XIV. Third Memoir on the Inverse Method of Definite Integrals. 

 By the Rev. R. Mukphy, M.A. F.R.S., Fellow of Cuius College, 

 and of the Cambridge Philosophical Society. 



i;;Read March 2, 1835.] 



INTRODUCTION. 



In the two preceding Memoirs on the Inverse Method of Definite 

 Integrals, the limits of integration had been fixed throughout at and 

 1, but in the sixth Section, which is the first of the present Memoir, 

 the integrations terminated by arbitrary limits are fully considered; and 

 when performed with respect to any function of the independant vari- 

 able, the proper methods for discovering reciprocal functions are given, 

 and it is remarkable that the forms thus obtained for the trigonome- 

 trical functions, for Laplace's and an infinite variety of other reciprocal 

 functions, are all similar, differing only by a constant. 



In identities obtained between the »"" differential coefficient of a 

 function not containing n, and its expanded value, we may, generally, 

 by changing the sign of n, obtain a corresponding identity between 

 the ra"" successive integral and its expansion, abstracting from the ap- 

 pendage of integration which ought to contain ?« arbitrary constants ; 

 this property however extends also to certain reciprocal functions which 

 contain n ; and this consideration leads in the same section to the com- 

 plete resolution of Laplace's equation for the reciprocal functions of 

 one variable, which are the coefficients in the developement of the reci- 

 procal of the distance of two points; the w*"" coefficient when multiplied 

 by an arbitrary constant, satisfies that equation, as is well known, but 

 as the equation is of the second order, another function multiplied by 

 ■^ Vol. V. Part III. Tr 



