of Algebraic Equations of the Fifth Degree. 557 



in the second member of which equation the element y, and 

 not /3, is common to the interchanges, a, /3, y are supposed 

 to be unequal. In X (^^^^ ("".F) ^" inversion of the inter- 

 changes can take place without disturbing the value of that 

 function. 



Further, it is clear from the equation (/3.) that 



and X (a y) (« f) = X (« ^) (/3 y) ; 



so that we shall have the three equal expressions 



X(«i)(^.7> X(^y)(ay), X(ay)(a^^); (/3'.) 

 which, if we continue to apply the equation (/3.), will reappear 

 periodically. 



22. With respect to X(a^) (y8) («?), on denoting it 



by Y we shall find, since X„ (e?/ = X^ 

 X(«^)(r8)*=Y(6?). 



If therefore we substitute* X (^yS^ C'^^) instead of 

 X('«/3^ CtO' ^"'^ affect with (^_?) both members of the 

 equation which will thence arise, we shall have 



X (r.O W (^..0 = Y = X (a^) (y.S) (s 5) ; 

 that is, we can operate with the first and second of the inter- 

 changes as if the third did not exist. In like manner it might 

 be shown that in operating with the second and third we may 

 neglect the first. 



And analogous results will be obtainable whatever may be 

 the number of the interchanges with which X is affected. 



Section III. 



23. Returning now to the equations (d.), and designating 

 them, in the order in which they occur, by^ = 0, f^ — 0, 

 . ,yj = 0, we find, on inspection, 



/.ei) =f. C.r). 



We also find, as might have been foreseen, 



/a (".?)=/- C.r). 



And it will have been observed that the two functions in- 

 * See (».) and (/3.). 



