198 Mr. Herapath on the complete Solution 



and eliminating vj/ax between (2) and (3) there results after 

 due reductions 



4,* = \fx + C {<px -fx. $ ctx] (4) 



C being an arbitrary constant which of course may be changed 

 into an arbitrary symmetrical function of x and ctx, or, which 

 is the same, into such a function of x that it does not change 

 by the substitution of ctx for x. Let *<f,x be such a function, 

 then, ^x= \f k x + <px. <f,x —fx. <p ctx. <f,etx. (5) 



Now since <p is perfectly arbitrary, it is evident that the li- 

 mitation of (f, to symmetry does not prevent the product 

 <px.<f t x from being likewise perfectly arbitrary. Hence 



\J/.r = \fx + <px —fx.qctx (6) 



is as general a solution as (5). 



This indeed might easily have been deduced from (4) by 

 including the constant C in the arbitrary function <p ; but I have 

 chosen this train of argument for the sake of showing that the 

 introduction of an arbitrary symmetrical function does not add 

 to the generality of the solution. 



Mr. Babbage has obtained the general solution of ( 1 ) in the 

 Philosophical Transactions for 1817, under a more complicated 

 form than (6) "by" what Mr. Herschel justly observes "a 

 most singular and ingenious consideration." In Spence's 

 Essays too, p. 163, Mr. Herschel has given for the complete 

 solution tyx = i> t x + ^x. % {*> **} 



" where ty t x and \J/ 21 r are any particular solutions of the re- 

 spective equations 



tyx -\-fx.^ctx =f t x and ^x -\-fx.fyctx = 0." 

 By taking 



■ty t x = \f { x + x — ctx and ^ 2 x = x — ax. 



Mr. HerscheFs solution coincides with our (5) if <p.r = x. 

 But if we compare the solution so obtained with our (6), we 

 shall find the former much inferior in point of generality. For 

 instance, Mr. Herschell's 



<fr* = \fi x + { x — ax \ - x (*» ax )> (?) 



whose first term is the same as the first of ours. Supposing 

 therefore this solution complete, 



9%\*, ctx) ><?x, 

 using the symbol > to signify greater, and < less generality. 

 Of course also 



X (■*> **) >—> 



* <p" or ■£ is employed to signify a symmetrical function of what follows, 

 for the convenience of the printer. 



which 



