Mobius's Theorem on the Reversion of certain Series. 539 



Conversely we may regard the results of §§ 22-28 as 

 affording evaluations of the determinants themselves, these 

 determinants being of some interest on account of the curious 

 law of their formation. Regarding the formulae from this 

 point of view, it may be noticed that case (i.) of Mobius's 

 theorem (§4) implies that the determinant 



2 r , 3', 



*', 



5', 



6',.. 



r, o, 



2', 



o, 



or 



, .. 



o, r, 



o, 



o, 



r, .. 



0, 0, 



l r , 



o, 



o, .. 



0, 0, 



0, 



1', 



o, .. 



0, 0, 



o, 



0, 



o, .. 



+N "V, accord] 



ng as wis d 



r 



= or (-r 



factor or is the product of N simple primes. The'second row 

 of the determinant is derived from the first by division by 2 r , 

 the third by division by 3 r , and so on. Similarly case (ii.) 

 implies that 



(2n + l) r 



3 r , 



5 r , 



r, 



s r , 



lr, . 



1', 



o, 



o, 



y, 



o, • 



o, 



1', 



o, 



o, 



o, . 



o, 



o, 



i r , 



o, 



0, . 



o, 



o, 



o, 



i r , 



0, . 



0, 0, 0, 0, 



r 



= or ( — ) n+N (2n + l) r , according as 2n + l is divisible by a 

 squared factor or is the product of N simple primes. 



Application of the Theorem to Elliptic Functions, § 32. 



§ 32. Most of the ^-series in Elliptic Functions are of the 

 form 



/(?) + W) + **M) + e,M) + &c. = F(k,k',K); 



and if we denote by k n , Jc n ', K n the quantities into which k, H , K 

 are transformed by the change of q into q n , we may deduce 

 from such a series a formula of the form 



/ (g) = F(k, V, K) -€ 2 F(* 2 , k' 2 , K.) - ei ¥(k 3 , k' s , K.) 

 -«»F(*„ V$ K.) + «,F(*„ k's, K,)- . . . 



