116 Mb MURPHY'S SECOND MEMOIR ON THE 



SECTION IV. 



Inverse Method for Definite Integrals which vanish; and Theory of 



Reciprocal Functions. 



1. When the equation fif{t).t' = (p{x) is supposed to be restricted 

 to particular values of x, then whatever may be the form of (p {x), 



J'{t) may always be determined ; the values to which x is restricted 



we shall suppose to be the natural numbers 0, 1, 2, 3 (w — 1), and 



the method here pursued will also apply if the values of n should be 

 different from those mentioned. 



2. * First, let f,f{t).t' = 0, the limits of t being always and 1, 

 and let us seek for f{t) a rational function of t of the lowest possible 

 dimensions, which shall satisfy this equation when x is any integer from 

 to n — 1 inclusive. 



Any value of f{t) which answers the proposed conditions may be 

 divided by the absolute term, and the quotient, it is evident, will 

 equally fulfil those conditions; we may therefore take the first or 

 absolute term in f{f) to be unity, and as the conditions to be satisfied 

 are w in number, we must have n coefficients in f{t), which will hence 

 be a rational function of the form 



1 + Alt + A,f + + Ant"; 



and therefore (j>{x) = + — -^ + — ^ + + "—-^, 



p 



or = T^r by actual addition, 



putting Q for (a; + !)(« + 2), (x + w + 1), and P representing a function 



oi X oi n dimensions. 



Hence P=0, provided x be any number of the series 0, 1, 2....(/i — 1); 

 these are therefore all the roots of that equation, P being of n dimensions ; 

 hence we must have 



P = c.x.{x-l){x-2) {x-n + \)\ 



c representing a constant quantity. 



* I have resolved this question in a different manner in the " Treatise on Electricity." 



