INVERSE METHOD OF DEFINITE INTEGRALS. 361 



Now, by the assumed equation we have 



ui — hu~i = k, 



du 

 and by differentiation («J + Am"*) -jr = 2m ; 



but also (m^ + hu~^) = k + 2hu~i; 



hence, {k + 2hu~i)'Tr=2u; 



and therefore, F„ = 2 the coefficient of ^"+' in u {l-2u(l - 2t) + u"} -^ 

 from which it follows that if we form the two equations, 



u'=h + (ku')i] . \U' =u' {l-2u' (l-2t) + u''}->^ 



} putting i 



u" = h + (ku")i] [t7" = «"{l-2«"a-20 + «"'i^^; 



then ^^j— = ^0 + J^^k + V^¥ +r,k'kc. ad inf. 



supposing that in the left-hand member h is finally put equal to unity. 

 It may be observed that the quantities u', u" are the two roots of the 

 equation u'^ — {2h + k)u' + h!' = 0. 



29. To expand a given ^function (pit), in terms of the transient 

 function \ ^ . 



Let the general term of the expansion be A„V„, then by the nature 

 of reciprocal functions we have 



= AJtPj", (Art. 24.) 



M.(W-l) 1 



= (-l)".^„ 



(w + !)(« + 2) (2« + l)' 



lience,(p(t)=Kft<l>(t)-^^rj,(p(t)t-\-^^.FJt<}>(t).f-&ic. 



