

(JOlt 



where ^j, a?2, . . a?^ are the roots of x"" + Aja?"*-^ + . . =0, and 

 yvVvVni of y"*+A'iy'""^+ . . =0; the equation in y will 

 take the form 



+ j~ @.(«P + /eQ+7R+ . . +XL)V' 

 - j^@ . (aP + /3Q + 7l^+ . . +^L)'r-' 



tt, A 7, . . \ being any integers. I have here substituted @ 

 for / thinking it desirable that the letter /'should be set apart 

 as the symbol of integration. 



The coefficients of this equation are, as I proceed to show, 

 merely condensed modes of expression formed, in accordance 

 with a well-known method of notation, by subjecting ®, a symbol 

 of operation to the same laws, with certain obvious limitations, 

 as would obtaih if @ were a symbol of quantity. Thus b^ 



@.(aP+/3Q + 7R+ ..+\L) 

 is represented 



(5aP + @y8Q + @7ll+ .. +@X.L; 

 @a, @/3, . . @X being the sums of the ath, ySth, . . \th powers 

 respectively of the m quantities a?,, a^g, . . a?^ . And, in general, 

 if we expand 



(aP + ^Q + 7R+ ..+XL)» 

 in precisely the same manner as we should do if We were opera- 

 ting on an ordinary algebraical expression, and then apply the 

 symbol @ to each term, bearing in mind that a, y8, 7, . . \ are 

 the sole elements of the functions thus characterized by @, we 

 shall arrive at the expression indicated by 



@. (aP+/SQ+7li+ . . ^-xL)^ 



The expansion in question will accordingly consist of as many 

 terms of the form 



i.2..axl.2..6x .. xl.2..^ 



