26 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 



the plane x%, and put a for the inclination of the direction of the force to 00, produced, we 

 shall have 



/=1,m = 0, n =0, I' = k= cos a, m! = 0, ri = sin a ; 



whence 



cos a ' / r\ cos a rl , „ , K , , 



v = o, 



?™/H)-^/ ? <'/<<-<H 



• (40) 



In the investigation, it has been supposed that the force began to act at the time 0, before 

 which the fluid was at rest, so that/ (t) = when t is negative. But it is evident that exactly 

 the same reasoning would have applied had the force begun to act at any past epoch, as remote 

 as we please, so that we are not obliged to suppose / (t) equal to zero when t is negative, 

 and we may even suppose/ (t) periodic, so as to have finite values from t = - « to £ = + o». 



By means of the formulae (39), it would be very easy to write down the expressions for 

 the disturbance due to a system of forces acting throughout any finite portion of the medium, 

 the disturbing force varying in any given manner, both as to magnitude and direction, from 

 one point of the medium to another, as well as from one instant of time to another. 



The first term in £ represents a disturbance which is propagated from 0, with a velocity 

 a. Since there is no corresponding term in r\ or £, the displacement, as far as relates to this 

 disturbance, is strictly normal to the front of the wave. The first term in X represents a 

 disturbance which is propagated from 0, with a velocity b, and as far as relates to this dis- 

 turbance the displacement takes place strictly in the front of the wave. The remaining terms 

 in £ and £ represent a disturbance of the same kind as that which takes place in an incom- 

 pressible fluid in consequence of the motion of solid bodies in it. If/ (t) represent a force 

 which acts for a short time, and then ceases, / (t — t') will differ from zero only between 

 certain narrow limits of t, and the integral contained in the last terms of £ and t will be of 

 the order r, and therefore the terms themselves will be of the order r -2 , whereas the leading 

 terms are of the order r _1 . Hence in this case the former terms will not be sensible beyond 

 the immediate neighbourhood of 0,. The same will be true if/ (t) represent a periodic force, 

 the mean value of which is zero. But if f if) represent a force always acting one way, as for 

 example a constant force, the last terms in £ and ¥ will be of the same order, when r is as 

 large as the first terms. 



28. It has been remarked in the introduction that there is strong reason for believing 

 that in the case of the luminiferous ether the ratio of a to b is extremely large, if not infinite. 

 Consequently the first term in £, which relates to normal vibrations, will be insensible, if not 

 absolutely evanescent. In fact, if the ratio of a to b were no greater than 100, the deno- 

 minator in this term would be 10000 times as great as the denominator of the first term in £ 

 Now the molecules of a solid or gas in the act of combustion are probably thrown into a 

 state of violent vibration, and may be regarded, at least very approximately, as centres of 

 disturbing forces. We may thus see why transversal vibrations should alone be produced, 



