III. On the Resolution of Equations in Finite Differences. By the 

 Rev. R. Murphy, M.A. F.R.S. Honorary 31ember of the Royal 

 Corl- Institution, Fellow of Cains College, and of the Cambridge 

 Philosophical Society. 



[Read Nov. 15, 1835.] 



When the degree of equations in Finite Differences does not 

 exceed the first, whatever may be their order, methods for their solution 

 in most cases have been furnished by analysts. With respect to those 

 of higher degrees, scarcely any thing has been done to assist in ob- 

 taining explicitly an algebraical expression for the unknown quantity*. 

 The utility of solutions for such equations, occurring, as they do, in 

 the theory of chances, is more apparent by the proof which they afford 

 of the expansibility of various kinds of successive functions on which 

 some doubt has hitherto existed. 



Tlie difficulties which those have encountered who attempted to 

 obtain expansions in an algebraical form, for functions which from 

 their nature may be denominated repeated functions, are known, such 

 are for instance 



(x times) 



a 



a 



log. log. log {x times) {«} 



sin. sin. sin (a; times) {«}, 



In the great work of Lacroix this subject is entirely passed over. 

 U2 



