44 MR. A. B. BASSET ON THE MOTION OP 



Throughout the present investigation terms involving the squares and products of 

 the velocity will be neglected. This is of course not strictly justifiable, unless the 

 velocity of the sphere is slow throughout the motion. If, therefore, the velocity is 

 not slow the results obtained can only be regarded as a first approximation ; and a 

 second approximation might be obtained by substituting the values of the component 

 velocities hereafter obtained in the terms of the second order, and endeavouring to 

 integrate the resulting equations. I do not, however, propose to consider this point 

 in detail. 



2. In the first place it will be convenient to show that the equations of impulsive 

 motion of a viscous liquid are the same as those of a perfect liquid. 



The general equations of motion of a viscous liquid are 



du du du du . 1 dp 



-jl + u- +V+W- -- X + -J 1 vv 2 u = 0, 



dt da dy dz p dx 



with two similar equations, where v is the kinematic coefficient of viscosity. 



If we regard an impulsive force as the limit of a very large finite force which acts 

 for a very short time T, and if we integrate the above equation between the limits 

 T and 0, all the integrals will vanish except those in which the quantity to be inte- 

 grated becomes infinite when T vanishes ; we thus obtain 



1 d 



1 d ( T 



U UQ+ - I pdr = 0. 

 dx} Q r 



Putting f pdr = us where CT is the impulsive pressure at any point of the liquid, 

 we obtain 



p (u u ) + ^ = 0, &c., &c., 



which are the same equations as those which determine the impulsive pressure at any 

 point of a perfect liquid. 



3. Let us now suppose that a sphere of radius a, is surrounded by a viscous liquid 

 which is initially at rest, and let the sphere be constrained to move with uniform 

 velocity V, in a straight line. If the squares and products of the velocity of the 

 liquid are neglected, Professor STOKES has shown that the current function t/ must 

 satisfy the differential equation 



where 



<P sin0 d 



and (/, 9) are polar coordinates of a point referred to the centre of the sphere as 

 origin. 



