106 Mr. John Herapath on Fimdional Equations. 



function of everj- possible change x can undergo, combined in 

 every possible way the n contemporaneous equations require. 



It should he observed, that one of the two above methods 

 may under certain very simple circumstances give a resulting 

 equation of identity, in which \J; x eliminates itself: but I be- 

 lieve this can never happen with both methods at the same 

 time. We maj' likewise add, that though no constants or sjm- 



metrical functions of .r, ex. x, cc x, . . . a"~ x appear in the pro- 

 posed equation, thej" mav nevertheless exist in it in any man- 

 ner we please, consistent with tlie conditions of possibiUty. 



I shall now exemplify the above results with two or three 

 cases, which will give me an opportunity of correcting an in- 

 ference in one of my former papers, and of introducing a me- 

 thod or two of obtaining the complete from a particular solu- 

 tion. We will commence with the latter method. 



Suppose ^fx is a particular solution of any functional equa- 

 tion whatever. Then it is plain there can be no expression 

 which may not be equated with fx. (p x or fx + ^ x, which- 

 ever we please. Consequently, if this product or sum be put 

 for -^ X m the given equation, the properties of this equation 

 will restrict the unlimited generality of (p just as much as the 

 complete solution requires. For example, in the equation 

 •\>x -\-fx.-^a.x = (8) 



in which «- .r= x and the condition of }X)ssibility isfx .fax=^ 1, 

 a particular example is 1 —fx. ^Multiplying this by ^ j and 

 substituting ( 1 —fx) ^ x for vj/ x and ( 1 —fa. x) fax for rj/ « x, 

 we shall at length find c x = (pux. 



Therefore the property of <p must be such that, whether it be 

 a function or any system of functions and operations of any 

 kind on x, it must give the same result when performed on .r 

 or on ux. In matters of such unbounded generality it would, 

 perhaps, be too much to say, that $ confined to what is usually 

 understood bv functional operations could answer every pos- 

 sible case. However, as we are speaking of functions, we will 

 suppose the conditions of $ may be effectually fulfilled b}' 

 some function. It will then of course be any symmetrical 

 function of jt and ux; and the complete solution will be 



4f X = (1 —fx). [$T + $«x] 

 which coincides with 



^ X = 1^ X — fx .fax 



obtained by the second of our general methods. For both 

 solutions have the same properties ; and if respectively divided 

 by 1 — fx, are arbitrary symmetrical functions of .r, a .r. 



As 



