Functions given by Groups. 331 



Since 0~^ IS in N^ and not in G^, it must be in some 

 derivate of G^ ; let this be AeG^ in {U)-. then we read, art. 10, 



A^« = 0a'0a=i, inW. id) 



Wherefore {c) becomes 



+ GA-'+y+iAe0«G^0-^+--- = Ne; {e) 



The left multipliers of G in {F) are 



I> lAej 2Ae,* V-lAeJ /Agj Z+lAg,* * * 'q-iAj. 



The left multipliers of Q^ in {c) are 



(l> lAe, aAg, . . . /-lAg, ^Ag, z+iAg + . . . Q_iAe)0a> 



or the preceding each multiplied on the right by 6a- 



We know that the first set are all different by article 4 ; 

 therefore, the second set 6o» lAe^aj &c., are all different, and 

 consequently, by {(T), all the sinister multipliers of G^^ in (e) 

 are different. 



Put now G?*', in {e) under S ; it becomes 



+ (3d0a' +/+lAe0a(Bd0j' + • • • = N, (/) 



We have demonstrated in article 9, equation (9), 



For 0i in article 9 means any 0, ^.^., 0J^, of the index- 

 group, whereby (/") becomes my new theorem, 



6^-=0a(5d+lAe0„(3<.+ •• +/-lA,0„(B^+(Brf 



This is (3d • ' in an order different from that of GJ " under 

 S in table ( A„). We see 0^ and Q - i values of it, the 

 values of Q — I different derivates of G^ ; for we have proved 

 that these derivates are made by Q - i multipliers of G^, as 

 different from each other as are those of Gg in (A„). 

 Thus in {h) it is demonstrated of G^ and Gg in {a) that 



u 



