acted upon by the Undulations of an Elastic Fluid. 363 



Without knowing the composition of the constant h, it is pos- 

 sible to point to circumstances under which its value may exceed 

 unity. For suppose m and \ for the incident vibrations to be 

 very large ; then as soon as the transverse vibrations are brought 

 into action on the further side of the sphere, the motion of the 

 fluid will partake of the character of direct and transverse vibra- 

 tions relative to an axis, the axis in this case being the prolon- 

 gation of a straight line through the centre of the sphere in the 

 direction of propagation. But for motion of that kind it has 

 been shown that the transverse vibrations have the effect of in- 

 creasing the condensation on the axis, compared with that for 

 the same velocity when the motion is in parallel lines, in the 

 ratio of /c 2 to 1. By a like transverse action the condensation 

 on the further side of the sphere might be so increased as to 

 exceed that on the nearer side ; in which case h would be greater 

 than unity, and the sphere would be drawn towards the origin 

 of the waves, being acted upon as by an attractive force. On 

 the contrary, for very small values of m and X the defect of con- 

 densation on the further side of the sphere might be only par- 

 tially supplied by the lateral confluence, so that h would be less 

 than unity, and the translation of the sphere would be from the 

 origin of the waves, their action resembling that of a repulsive 

 force. In the Theory of Dispersion the term involving 1— A had 

 to be neglected, probably because for luminous undulations h = \ 

 nearly, and the translating power is very feeble. 



I do not purpose at present to discuss further the inferences 

 that may be drawn from the above determination of the accele- 

 ration of the sphere, and shall only remark, in conclusion, that 

 although the principles of the present solution of this problem 

 are fundamentally the same that I have previously employed, I 

 consider the proofs of the preliminary propositions to be more 

 exact, and the details of the general argument to be more logi- 

 cally arranged than in former attempts. As I have already said, 

 this communication professes to contain nothing but mathema- 

 tical reasoning based on admitted premises, and both the reason- 

 ing and the results may be regarded quite apart from any ulte- 

 rior physical applications of them. I cannot, however, but be 

 of the opinion that if the undulations of an elastic fluid can be 

 proved to be capable of causing permanent motions of translation, 

 this fact must materially influence the view that we take of the 

 nature of the physical forces. 



Cambridge, April 11, 1866. 



