Arrangements and Groups. 127 



Multiply both numerator and denominator of this last fraction 

 by r and remember that rlr-^r^lr and that |;i— r-|-i« 

 (n—r-^i) \n^r, we get 



r\ n 



Comparing (^13 j and (%) we easily get 

 From f 8 j it easily follows that 



G(")=^-^'Ga'-.)- ■ ri4; 



«— 1 



r— I \n^r 

 From (^15^ and (^gj we get 



G(r>)+Ga':-})= 



(n—r) I ;e— I 



; U?— ^ 



which by Art. f8j equals G("), hence 



Gc;)=G(r^)+GC;z}). ri6; 



We have obtained a few relations connecting arrangements 

 with arrangements in equations (\), (2), (6), (y), also a few 

 relations connecting groups with groups in equations Tii^, (14), 

 (16). We now obtain a few relations involving both arrange- 

 ments and groups in the same equation. 



We have already found in Art. 4 



AC)=|rG(;), ri7; 



and as I r=AO we may write (ij) in the form 



Ac;)=AOGG'). ris; 



From (y), A(Q=A0_i) and writing this value in (^18^ we get 



Aa')=A(;_,)G{';). ri9; 



In (iS) substitute the valufe of G(") given in (^i6j and we get 



A(';)=A(;:) [G(rM+G(;'z})]. r2o; 



But it readily follows from (^i8j that 



A(r^)=A(r)G(rM, 



^^^ ^=- A(Tr- 



Substitute this value of G(" M iti f 20J and we get 



A(;')=A(rM+Aa) G(;'zj). r2i; 



