358 Mr. J. Cockle on the Theory of Equations 



Y( r, ' • )• ^Tien the affix of substitution is of the form 



( , ), it is replaced by ( . . )j which indicates an interchange or 



transposition of the elements a ^nd b. From the nature of an 

 interchange we have 



13. I premise the following known results of the theory of 

 substitutions : — 



I. The repetition of an interchange restores the function to 

 its original state. Consequently 



Y /ab\ /'«^ \ _ Y f'** y = Y 



II. Let Aj, Aj and Ag denote any three forms or states which 

 the elements are capable of assuming from changes of arrange- 

 ment. Then 



The substitutions on the left are called contiguous. 



III. A limited number of repetitions of any substitution 

 brings us back to the original function. This number (the 

 degree of the substitution) cannot exceed the number of elements 

 or suffixes contained in the affix when reduced to its most simple 

 form. 



IV. Every substitution of the nth degree may be represented 

 by a substitution of the (n— l)th degree followed by an inter- 

 change. And any substitution among n quantities may be repre- 

 sented by, at most, n—1 interchanges. This is demonstrated 

 by the introduction of contiguous substitutions. For example, 



yl'abcd\ _ ^/abcd\ /bcad\ _ ^/abc\ /cd\ 

 \bdao/ KbcadJ \bdacj ~ \bca) \"/' 

 and 



<:::)=<:::)C::)=H-')(--)' 



therefore 



V. It is indifferent in what order independent interchanges 

 are taken. Thus 



Y(!^)(!f) = Y(t^)(«^). 



