487 



III. " On the Calculus of Functions." By W. H. L. RUSSELL, 

 Esq., A.B. Communicated by A. CAYLEY, F.R.S. Received 

 October 31, 1861. 



(Abstract.) 



One of the first efforts toward the formation of the calculus of 

 functions is due to Laplace, whose solution of the functional equation 

 of the first order, by means of two equations in finite differences, is 

 well known. Functional equations were afterwards treated syste- 

 matically by Mr. Babbage ; his memoirs were published in the Trans- 

 actions of this Society, and there is some account of them in Professor 

 Boole's Treatise on the Calculus of Finite Differences. A very im- 

 portant functional equation was solved by Poisson in his memoirs on 

 Electricity; which suggested to me the investigations I have now 

 the honour to lay before the Society. 



I have commenced by discussing the linear functional equation of 

 the first order with constant coefficients, where the subjects of the 

 unknown functions are rational functions of the independent variable, 

 and have shown how the solution of such equations may in a variety 

 of cases be effected by series, or by definite integrals. I have then 

 considered functional equations with constant coefficients of the 

 higher orders, and have proved that they may be solved by methods 

 similar to those used for equations of the first order. I have next 

 proceeded with the solution of functional equations with variable co- 

 efficients. In connexion with functional equations, I have considered 

 equations involving definite integrals, and containing an unknown 

 function under the integral sign ; the methods employed for their 

 resolution depend chiefly upon the solution of functional equations, 

 as effected in this paper. The calculus of functions has now for a 

 long time engaged the attention of analysts ; and I hope that the 

 present investigations will be found to have extended its power and 

 resources. 



IV. " On Tschirnhausen's Transformation." By ARTHUR CAY- 

 LEY, Esq., F.R.S. Received November 7, 1861. 



(Abstract.) 



The memoir of M. Hermite, " Sur quelques theoremes d'algebre et 

 la resolution de 1' equation du quatrieme degre," Comptes Rendus, 



