Mobius's Theorem on the Reversion of certain Series. 527 



The Functions cr r (n), A r (n), Sfc, §§ 12, 13. 



§ 12. The five functions oy(w), A r {n), A'(n), E r {n), E/(n), 

 have been specially considered in the six preceding sections, 

 because, for certain values of r, they occur as coefficients in 

 some of the fundamental ^-series in Elliptic Functions. 

 When n is uneven the three functions <r r (n), A r (w), A/(n) 

 become all equal. 



The first of the two formulae in § 6, viz., 



was given by H. J. S. Smith in vol. vii. p. 211 of the i Pro- 

 ceedings of the London Mathematical Society ' ; but so far 

 as I know, no special reference has been made to the other 

 formulae. All the formulae may of course be established 

 without the aid of Mobius's theorem ; but the duality of the 

 results and their connexion with Mobius's theorem seemed to 

 me to be of interest apart from the results themselves. 



It will be observed that the double result is connected with 

 the double form of the g-series in Elliptic Functions. Thus, 

 taking for example the theorem in § 6, in the case of r= 1, we 



00 



have Sj a(n)x n 



x 2x 2 Sx d 4^ 4 B 



and, by expanding the terms in rows and summing the 

 columns, we transform this series into 



X x 1 X d # 4 - 



(l-x) 2+ {l-x 2 ) 2+ {l-x 3 ) 2+ (l-*) i + 

 These two forms of the series correspond to 



f(x) + e 2 f(x 2 ) + e 3 f(x*) + e 4 f(x*) + &c. 

 and 



respectively. 



In order to apply Mobius's theorem it is unnecessary to 

 actually sum the columns of the developed series, as in the 

 transformation of the g-series. 



§ 13. With reference to the quantities e n (and therefore 

 also rj n ), it is to be noticed that they must be such that 

 e m x e n =e mn for all values of m and n. Thus, for example, we 

 cannot put e n — ( — l) n "V", viz. en=n r if n be uneven, and 

 e n — — n r if n be even, for if e 2 — — 2 r , we must have 



£ 4 =-2 J 'x — 2'= +4*\ 



