626 DR WALLACE ON A FUNCTIONAL EQUATION. 



4. There being a necessary connection between the form of a function and 

 its properties, by which, from the former, we may deduce the latter, it follows 

 that, reversely, when the properties of a function are known, we may from these 

 deduce its form. 



Hence it appears that, relatively to the form and properties of a function, 

 there may be a direct and an invei^se theory ; by the one, the properties are de- 

 duced from the form : and by the other, the form from the properties. These will 

 be analogous to other reverse theories ; as involution and evolution, or the direct 

 and inverse methods of fluxions, &c. But here again it happens, as in the theo- 

 ries just mentioned, that the difficulties to be encountered in the inverse are 

 greater than those presented in the direct theory. 



5. Let us take as an example the function y=\ogx; from this, by the defini- 

 tion of a logarithm, x=a}', a being a given number ; we have similarly y=loga^, 

 and af=ay' ; hence, xaf=a}' . ay'=a^^y' , and y +y=log (xxf) ; that is, 



\ogx + \ogxf = \og{xx') . (1) 



Here we have easily deduced a property of the function from its form. 



The reverse problem requires that we find a function of a; whose form is un- 

 known, and which, being expressed by the sjmbol /"(«), has this property. 



/(x) + f{^)=f(x^). (2) 



But the algebraic analysis that so readily applied to the former problem, does not 

 so easily apply to the latter. This last equation (2), in which the form of the 

 function/ (^) is unknown, is called a, functional equation ; and it is resolved, when 

 the equation (1) has by a legitimate process been deduced from it. 



6. In physical inquiries, functional equations may occur, by the solution of 

 which the physical laws and then* consequences may be discovered. I propose 

 in this memoir to give two examples of such an application of this theory, to 

 the doctrine of statics. In the first I shall deduce the known law of the equili- 

 brium of three pressures applied at a point from a functional equation ; and, in 

 the second, from the same equation, investigate some elegant properties of curves 

 of equilibration, which are applicable to the construction of bridges. 



7. Let cc denote a variable quantity, and/(^) a function of the variable ; also 

 let x^ and x^ denote two values of a;, which are entirely independent of each 

 other, and c a constant quantity. Let us suppose the function /{x) to be such 

 as satisfies this equation, 



/(x:).f(x:) = c[f(x^ + x,)+f{x^-x,)} ... A 



It is proposed to determine all the possible forms of the function. 



8. There is a very simple property of a function, from which I propose to 

 deduce the solution. It is this : 



The partial differential coefficient of a function which is the sum of two inde- 



