﻿292 Mr. G. J. Stoney on the Cause of the 



consists of plane waves. Then, whatever the form of these waves, 

 the relation between the displacement of an element of the sether 

 and the time may be represented by some curve repeated over 

 and over again. This curve may be either one continuous curve, 

 or parts of several different curves joined on to one another. In 

 the latter case (which includes the other) one of the sections of 

 the curve may be represented by the equations 



y=:<b Q (x) from x — to x—x v 



y=z$ x {x) from^? = a? 1 to x=x 2 , I (i) 



and so on to 



y=cj) l {x) from x~=.x l to x-=-2ir> 



y being the displacement, and x being an abbreviation for 2-7T -, 



where Tis the complete periodic time of one wave. 



The undulation in vacuo will then be represented, according to 

 Fourier's well-known theorem, by the following series : 



y=A + Aj cos# + A 2 cos2# + . . . "1 ■ 



+ Bj sin x + B 2 sin 2x + . . ., J 



where the coefficients are obtained from equations (I) by the de- 

 finite integrals 



f*2ir 



I ycosnx, dx = irA n , 1 



\ • • • • (3) 



f 



y sin nx, dx = 7rB, 



'J 



Equation (2), the equation of the undulation before it enters 

 the glass, may be put into the more convenient form 



y — A =C 1 sin(^ + a 1 )4-C 2 sin(2a?-f-« 2 ) + ..., . (4) 



where y — A is the displacement from the position of rest, and 

 the new constants are related to those of equation (2) as follows : 



C„=^Af+Bs, «„=tan--'^ (5) 



The first term of expansion (4) represents a pendulous vibration 

 of the full period t ; the remaining terms represent harmonics 

 of this vibration; i. e. their periodic times are Jr, §r, &c. All 

 of these also are pendulous ; so that equation (4) is equivalent to 

 the statement that whatever be the form of the plane undulation 

 before entering the glass, it may be regarded as formed by the 

 superposition of a number of simple pendulous vibrations, one 

 of which has the full periodic time t, while the others are har- 

 monics of this vibration. 



