INVERSE METHOD OF DEFINITE INTEGRALS. 123 



For if we put ti = in the equation u{l — hU) = t we get 1 = 0, 

 and putting u = 1 we have by supposition U = and therefore t = I, 

 hence the limits of u are the same as the limits of f. 



But j;Q„f = the coefficient of h" in f^^.f, 



^i U/t 



and l^^.if = JJ^ = !„u''{l-hUr 



expanding the part under the sign of integration, and taking the co- 

 efficient of h" we obtain 



hHnt - 1.2.3 n -(-l)^^ ■^- 



11. If U he a rational and entire Junction of u which vanishes 

 when u = \, and if Q„ be the term independent of u in the product 



U"- \\— —\ , then shall Q„ be itself a rational and entire function of 



t possessing the property of ftQj'^ = 0, x being any integer from to 

 n—\ inclusive. 



For it has been proved in my former Memoir on the Resolution 

 of Equations*, that the root of the rational equation <f){x) = is the 



coefficient of - in — h. 1. ^— , hence the value of u in the equation 



u(\ -hU)=t, is the coefficient of - in -h.l. j(l--] -hul, and 



differentiating, it follows that the value of -tt is the term independant 

 of u in 



u) 



* Camb. Trans. Vol. iv. p. 131, 



