303 REV, T. p. KIRKMAN ON THE THEORY OP 



G=i + P + P2 + P^. 



pn-l^ Ki^2--g«-lg;t-2-- 



PiP-i'-qa+iqa+r- ' 



we have 



JJ^pm-1 _ F-\r-2 j"-3 • • y-(a+i) y-(a+2) • • 



P\ Pi PZ ■ • qa+1 %+2 ■ ' 



which has no factors but of the second order, as we have 



just proved. 



The number of these derivants 'R, all beginning in the 



denominator and in the numerator, with the same element 



1 of 



p^= labc- • 

 is 



for every factor 



may have A different or B different exponents, according 

 as it is a j!? or q, &c. 



Let M < A; (theorem C) ; we have 



JJ^pM _ P-l P-2 P-Z ' g-(g+l) y-(«+2 ) 



Pi p^ p% " qa+\ qa+% 



which cannot be identical with R, unless 



M = Ae=Be=Ce=-- &c., 

 which is impossible, because k (Art. 12) is greater than 

 M. 



Wherefore RG has k different substitutions. 



The derived group RG comprises the e substitutions 



RP ^, RP2^, . . RP ^^ 

 where 



k 



which differ from R only in the exponent of 



in the numerator. There are, therefore, e+ 1 systems of 

 exponents among those enumerated which give the same 

 derived group, 



RG = RP'^G = RP^^G = . . = RP'^'^G. 



