Definite Integrals, with Physical Applications. 401 



Now the function — is itself of a form proper for <p (x), let the correspond- 



T 



ing value of f(t) be put = — when determined by the methods given 



in Section (1). 



1 R 



Hence T= coefficient of- in — .t" 



x x 



= term independent of x in Jit' 1 ; 



rT R 



.-. — \ — =term independent of x in — . f" 

 -'( t x 



= coefficient of x in Rt". 



r 1 r T 

 Similarly / — / — = coefficient of a? in Rt'' 



— [—. f — . f — = coefficient of x 2 in Rt~". 



&c. = &c. 



Having thus obtained the absolute term and the coefficients of ;»'. x\ x\ &c. 

 in Rt'*, we have therefore, 



*-'- J f7 + "^jff-*jfjjf7jf7***" 



(S) 



all the integrals being made to vanish when t = l, for then Rt~' = R, 

 which contains no coefficient of x, x-, x % , &c. ; the value of T is then 

 = term independent of x in R ; that is, A. 



liy differentiating the Equation (3) and substituting S for its value 



we get 



dS dT xS t dS a M (IT 



(It (It t ' •' l (It +Xt A ~ l (It ' 



dT 

 and integrating we have St' - f,t" -j- , the integral being corrected bj 



the condition S=A when t=\; the value of S is therefore fully discovered. 



