of Algebraic Equations of the Fifth Degree. 555 



we continue to operate with ( a M we shall merely reprod 



uce 



the same series of jo terms disposed in the same order as be- 

 fore. 



Thus we shall obtain, in the form of an equation, 



a and r representing any integers, zero included. What is 

 termed the degree of ( . M is indicated hyp, which is, as might 



easily be shown, equal to the number of the indices a, /3, y, . . 

 contained in the affix when reduced to its most simple ex- 

 pression*. 



18. An important theorem on the decomposition of substi- 

 tutions here presents itself. 



Observing that if 



/AA _/a/3y..^>,\ 

 we shall have 



where \q) indicates an interchange or transposition of the 

 elements a and /3 ; and that if 



we shall have 



the operation denoted by ( a\,) being in this case equivalent 



to the two interchanges ( ^ '^ | and I'^X] taken in succession : 



we are at once led to infer that every substitution may be re- 

 presented by a succession of interchanges. 

 And in effect if 



/AA _ /«/3y..?A 



^ Examples of such a reduction will appear in (18). 



