﻿544 Mr. W. Ellis Williams on the 



of space outside the sphere, and its value is 



^Val J I r(cos 2 6+ ± sin- 0)drd0d(f> 



* Jr=a J 0=0 J 0=0 



This is evidently infinite even when V is infinitely small ; 

 now this momentum is produced by the force which act- 

 on the sphere maintaining its velocity against the resistance 

 of the fluid ; and since this force is proportional to the 

 velocity and hence infinitely small, it must have acted for a 

 time which is of the second order of infinite quantities. For 

 the solution to be valid the sphere must therefore have 

 moved from infinity with the same infinitely small velocity, 

 otherwise the motion is not steady and equation (4) does not 

 apply. 



Now it is evident that if the velocity is not infinitely 

 small, steady motion cannot be established, even when the 

 sphere starts from infinity, as the time during which the 

 force acts is now infinite of the first order only. The 

 precise extent of the departure from steady motion may be 

 found by considering the case of a sphere starting from rest 

 and proceeding with a small uniform velocity. This case 

 has been solved by Bassett (Hydrodynamics, ii. p. 286), 

 whose solution may be written 



3V« sin 2 ^|2^ r „ 



^=^o J-~ \ [*2XV + 2aX v' j^/tt + Ka 2 -r 2 )K A "d\. 



r V 73 " J* 



When r is small and t great the second term vanishes and 



the motion is the same as in the steady state, but when r is 



large the integral does not vanish even for very large values 



v 

 of t : for the motion is onlv steadv when — _ vanishes, and 



s/t 



hence at points very distant from the sphere the motion 

 never becomes steady. The conditions are not very much 

 improved when the fluid is confined by an outer boundary, 

 so that only small values of r need be considered, for the 

 time during which motion is possible without completely 

 altering the boundary conditions diminishes in the same 

 proportion as the dimensions of the boundary. The only 

 case where a steady motion is experimentally realisable is 

 when the outer boundary is a tube or elongated vessel, for 

 then the motion of the fluid is confined to the portion of 

 the tube in the neighbourhood of the moving sphere, and 



