136 ON THE SOLUTION OF 



In order to solve it, assume 



^ = ** ; 



t , a a , x yx 

 then ilr 7- = -5- and = ^- . 

 Tjra; ^x ^a; jfx 



Let a? = w e , x# = w m , and therefore 



Then = = -~ , where C is arbitrary ; 

 u*n V 3 



therefore %a? = (7a?. 



is Ibfunction of 0, which does not change when z + 1 is 

 substituted for z. 



We confine ourselves to the only simple case, that in which 

 it is an absolute constant ; then 



r -~- - ... 

 Cx x 



and tyx b log - ... c being an arbitrary constant. 

 c 



On substitution, we find b ae; therefore 



# = aelog^ ........................... (11) 



o 



is a solution of the problem. 



This is the equation of a logarithmic curve, which has there- 

 fore the required property. The method employed to resolve 

 the equation in yjc, namely, 



is applicable to every equation of the form 



.................. (12). 



Every such equation may be at once reduced to the follow- 

 ing equation in finite differences, 



F(u K u K+l ...u^ n }.= Q ...... . ............ (13). 



This reduction is in reality a particular case of an important 

 transformation due to Mr Babbage, which often enables us to 

 solve functional equations of the higher orders. 



