Functions formed on Groups. 43 



G, D . Oin therefore has only 10 values, which are 6 

 times repeated in (&£/ \ Wherefore G, D , and every group 

 marked out by it give that 10- valued function, and G 2d is 

 retained. 



Under 2 = apyyyy, 



we have won three functions : 



(5 d = (356142) has six terms and fifteen values : 

 D = (234561) has twelve terms and sixty values : 

 0. 2D = (342615) has twelve terms and ten values. 

 2 = «/3/3yyy . 

 10. The maximum groups 



Gi= 123456 132546 G 2d = 123456 132546 



465213 5 6 43i2 251364 341265 



231645 321654 562143 463^2 



654321 645231 645231 654321 



312564 213465 436512 526413 



546132 456123, 3^625 215634 

 have each 132546 in common with I 12 , which here is 



I (+ i = Ii2= 123456+ 132456= 123456x123456 = HJ 2 . 



61123564 132564 123564X1 132546 =e 



62123645 132645 123645X3 



8 i 23465 132465 123465X3 



64123654 132654610 123654X4 



65123546 132546611 123546X5 



< Taking 0^ = 465213, above written, we get, by aid of 



He, 



XiGjX" 1 , X 2 G„X7\ X 3 G d X 3 ~\ X&aK 1 , X 5 G d Xo rl , which are 



546321, 465321, 456231, 564231, 45 6 3 12 , or 

 G e i G e2 . . . G e6 j 

 to be marked out by G dt as all giving G£ ' ', all functions of 

 6 terms. (F.G. 13). 



It is evident in these five groups that every substitution 

 of H 6 is a C, no Q, making (art. 3). 



06,1=6. 

 a true, so that O d is six times repeated in G j • • under 2 and is a 

 ten-valued function, as are all the five G,7~i ' ', &c, marked out. 



