320 The Rev. Thos. P. Kirkman on 



enumerated and dictated when there are n indices, no one 

 repeated. It is only when some are repeated that difficul- 

 ties begin; for the number of different functions is then^ 

 and then only, < Q. 



8. Our problem is correctly stated (M.M. p. 342), thus : 

 to determine how many of the Q groups Gi G2 &c., give> 

 under S, distinct Q-valued functions *i = (3i, $2 = (32, &c., of 

 which no function is a value of another. 



When our selected S is written over the groups of (A„) 

 and over their derivates, or supposed so written, (A„) be- 

 comes (HJ, z.e., An under S. Our first business is to erase 

 in (H,j) all the functions (3 that have fewer than Q values. 

 If (3fc has fewer, it has g~^Q values (q>i), and (3^ will be 

 algebraically identical with g—i derivates of G^ under S, 

 and will be read in the line of G^ ^— i times. Thus 

 simple inspection of (H^) enables us to exclude the 

 Q — r functions that have fewer than Q values; and the 

 remaining r groups of (AJunder S {MM. p. 342, line i) 

 are all that we have to handle. 



It is thus to be understood that when we speak of (Hn)> 

 we are dealing only with the r functions in it that have Q 

 values. 



It will appear in our handling of a definite table (A„) of 

 maximum equivalents, that this sifting out of the Q — ^ 

 functions in (H„) that have fewer than Q values, can be 

 done without actually writing out the groups and their 

 derivates; but that G^ above is sufficiently given for this 

 purpose in the form 



G;t = 0Gd<^"^ 



Grf and ^ being known. 



I regret that I did not use in pp. 342, 343, the symbol 

 (3d to denote what there is called ^, the function given by 

 G^ under S; nor use 6 and for distinction as below 

 used in I^+i. 



