342 llEV. T. P, KIRKMAN ON THE THEORY OF 



There remains a certain number r of the equivalent 

 groups, to which the system of exponents is applicable 

 when all those to which it is inapplicable have been 

 rejected. 



It remains to be determined how many of these r 

 groups give distinct functions ^, of which no function is a 

 value of another. 



Any one G^ of these r groups gives the same function 

 with the derangements 



G-rf^l, Gr,^^2, Gc,i9^ ' • (ja^t, {d) 



where 



/= [Tra-irh'Ttc- •) — t, 



which is the number of dilFerent arrangements of N letters 

 possible by exchange of letters bearing the same expo- 

 nents. For the function ^ given by (y^Om cannot differ 

 algebraically from that given by G^^. 



58. It will generally be the case that G^^ and^-i others 

 of the r groups have each m of these t substitutions 

 6^6.2- 'Of. 



Let us suppose that no group contains more than m 

 of them, which are in G^, 



We write 



Ga=G,e, = G,e,= ^^=GJr>^ (e) 



The number of different derangements in the series [d) 

 is in general reduced by these equations (e) ; that is, we 

 have reductions of the form 



and instead of t-m derangements {d) we find that there 



are but 



t^-m=zt' , 



and that the function ^ given by G^ is formed also on t' 



derangements and no more of G^^ . 



Now these t' derangements of G,; are each the derived 



of a group equivalent to G,/ . We conclude that this func- 



