VI. Second Memoir on the Inverse Method of Definite Integrals. 

 By the Rev. R. Muuphy, M.A. Fellow of Cuius College, and of 

 the Cambridge Philosophical Society. 



[Read Nov. 11, 1833-3 



INTRODUCTION. 



The object of my former Memoir on the present subject, pub- 

 lished in the Fourth Volume of the Society's Transactions, was to 

 investigate the principles by which we might revert from a function 

 outside the sign of definite integration, to the function under that 

 sign, whenever the latter belonged to any of those classes usually 

 received in analysis. In that case the function outside the sign of 

 integration possessed the characteristic property of converging to zero 

 when a variable quantity x was made to increase indefinitely ; in the 

 present Memoir I have endeavoured to complete this theory, by the 

 research of the forms and properties of the functions under the sign 

 of integration, when the characteristic above mentioned is not pos- 

 sessed by the function resulting from integration : and as the subject 

 increased in difficulty, those methods of analysis which possessed greater 

 simplicity and uniformity have been most adhered to, in the follow- 

 ing investigations. 



The fourth Section is devoted to the research of the nature and 

 properties of the function under the sign of integration, when the 

 integral always vanishes between the limits (0 and 1) of the indepen- 

 dent variable which have been uniformly adopted in this as in the 

 first Memoir. The class of functions thus investigated possess the re- 

 markable property of vanishing an indefinitely great number of times 



