ESSAYS 511 



by Fourier's theorem we obtain 



<j, m (a) = ~f o Cos ax [x(x)]>»dx, 



where x( x ) >s an arbitrary function subject to certain conditions. 



Herschel J has shown that ainumber of very general types of functional equations 

 can be reduced to difference equations by means of an ingenious artifice, and it 

 seems likely that this method can be extended so as to give a reduction to equa- 

 tions involving distributive operations of other types. A few of Herschel's 

 examples will now be given to illustrate his method. 



(1) If P and Q are given functions of x, the functional equation 



f[x, P(x)] = +[x, Q(xj] 

 is satisfied by 



W*,y) -/W + O - n*)Iy - QLx)Hx>y), 



where f(x) and x( x , y) are arbitrary functions. 



(2) To solve the equation -^(x, x) - ^r(x, o) = a, where a is a constant. 



Assume ty(x,y) = <f>(*> h + y\x), 



then the equation becomes 



$(.r, h + 1) - <\>{x, h) = a, 



the solution of which is <£(.r, h) = ah + F(x, h), where F is a periodic function 

 of h of period 1. Substituting for <p and putting /« = owe get 



\j,(x,j,) = aZ + F(xS), 



X \ Xj 



where F(x, 1) = F(x, o). According to (1) the solution of this is 



F(x,z) =/(*) + z(z - i) X (x t z). 

 Hence, finally, we get 



+(x,y) = a y- +/(x) + y(y - x)G{x,y\ 



where /and G are arbitrary functions. 



(3) Solve the equation F{x,y\r(x,P),^(x,Q)} = o. 



Let 6{x,y) be a function such that 6(x,P) = 0, 6(x,Q) = 1 ; these conditions 

 may be satisfied by putting 



*?#* == i^7> + {y ~ p){y ~ 2)*(- r >-^- 



Now write ^(x,y) = <j>[x,/i + 6(x,y)], then the functional equation is replaced by 

 the difference equation 



F{x,<f>(x,/i), cf>(x,h + 1)} = o. 



For further information on the calculus of functions we may refer to Herschel's 

 Calculus of Finite Differences (Cambridge, 1820); Boole's Finite Differences 

 (i860); D. F. Gregory's Mathematical Writings (1865) ; G. Oltramare, Calcul de 

 generalisation (Paris, 1899); Abel, CreUe^s Journal, 1826, 1, II ; 1827,2; A. R. 

 Schweitzer, Bulletin of the Americati Mathematical Society, 1912,18, 299; 1912, 

 19, 66 ; 1914, 21, 23. 



The theory of distributive operations is too extensive to summarise here. 



1 Cambr- Phil. Trans. 1820. 



