Mr. John Herapath on Functional Equalions. 105 



analogical views, which have been the means of misleading 

 probably every other individual who has attempted the dis- 

 covery. 



This determination of tlie number of arbitrary functions, 

 which has been so great a desideratum, might be differently 

 and more concisely demonstrated, but I have preferred a more 

 exjilicit method, lest, by not comprehending every prominent 

 point in detail, doubt should rest on the result. I shall pre- 

 sently give some examples of its accuracy. 



On the direct complete Solution of {2) and (6). 



In the two following methods, by which the complete solu- 

 tion may be obtained, each having under peculiar cii'cum- 

 stances its advantages, I shall state them separately. 



First method. — For \J/ « a: in (2) put \|/ « j- . <p or", which is equal 



to -^lax when u = 0. Substitute ctx, a? x, . . . «"~ x succes- 

 sively for .r, and from the 7i equations eliminate •^a.x, \J/«^^, 



... 4/ a"~ X. The result will be an equation of the form 



/. r 5 n—\ , r J) n—i v\ n. 



Jt \^i ax, or X,. ,. en. a', \J/ o^, ip x , (p a x , . . . f #£ r f =s 



Differentiate this with respect to v, divide the result by d u, 

 and put u = 0, which gives an equation of the form 



J'.^\x,ux,.,.a. x,yl/x,<px,ipxx,...<pot xj=0 (7) 



comprehending the log. in ip. From this equation the form 

 o( y^x may be determined. 



Second method, — For ^ ctx m (2) put h-^ a.x + v f x, which 

 coincides with \I/«.r when v = and Z> = 1. Pursue then the 

 same course of substitution and elimination as in the first me- 

 thod ; and after differentiating with respect to b and v, dividing 



by db, and including --y in the arbitrary function, we shall 



have an equation similar to (7), from which vj/jr is to be deter- 

 mined. 



In (6) we have merely to put 



\J/ a ^~ X, ^ cc^~ X, . . . \J/a"~'4/~ x 



for ^ X, (3-.r, ... (3"~ a: 



and then putting \I/a' lor x, the equation comes under the form 

 of (2), from which \|/.r may be determined, and of coiise(|uciice 

 /3 x; because ux is any periodic fund ion of the 7ilh order taken 

 at pleasure, provided no order of it ux, u'x, . . . become = x, 

 unless it be the 7ith. 



These solutions are complete; for they contain an arbitrary 

 Vol. G6. No. 328. y/M":. 1825. O function 



