GROUPS AND MANY-VALUED FUNCTIONS. 369 



80. The group G', with N final through out^, can be 

 written as the modular group of (N - i ) (N - 2) substitu- 

 tions, 



G'=^'o + A/o + Ayo+ . . +A^-Vo, 

 where A^o &c. are N - 3 derived derangements of g\ 

 = 1+^ + ^+.. 



(^=234».(N-i)iN) 

 by the substitutions of g\. 



This group G' consists of the cyclical permutations of 

 the N - 1 first places, of unity, of A, of A^, of A^ &c. 

 The didymous radicals of Ji^ are 



«i, «2 = %A, G!3 = aiA^, a4 = 0iA^, &c. 

 If then we write under a^, 



ttiO, ai6\ a]ffi' ', 

 the cyclical permutations of the first N - 1 places of Uy ; 



under «2> 



a.i$, a.-0^, a^6^- • ; 

 and under %, 



a^6, a^6^, a^O^- •, 



&c.; we shall form a^G' the derivate of G' by ay, consisting 

 of (N-i)(N-2) substitutions all ending in 1. And this 

 is the derivate of G' by every substitution in a^G'. 



In the same way we can complete the derivates ^^G', 

 c-Sjt , &c. of G' by b\ by c , kc, which have a different 

 final vertical row of one letter. For G' can be written 



G'=/o + B/o + Byo+'-, 

 where BB^B*- • are the substitutions oi g'^. 



81. It can be proved, by a simple tactical argument 

 founded on the property stated in Art. 79, concerning the 

 position of every duad xy, that if «o and b be any two of 

 the square roots of H', baj) = c is another. 



Let b be written below a^, and under b write the sub- 

 stitution 



° b ' 

 then c is the third of the series a^C' • of the didymous 



SER. III. VOL. I. 3 b 



