3i8 The Rev. Thos. P. Kirkman on 



For /2= 5, S may have the following forms only : — 



S = a^yll^ or a/3yyy, or afij3yy, or aa/3/3/3, or a(3fiPl3. 

 No two functions (B and (5', with their Q — i values each, 

 are allowed as different functions, when one is obtained 

 from the other by a different arrangement of the indices of 

 the same S, whether with or without breaking of the 

 repeating clusters aa^ as, &c. ; much less when one is obtained 

 from the other by altering the numerical values of ajSy.... 



It is proved in M.M. p. 344, that no additional functions 

 can be won by permuting indices of S in (H»)- 



The proof given in p. 344 is that the writing of S' = ayjS. . . 

 for S = ajSy. . . over x^x^x^, . . gives no additional function $ = (5'. 

 Here is one transposition only made of two different indices. 

 This is proof sufficient, because every permutation of ajSy. . . v 

 can be effected by a succession of such transpositions. In 

 that page 344, lines 1 1 and 1 2, A and B should be A' and 

 B' ; and, in lines 12 and 14 G should be G', an equivalent 



of G, namely |f G|f- Vide Note A at the end. 

 23 23 



6. The Index group, I^+i, which is always determined by 



S, is of the order 



t^-\^a\h\c\...p\. (3) 



If /2 = 6, If+i for S = aj3778S or 2 = 0/3/3777, ^^ 



aj3yy^d ajjfDyyy 



1^4.1== 14= 123456 or It+i = 112=123456 

 124356 123564 



123465 123645 



124365 123465 



123654 

 '23546 

 132456 

 132564 

 132645 

 132465 

 132654 



132546 

 of which the orders are i ! i ! 2 ! 2 !, and i ! 2 ! 3 ! 



