XIII. On the Inverse Method of Definite Integrals, 

 with Physical Applications. 



By the Rev. R. MURPHY, B.A. 



FELLOW OF CAIUS COLLEGE, 

 AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. 



[Read March 5, 1832.] 



INTRODUCTION. 



The mass of theory, on the subject of Definite Integrals, con- 

 tributed by Fourier, Poisson, Cauchy, Gauss, and other modern 

 analysts, is very considerable. Many of their properties, and 

 applications have been pointed out, and the calculation of their 

 numerical values facilitated. We may, therefore, regard the 

 direct calculus of Definite Integrals as already formed, to a 

 certain extent. 



But as subjects are generally best understood, when examined 

 on all sides, it would be advantageous, even in this point of 

 view, to possess an inverse method of Definite Integrals, by 

 which we may re-ascend from the known integral, to the un- 

 known function under the sign of definite integration. At the 

 same time objects of importance in the pure, and physical mathe- 

 matics would be attained. Euler and Laplace have valued the 



