Definite Integrals, with Physical Applications 371 



The corresponding term in f(t), is 



1.2.3. ..n ' /•/, , l\'" i 



jf" 



i 1.3.5...(a«-l) ,,, , _ /- 1.3.5...(2w-l) 



t» i /-/u i iV" 5 1.3.5...(2m-1 ,,, , .,_, ,- 



thus the general term of f(t), abstracting from the sign, becomes 



(2«f.fh.l.i 



t m-l V--/ •!— ^ 



vCilm ' 



2.3...2W 



r- 1 cos.J2«\/h.l. Ql 

 from whence /"(f) = - 



\Ai,i.(i) 



13. In applying the above principles for recurring from <p(x) 

 to f(t) in the equation f,f(t).t T = <p(x), we should attend to the 

 extent of the values to which x is limited. 



When <p{x) remains finite from x = to x=<x> we may assign to 

 x any value from x=— 1 to x=<x>. (Art. 2). 



When </>(#) remains finite from x = h to #=*, i.e. if h be the 



greatest real root of the equation , , = 0, we may assign to r 



any value from x = h — 1 to x=cc. For the equation may be put 

 under the form f l f(t).t".t'- h = (l>(x); put x-h = x, and f(t).t h = F{t), 

 :. f,F(t).t x ' = (f>(x' + h), which remaining finite from x' = to x'=<x> 

 we may assign to x' all values from x'=—l to x'= + x, i.e. we 

 may put for x any number between h — 1 and + « . 



Fo/. IV. Part III. 3B 



