INVERSE METHOD OF DEFINITE INTEGRALS. . 123 



and it is, moreover, evident that either of those values are identical with 



1.2...nde' ' 



which is included in the general form given in Art. 5. viz. 



d\ {ft'" V) 

 dj" ■ 



12. 2'o find a rational and entire function of f of h dimensions, 

 which if multiplied hy a rational and entire function of t' of less than n 

 dimensions, the integral of the product may vanish between the limits t = 

 and t=l. 



Let the required function be represented by (p, q),„ so that 



{p,q\^l + A,t^ + A,f-f + A,,f^, 



and by the proposed conditions we must have 



lAp, qXt-" = 0, 



ni being any integer from to « — 1 inclusive, put t^ = T, the limits 

 of 7' are the same as those of t. 



Hence J^ip, ?)» T~^~' ■ T''= 0. 



1-1 

 Now ij), q)„ T" , is a function of T of which the indices are in 



arithmetical progression, - being the common difference, and T' the 



first term ; and as the nature of the question affords m independant 

 equations for the determination of the n coefficients Au A-,...A„, it 

 follows that there is only one function of the kind, which will satisfy 

 the proposed conditions, and by Art. 5, it is evident that the function 



answers those conditions, and is manifestly of the required form, it 

 Vol. V. Paet II. R 



