of Algebraic Equations of the Fifth Degree. 553 



(a b c \ 



of substitution. 



14. The number of different values which X can receive 

 when we change the order of the elements on which it depends 

 cannot exceed the product 1 .2.3 . .n; but the affix of sub- 

 stitution will admit of 1 . 2 . 3 . . w x 1 .2.3. .ti differently de- 

 rived expressions. 



Thus if we denote by A^ A2, . . A,.2.3..». the different 

 forms or states which the five indices (1, 2, 3, 4, 5) are capable 

 of assuming from the several changes of arrangement to which 

 they are supposed to be subjected, the values of X may all of 

 them be expressed by 



x(i;>^(i:)'-(t)'-MA;> 



V denoting the product 1 . 2 . 3 . . « ; but in this system we may 

 successively substitute A^j A3, . . A, instead of Aj : whence 

 will result (" — 1) other systems, each of them consisting of y 

 terms. 



15. Suppose X to be such that the number of different 

 values of which it is susceptible shall be less than v. 



Here certain terms in the system 



-a;)--a:)'--a:)' 



must be equal to each other. 

 Let therefore 



x(i;)=x(i;) = ..=x(y. 



On submitting each of these /x expression to the substitu- 

 tion denoted by ( a ^ j, and observing that instead of an 

 expression of the form 



where X has been subjected to two successive substitutions, 

 we may write 



we shall have no difficulty in perceiving that the new set of 

 equal quantities which will arise may be represented by 



