of the Rays of Light considered theoretically. 171 



tions from a given point, which is usually taken for the funda- 

 mental principle, is deducible from this. 



Having thus found in what manner a series of undulations 

 is affected when incident upon an immoveable partition, let us 

 consider what will take place if the reflecting plane be suscep- 

 tible of motion, in such a manner, however, that its velocity 

 is always very small compared to the velocity of propagation. 

 Suppose the plane to be acted upon by a force n x, varying as 

 the distance x from the place of rest. Its velocity at the time 



t is — , and the condensation on the side opposite to that on 

 which the waves are incident will be — — . Hence a portion 



a at * 



equal to this of the condensation of the incident waves will not 

 be reflected, but will be transmitted by the motion of the 

 plane: and thus the excess of condensation on the reflecting 



side of the plane will be — (m sin — — \ Consequently 



the plane is acted upon by an accelerative force 



. , / . vr at dx \ 



-nx + k(msm— r- ^-), 

 k being a constant depending on the quantity of matter moved. 



Tj d 2 x , dx j -prat 



Hence, -— — h k -7— + n x — k m sin — — = 0. 



di? dt X 



This equation integrated in the usual way, gives 



-kt 



x = e 2 (c cos h t + c' sin h t) — 



km /,-jfia' 1 - •. . vat , *ak 



(—-»> + 



x*a*k* 



C-Tfia" 1 .. • nat , *ak vrat\ 



_ _ n) sin _+ _ cos — ; 



a 9 



— kt 



The term involving e 1 will soon disappear on account of 

 the diminution of this factor as the time increases, and the 

 motion will then be given by the equations, 



k m 



( — - M > + 



a- 2 o* h* 



X* 



Ka- 4 a 1 \ . -it at <x ale trat \ 



— -V sm — + — cos — ) 



d x a" a km 



dt . X / a- 2 a* \ a^a'A? 



( — s n )H -„ — 



\ X* / X* 



t/T 2 a 2 \ it at 7/ a k . n at \ 



\ ( — — — n ) cos • sin ) 



I \ x 2 / x x x J 



Z 2 Suppose 



