60 Proceedings of the Royal Irish Academy. 



shall equal /5 only. If the first term of any proposed series for [<^„]~^ 

 be taken greater than ^S", then 



[c^,][<^,J-^ and [<^J-^[<^J 



will not contain /5 at all ; and if the first term be taken less than yS'*, 

 we must have a series in which the exponents of vary from - co to 

 + 00 ; while it is not easy to see how the coefficients can be determined. 



22. The Solution of the Rational Integral Equation = y, given 



hy [xlv^y? ^^^5 71 values^ which are the n Roots of the Equation. — Let the 

 equation be 



X'' + p.^x''-^ + p.ox''-' . . . p.^^iX = y, 



the highest term being x", and the lowest term not lower than p.^^^ix; 

 that is. there are no negative powers of x. Then the equation has n 

 roots. Let xp,, denote the operation performed on x, so that now 

 has the restricted meaning that it shall contain no term less than 

 p.,,,,li. Then, by § 16, 



But 'yy has n values ; let them be denoted by a, h, c, d, . . . ; and let 

 X, = [x] «, xo = [x] b, X, = [x] ^, . • . 



Then we have to show that Xy, Xo, x^, . . . are really and exactly the 

 71 roots of the equation 



\J/„']x-y = Q. 



This may be surmised to be the case ; but it will be advisable to 

 seek further proof. If Xi, x^, ^3, . . . are really and exactly the roots 

 of the equation, then must 



{x - x-^ {x - X2) {x - x-i) . . . = 0. 



In other words, Xi, X2, Xo, . . . must be quantities such that the sum of 

 their products, taken successively one, two, three ... at a time, shall 

 equal the successive coefficients of the original equation, with the 

 necessary changes of sign ; that is, by actually carrying out the 

 multiplication of 



we shall arrive again at the original equation. 



