48 MR, A. B. BASSET ON THE MOTION OP 



This equation gives the value of i/ after a sufficient time has elapsed for the motion 

 to have become steady, and agrees with Professor STOKES'S result. 

 5. Let v t be any solution of the partial differential equation 





Then, if v = 0, F(< r)v r dr, where F(T) is any arbitrary function which is inde- 



J o 

 pendent of r and t, and does not become infinite between the limits, will also be a 



solution of (14) ; for, substituting in (14), the right-hand side becomes 



F(0)v, + f F(< - r)v r dr = F(K + f F(< - r) f* dr 



Jo J o 



if f = 0. 



6. The second expression on the right-hand side of (13) is the value of \jj. z sin 2 6 ; and 

 it is easily seen that this expression vanishes when t = 0. Hence it follows that the 

 expression which is obtained from (13) by changing t into r and V into Y'(t T) dr, 

 and integrating the result from t to 0, is also a solution of (1). Now, if F(0) = 0, it 

 will be found in substituting the above-mentioned expressions in (2) and (3) that F(t) 

 is the velocity of the sphere, supposing it to have started from rest ; hence this expres- 

 sion gives the current function due to the motion of a sphere which has started from 

 rest, and which is moving with variable velocity F(t). 



In order to obtain the equation of motion of the sphere, we must calculate the 

 resistance due to the liquid ; but in doing this we may begin by supposing the velocity 

 to be uniform, and perform the above-mentioned operation at a later stage of the 

 process. 



If the impressed force is a constant force, such as gravity, which acts in the direction 

 of motion of the sphere, and Z is the resistance due to the liquid, it can be shown, as 

 in Professor STOKES'S paper, that 



and that 



Z = braUpa cos - p ^ sin 2 0} sin d6, 

 *oV *** / 



where p is the density of the liquid ; also, since 



[p cos sin edO=-% f" sin 2 ^ d0, 



J o J Q uv 



