50 Proceedings of the Royal Irish Academy. 



lowest power oi x\b> x\ and any equation of which the highest power 

 of X is x''\ and the next power is x'"'^^ such as 



x^ + ax''^''^ + hx"" . . . = y 



becomes 



X 4- + . . . - = - « 



on division by x'"'^ ; that is, becomes the second form. But there is 

 another way of reducing any equation, say, 



to one of these forms. For, taking the ^ power of both sides of the 



equation, and expanding the left-hand radicle by the multinomial 

 theorem, we shall obtain the equation in the forms 



X + piX' + ^2^^ • • • = '^y? X ^ pi-\- p^x'^ . . . = -^y, 



as required. And the solutions will have n values. 

 (9) Solve 



log (1 + a;) = and xe' = y. 



and 



16. General Expression for the Invert of a Linear Algebraic 

 Operation. — Let 



^" + j^iO?""^^ + PiX''^"- . . . = y, 



+ ^.i^r**"^ + p_2X'^~'^ . . . = y 



be the general forms of the equation requiring solution ; and let 



the number of terms being quite indefinite. We have then to find 

 expressions for [<^n]"^ and 



Suppose 12 3 4 



and let 



{^nY = (S + a,3' + a,(3' . 



