acted upon by the Undulations of an Elastic Fluid. 359 



The resultant of the pressures at all points of the surface, esti- 

 mated in the direction of the incidence of the vibrations, is 



2t7t 2 jV(o— a } )d0 sin d cos 0, 



taken from = to #=7T. Between these limits the integral 

 relative to the term containing cos 2 6 is evidently zero, and the 

 resultant pressure is consequently 



27rm!qa <2 c 3 , , ± . 

 f cos q (a't + c ) . 



Hence, supposing the mass of the sphere to be — = — , the acce- 



o 



lerative action of the fluid on the sphere is 

 -j^r cos q{dt + c ), 



which, it may be remarked, is independent of the magnitude of 

 the sphere. 



In order to complete the determination of the resultant of the 

 pressure on the sphere, other considerations have now to be en- 

 tered upon, from which it will appear that the action of the fluid 

 on the second hemispherical surface is not, as the reasoning so 

 far indicates, of exactly the saaie amount as that on the first. 

 The circumstance which modifies the pressure on the second he- 

 misphere is the composite character of the incident vibrations, 

 in consequence of which, as soon as they are propagated beyond 

 the first hemisphere and direct incidence ceases, the transverse 

 vibrations come into play, being no longer neutralized. The 

 consideration of this particular action of the fluid is an essential 

 part of my Theory of the Dispersion of Light, and I shall there- 

 fore have occasion here to say little more than what is contained 

 in the Supplementary Number of the Philosophical Magazine for 

 December 1864. In the extreme case of vibrations so rapid that 

 the value of A. is small compared to the radius of the sphere, the 

 transverse action might have the effect of preventing the undu- 

 lations extending more than a short distance along the second 

 hemisphere, so that the fluid in contact with the greater part of 

 its surface would be at rest. This is not the case of the present 

 problem, because c has been assumed to be extremely small com- 

 pared to X, and the excursion of a given particle of the fluid may 

 be supposed to be equal to, or even much larger than, the radius 

 of the sphere. But in all cases the transverse action will cause 

 the state of the fluid" on the further side of the plane through 

 the sphere's centre to differ from that on the other, and the cal- 

 culation of the exact amount of the difference should be within 

 the reach of analysis. This part of the problem, however, which 



2B2 



