388 Mr MURPHY'S THIRD MEMOIR ON THE 



the analytical processes of calculation require the solution of definite 

 integral equations: some of these have been resolved by Laplace and 

 others, by means of particular artifices by which the unknown functions 

 were subjected to differential equations ; but as no general method 

 existed for this purpose, the resolution of such equations has been ex- 

 tremely limited, and apparently simple physical problems, such as the 

 distribution of electricity on surfaces, (with the exception of a very 

 few cases) have consequently defied the powers of analysis. Besides, 

 an abundance of facts connected with the interior arrangement of the 

 molecules of bodies are of such a nature, that mathematics possessed 

 but little power of reducing them to analytical forms, calculated to 

 produce any valuable inferences ; these facts are daily increasing in 

 number, and the analyst is far behind the cultivator of Experimental 

 Physics. The Memoirs on the Inverse Method of Definite Integrals 

 which are now concluded, and which have been pursued when the 

 absence of ordinary engagements permitted, originated in the belief 

 that by proceeding gradually from the simplest classes of Definite 

 Integrals to the more complex, the general principles of an Inverse 

 Method would be discoverable. The formation of all possible classes 

 of Reciprocal Functions, and the Transient Functions included amongst 

 them, have at length furnished means for the resolution of equations 

 to Definite Integrals. The author is however well aware that there 

 must exist numerous imperfections in the manner in which his design 

 is executed, but believing also that by those endeavours, however weak, 

 some fresh powers have accrued to analysis, as an instrument of investi- 

 gation, he trusts they will deserve the approbation of the Society. 



R. MURPHY. 



Caius Colleob, 

 Dec. 24, 1834. 



