MATHEMATICAL NOTE*. 



Solution of a Functional Equation. 

 To solve the functional equation 



,+oo 



<t> m x<f> n (a. -x)dx = (^a, 



J -00 



provided <f>x = <f> ( x). 



Let I <f>QLCOSOLzda = '\Irz. Then 



J -00 



^Wt.3 = I da I dx<t> m x(f> n (a n) cos ctz. 



J -CO J -00 



Write for a, a + x, then cos a.z becomes 



cos aiz cos xz sin az sin #2, 

 and as the sines change sign with their arcs, we get 



/.+00 /.+ 00 



^mfn^ = I dy. I dx(j> m x<f> n a. cos a^ cos a?z. 



J -00 J -00 



Or <*lr m+n z = ^ m ^ n z t 



whence ^=X^ 



% being independent of m, and therefore, by Fourier's theorem, 



1 



z cos ^^ 



"^^o 



The required solution. 



* Cambridge and Dublin Mathematical Journal, Vol. vn. p. 103, 1852. 



