434 



t)r. Q. A. Miller ore the Substitution 



Value 

 ofS. 



Number of 

 Groups. 





Number of systems of two elements 

 in the substitutions. 



1 



1 



1 n-1 



n 



2 



2 



J2 7i-2 

 \2 n-1 



n 

 n— 1 



3 



2 



/3 n-3 

 \3 n-2 



n 

 n-1 







f 4 n— 4 



n 



4 



3 



-? 4 n-3 



n-1 







(4 n-2 



n-2 







(5 n-5 



n 



5 



3 



^5 n-4 



n — 1 







(5 n-3 



n-2 



• 



• 



• • 



• 





r- 



Tm n— ra 



n 





1 \ m 1 1 



(m even) ~- + 1 



m n — m + 1 

 1 : • 



n-1 



m < 





m 

 \m re- 2 



Tm n — m 

 | m n— m + 1 



m 

 n 





(m odd) — ~ — 



n-1 





k. 





m + 1 

 Im n 2 



m — 1 

 re r 



The groups for the same value of S are all distinct ; but it 

 may happen that two groups which correspond to different 

 values of S are identical. This can, however, not occur so 

 long as the value of S satisfies the relation 



Q =n 



b< 2* 



Identical groups can therefore only occur when the value of 

 S is such that 



n Q == 2n 



2 <b< "3"' 



To the successive values of fS which satisfy this relation there 

 corresponds one of the following series of identical groups : — 



( a \ ( 2, 4, 6, 8, . . . If n is even. 

 W -^1, 3, 5, 7, . . . „ „ odd. 



To find the number of all the given groups which corre- 



