Functions formed by Groups. 41 



Since >; in K 2 (art. 8) is in I t+] and is no 6, G d is bound 

 (F.G. 8, 5) to mark out the group (F.G. 10) 



rjG a r)- 1 = 125634 -356142- 125634 = 536241=02,5 above. 

 Wherefore 



and we have proved that 



(B^--=(B 2 + e6'-, 



whence 



and all the 2*6 functions under 2 = aj3yyyy, 



et-, (Be + l"...,(Bi--(32el..., 



are the same function. 



For the function G rf , i.e., 356142, of art. 6 under 2, is 



a/3 a/3 a/3 a/3 a/3 a/3 = 



2[(i2 + 21X3456) +(35 + 53)(i246) +(64 + 46X1235) J (y>o). 



This is invariable by each of the substitutions 

 1, 125634, 123654, 125436, 

 which are all in I m and none of them in G d (art. 6). Any 

 one of them is C giving (art. 3). 



C0 d = (3 d , 

 wherefore (B d is four times repeated in &%' '> an d all the 

 aforesaid 12 are the same 15-valued function with its 

 values. 



9 2 . The cyclical G D = 234561 (2=aj3yyyy) has no 9 in 

 common with L t , which is 



= H 24 x Ji 



