342 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE 



the others, our approximation will be exact to at least terms of the 

 second order. 



To determine the arbitrary constant in the equation above, let us 

 suppose that when t = 0, p = 1. Hence 



Ca 



—t- + ma sin a cos a = ; 

 r 



j 2 xt i f ~T . (2-n-at \) m* . n %wat 

 and a Nap. log. p = ma sin a I — cos o e r + cos I a > — — sin- . 



It is worthy of remark, that if we confine ourselves to quantities of 

 the second order, the above result coincides with that obtained in Art. 3, 

 for an incompressible fluid. For to that degree of approximation 



2ttT 



sin a = — — ; and if p « I + <r, a: Nap. log. p = a* a. Also the term in- 



at 



volving e~ T will disappear after a very short time on account of the 



great magnitude of -. Hence 



„ ■ 2ira Qirat m? . .Qicat 



are = m . . r cos — sin* — - — , 



X X 2 X 



which, by putting b for , evidently coincides with the result ob- 



X 



tained in Art. 3. 



5. Prior to the consideration of the dynamical action of the fluid in 

 vibration on a small sphere, it will be convenient to determine the pressure 

 on the surface of a small sphere performing small rectilinear vibrations in 

 the fluid at rest. 



The sphere is supposed to be perfectly smooth, and therefore incapable 

 of impressing motion on the fluid in directions perpendicular to the radii. 

 Hence the motion given to the fluid by the motion of the sphere is 

 directed to or from a moving centre. If V be the velocity of the sphere at 

 the time t, and 9 the angle which a radius to any point of the surface makes 

 with the straight line in which the centre is moving, VcosO is the velocity 

 impressed on the fluid at that point at the same time. Now as this normal 

 velocity varies at a given instant from one point to another of the surface, 



