372 Mr Murphy on the Inverse Method of 



When <p (x) remains finite for all real values of x, we may in 

 the applications give x any value from - « to + x , and may 

 then use formula of the form 



f t f(t).(t'±t-') = <p(x)±cj>(-x). 



If <p(x) remains finite for all values of x from x= — b to x = b, 



r 



then putting t = f and x= — the equation becomes 



which integral remains finite from x'=— nb to x' = nb, and if we 

 increase n indefinitely these limits tend to - oo and + » , and 

 therefore we may use the formula 



for all values of t' between -nb and +nb, or which is the same. 



we may use 



f,f(t).(t'±t-*) = <p(x)±<p(-x), 



for such values of x as lie between —b and +b. 



*Ex. 10. Given 



//($.{#■-#-)*> |tang«), to find/W- 



•jr, /ir \ 1 1 1 1 



Since , tan (- *) = j^ - ^ + j^ - ^ &c. 



= (a?) -^ (-#)... putting 0^') = -{y^ + ^ +&c.j, 



and it remains finite from x= — 1 to +1, then deducing /(() from 

 <p(x) by the usual method, we have 



/ft._{i+*»+/+&c.i-jJL t . 



* Vide Cauchy's paper, Annate* de Math. Tom. XVII. Formula 115. 



