A SPHERE IN A VISCOUS LIQUID. 55 



different from the actual motion ; and if this should turn out to be the fact, the 

 solution of (18) applicable to this case must be obtained by some different method. 



Equation (39) shows that after a very long time has elapsed the acceleration 

 vanishes, and the motion becomes ultimately steady; in other words, the acceleration 

 due to gravity is counterbalanced by the retardation due to the viscosity of the liquid. 

 When this state of things has been reached, the terminal velocity of the sphere is 



This agrees with Professor STOKES'S result, who applies it to show that the viscosity 

 of the air is sufficient to account for the suspension of the clouds. 



10. We shall now consider the motion of a sphere which is surrounded by an 

 infinite liquid, and which is rotating about a fixed diameter. 



We shall begin by supposing that the angular velocity of the sphere is uniform 

 and equal to o>, and shall endeavour to obtain an expression for the component 

 velocity of the liquid in a plane perpendicular to the axis of rotation, on the supposi- 

 tion that no slipping takes place at the surface of the sphere. 



Assuming that the motion of the liquid is stable, it is easily seen that none of the 

 quantities can be functions of ^, where r, 6, and <f> are polar coordinates referred to the 

 centre of the sphere as origin. If, therefore, we neglect squares and products of the 

 velocities, the component velocity, v, of the liquid, perpendicular to any plane con- 

 taining the axis of rotation, is determined by the equation 



dt ~ 



and if in this equation we put v' = r sin 0, where r is a function of r and t only, the 

 equation for v is 



Pp 2 dr 2o 1 dr 



jj + . , = - -r (**) 



The value of the tangential stress per unit of area which opposes the motion of the 

 sphere is 



_, / 1 rfR <fi/ J\ 



T = vp( . -,. + : I, 



rf$ dr T i 



where R is the radial velocity ; but, since R is not a function of <ft, the value of this 

 stress depends solely on that of f*. Now Professor STOKES has pointed out that 

 unless the motion of the sphere is exceedingly slow, the motion of the liquid will not 

 take place in planes perpendicular to the axis of rotation, but the velocity of every 

 partu-le will have a component in the plane containing the particle and this axis. But 



