270 Prof. Challis on Hydrodynamics. 



pressure on the opposite surface, and estimated in the direction 

 of the incidence of the waves, is 



27rjV(o-- S )c 2 sin cos dd6, from 0= | to 0=7r. 



This pressure is found to be 



16Afl 3 c 4 Sm\. 



27\ b '"m ' 



Affrr^ A 

 and the mass of the sphere being — - — , the accelerative force is 



o 



4<7r 5 fca 3 c Sm'j 



9/V 5 A m~" 



By the considerations already applied to the condensation S, it 

 follows that in this case also the waves tend to produce a perma- 

 nent motion of translation of the sphere, and that the acceleration 

 from a centre varies inversely as the square of the distance. Since 

 the above expression contains c,the acceleration is not independent 

 of the magnitude of the sphere. Also the direction of the motion 

 of translation is from the origin of the waves if m\ be a positive 

 quantity, and the movement is like that produced by a repulsive 

 force. 



I have now reached the point to which I proposed to carry 

 these researches in the present communication. It was required 

 to ascertain whether or not the undulations of an elastic medium 

 are capable of causing permanent motions of translation of a 

 small sphere ; and the question has been answered affirmatively. 

 The investigation has shown that the effect is due to the circum- 

 stance that, on expressing to terms of the second order the rela- 

 tion between the condensation and velocity of a wave, the amount 

 of condensation is found to be greater than that of rarefaction, 

 and that, while the excess of condensation causes the excursions 

 backwards and forwards of a given particle of the fluid to be 

 exactly equal, the excursions of a spherical solid submitted to 

 the action of the waves are by that very excess made unequal. 

 Although the mathematical process by which this conclusion 

 has been arrived at is intricate, the rationale of the result may be 

 understood from common mechanical principles — it being evident 

 that as the solid does not change its density like the fluid, the 

 action upon it of the greater condensation cannot be counteracted 

 by that of the smaller rarefaction. 



The foregoing mathematical reasoning is incomplete, inasmuch 

 as it does not determine the composition of the constants m x and 

 m' 1} and the relation of the one to the other in a given case of 

 motion. Tor the present I reserve the consideration of these 

 points. In the meanwhile the reasoning, as far as it has gone, 



