 



e a = cj)(a), e b = 'f(b), e c -x(c), &c. 



§ 2. In order that we may have e?i = <j)(n) for all values 

 of ft, the function <j> must be such that <f>(rtin) =<£(m)0(n), 

 whence (j>(n) = n r . 



Unless, therefore, 



e a =a r , e b = b r , e c = c r , . . . 



(where r may have any value whatever), e n cannot be the 

 same function of n for all values of n. 



* Communicated by the Author. 



