INVERSE METHOD OF DEFINITE INTEGRALS. 115 



The theory of reciprocal functions is applied in the fifth Section 

 to the complete solution of the question, which was the object of this 

 and the preceding JMemoir, namely, to revert from any function what- 

 ever to that under the sign of definite integration, those reciprocal 

 functions being employed which are most convenient in each particular 

 instance. 



The last application in this Memoir of the theory of reciprocal 

 functions, is to the development of given functions of x in descending 

 powers or other forms which vanish when x is infinitely great; the 

 results of which may be applied to the valuation of functions of 

 very great numbers, and to a great variety of physical problems. 

 These series have also the peculiarity, generally, to terminate for the 

 functions of integer numbers. 



