356 Prof. Challis on the Motion of a small Sphere 



containing cos 6 in the value of a'—cr^ Hence 



— {m x sin qa f t + m\ cos qa't) = — cos q{a!t -f c ), 



3 



3m' . , 3m' 



/ _ 



mi = 2fa' Sm ?C °' m i = ~ 2^V C0S QC °' 

 Substituting these values of m l and m\ and putting c for r, we 

 have finally, for the condensation at any point of the surface of 

 the sphere, 



cr=<r l -\- ~jm! cos q (a' t + c ) cos 6. 



Also from the equation —j*- + -jrr =0 we find, for the velocity 



along the surface, 



3 m' 

 W= — - sin q(a't + c ) sin 6. 



Since, as has been already remarked, the differential equations 

 to this order of approximation are the same whether the origin 

 of the polar coordinates be fixed or moving, the preceding results 

 may be made applicable to the case of a small sphere oscillating 

 in the fluid at rest. In fact, if the vibrations of the fluid be 

 counteracted by impressing equal and opposite vibrations, and if 

 the same vibrations be impressed on the sphere, we pass from 

 the problem above considered to that of the oscillating sphere. 

 But in consequence of these impressed velocities W is evidently 

 diminished by m'sin q(a , t + c ) sin 6, and cr is diminished by the 

 condensation due to the state of vibration of the incident fluid — 



that is, by a-^^ ^ — ~ c cos 6. After subtracting these quantities, 



the remaining values of a and W are precisely those which are ob- 

 tained by the usual mode of solving the problem of the disturb- 

 ance of the fluid by the sphere. Now it is to be observed that 

 in the case of each problem the velocities and pressures are the 

 same, as far as regards approxiuiations of the first degree, whether 

 the fluid be considered very elastic or wholly incompressible. 

 But if the fluid be incompressible, its action on the sphere at rest 

 will be the same as if a cubical portion of it, rigidly separated 

 from the rest, and having the sphere near its centre, were com- 

 pelled to oscillate bodily in a direction perpendicular to a separa- 

 ting plane, provided always the rigid boundary be far beyond the 

 limit of any sensible effect of the sphere's reaction. It is evident 

 that under these circumstances as much fluid passes a plane 

 through the centre of the sphere perpendicular to the direction 

 of incidence as would have pased if the sphere had not been there, 

 which is exactly what the preceding analysis indicates. Con- 



