INVERSE METHOD OF DEFINITE INTEGRALS. 147 



From , this formulae it appears that the equation T„ = has n real 

 values of t all negative; and therefore n corresponding values of t, 

 which are all included between and 1. 



Moreover, if we form the equation 



u = T + hu, or u 



1-h' 



it follows by the theorem of Lagrange, that T„ is the coefficient of h" 



de" e'~* 



in ^'^•-j-> that is, in - — y, and putting t for e% T„ is clearly the co- 



h 



efficient of h" in the expansion of the function y . 



Conversely, we may now prove that the coefficient of h" in the ex- 



h 



pansion of - — - is a reciprocal function; for when h = 0, this function 



A ""■ ft 



is reduced to unity, we may therefore put generally 



= ro+T,^+T,A^ + &c. where T, = \. 



A 

 fX-k 



1-h 



Let h' represent any other arbitrary quantity, and we have 





1-h 



j= T,+ T,h'+T,h" + &ic. 



Multiply both series term by term and integrate, the result in the 

 left-hand members is 



{i-h){i-h')^' ~ i-hh" 



/which expanded becomes 1 + hh' + h^h'^ + kc.; which being identical with 

 the integral of the product of the right-hand members, will necessarily 

 require that the integrals of those terms which are not in corresponding 

 places in both series must vanish, and the integrals of the products of 

 the corresponding coefficients to be unity, which are the same properties 

 that have been demonstrated in Art. 24. 



