Definite Integrals, with Physical Applications. 373 



When the highest negative power which enters /(() is t~ h , then 

 f,f(t)t r (with the limits of t, and 1), is only finite when x>{h — 1), 

 but if x be <h-\, the parts of the integral which then become 

 infinite retain their former finite values by making the limits of t 

 to be » and 1 ; and all the principles established in this section 

 will apply to them with the new limits. 



14. Let f now denote a function of any number of variables 

 /, t , t" &c. and <p, a function of other variables x, x, x" &c. under 

 the same restrictions, with respect to each of the variables as 

 before, and let all the limits of integration be and 1 ; then if 

 there is given the equation 



fj<fr.....,f.r.t"'.r"" =<p, 



we may revert from (j> to f, by the following theorem : 



"Take the coefficient of 



, I , in A.t-'.r"J'"" 



XX X &c. 



and divide it by tt't"...S<.c. j the quotient will be,/?' 

 For let t.<p' be the coefficient of - in <p.t~ r , 



t'.d>" -in d>'.t'" or of — ; in d>f't'~'> 



X r XX T 



and so on, then (Art. 5) [t<p'.t r = (p, 



f t <p"J' r ' = <p', .: fjftf'.t't'^cp; 



and the same reasoning evidently holds for any number of va- 

 riables. 



It will not be necessary to annex any examples, to elucidate 

 the applications of this theorem, as they are similar to the ex- 

 amples for functions of only one variable. (Vid. Note B.) 



3 D2 



