424 Mr. Herapath on the Solution of ^ n x = x. [Dec. 



tkXz 2k"\z . , f. . . 2 fcX * 

 , tan , .... are respectively iunctions of sin ; 



n n i J n ' 



and consequently whether we admit <p , <p 4 , .... to be compre- 

 hended in <p or not the whole expression (35) sinks to an arbi- 

 trary function of sin -, that is to <p sin . And the 



same coincidence might easily be shown to be true of any other 

 apparently different integral, which from the nature of (34) must 

 contain circular or equivalent periodic functions, so tnat (34) 

 has really but one integral, involved, however, in an arbitrary 

 function. This is likewise confirmed by a very simple, direct, 

 and general method that has occurred to me of integrating equa- 

 tions of differences of, I believe, all orders and degrees; and 

 which method is applicable to the direct numerical resolution of 

 algebraic equations of all degrees as well as to other purposes, 

 of which I may say more hereafter. Hence (25) or (26) does 

 not admit of variety from variety of integral, but being obtained 

 directly and without any limiting conditions from the integration, 

 it must possess an equal degree of generality with the integral 

 itself, that is, the integral being of the most general kind, the 

 functional solution must be so too, or in other words complete. 

 A complete solution, therefore, of ^" x = x has only one arbi- 

 trary function. 



A direct proof that our (25) is the complete solution may be 

 given thus. Let <p a" <p~ l x denote (25), or any other solution 

 wherein a" is the particular form of \J/" x when $ x = x. Sup- 

 pose / " x is also any solution whatever that will fulfil the condi- 

 tion/" x = as. Put/" x = <p a. r <p~ l x, and changing x into <p x, 

 we have/" <p x = <p a" x, in which <p is the function to be deter- 

 mined. Now because f and « are periodics of the same order, 

 this equation is always possible, whatever be the forms of / and 

 « ; and indeed I have in another place given general solutions of 

 this very problem. It is, therefore, evident, that such a form 

 can be given to the arbitrary function in (25), that this solution 

 shall coincide with any other solution whatever. Consequently 

 (25) comprehends every solution, and is, therefore, the complete 

 solution. 



This I believe is the first direct and legitimate demonstra- 

 tion that has been given of the completeness of a functional 

 solution. 



■ Hence it follows, that the arbitrary constants a,., b r , c r , in (21) 

 form a part of the arbitrary function in (25). This is further 

 confirmed by the assumption with which we set out at the com- 

 mencement of § 3, which, being arbitrary, of course gives an 

 arbitrary form to the functional root. Arbitrary functions, 

 therefore, of whatever kind or quantity they may be, substituted 

 for these constants, merely clog the solution without at all con- 

 tributing to its generality ; since these functions, and the 



