Mr. J. G'Kinealy on Fourier's Theorem. 95 



and tell their own tale. The conditions under which the phe- 

 nomena invariably occur will give the laws ; and the theory will 

 follow without much difficulty. To use the eloquent language 

 of Sir Humphry Davy, "When I consider the variety of theories 

 which may be formed on the slender foundation of one or two 

 facts, I am convinced that it is the business of the true philo- 

 sopher to avoid them all together. It is more laborious to accu- 

 mulate facts than to reason concerning them ; but one good ex- 

 periment is of more value than the ingenuity of a brain like 

 Newton's/' 



XIV. Fourier's Theorem. 

 By James CPKinealy, Bengal Civil Service*. 



THE proof given of Fourier's theorem in all the text-books I 

 know of, is a modified form of that first given by Poisson. 

 What is at present proposed is to prove it by an analytical pro- 

 cess for periodic functions, and to show that it is simply the 

 solution of an exponential differential equation. 



If f{x) =zf(%+X), where X is the wave-length, we have, putting 

 it into the symbolical form, 



or 



It is a Well-known theorem in differential equations, that if we 

 get an equation of the form F(DJ/(a')=0, and can find the 

 roots of F (DJ=0, the equation can be put in the form 



(D,- fl )(D.- fll ) (D,- ff2 ).../W = 0, 



where a, a v # 2 , &c. are the roots of F(DJ = 0, and that the 

 solution will be 



/(a?) = Ae™ + A ] e f; > 3 ' + A 2 €°** , 



A, Aj, A 2 , &c. being constants depending on the nature of/(#). 

 In the present case F(DJ is e Al> ^— 1. Assume D x =zz, and 

 the equation to solve is 



e ^_l=0, 

 or 



e ¥~_ 6 -2*=0, 

 or 



or 



= 0, 



* Communicated bv the Author. 



