1887.] 



Motion of a Sphere in a Viscous Liquid. 



175 



+/W^6-^(l-i\/), 

 . = ( (1— ) + V<-»-fla { (i* *) 0(0 } 



where 



/=^17- x = *"> 0(O = JU(^)-^, 



/> being the density of the liquid, <r that of the sphere, and a its 

 radius. 



It thus appears thai after a very long time has elapsed, the accelera- 

 tion will vanish and the motion will become steady. The terminal 

 velocity of the sphere is /A- -1 , which is seen to agree with Professor 

 Stokes's result. 



If the sphere were projected with velocity V, and compelled by 

 means of frictionless constraint to move in a horizontal straight line, 

 the values of the acceleration and velocity would be obtained from the 

 preceding forrnulee by expunging the terms fe~ Kt , //\. _1 (1 — e -Ai ), in 

 the expressions for v and v respectively, and then changing f into 



-vx. 



The preceding results can only be regarded as a somewhat rough 

 representation of the actual motion, for (i) the square of the velocity 

 has been neglected ; (ii)no account has been taken of the possibility of 

 hollow spaces being formed in the liquid ; (iii) if the velocity of the 

 sphere became large, the amount of heat developed would be sufficient 

 to vaporise the liquid in the immediate neighbourhood of the sphere, 

 and the circumstances of the problem would be materially changed. 



In the latter part of the paper I have considered the problem Of a 

 sphere, surrounded by a viscous liquid, which is set in rotation with 

 given angular velocity, Q, about a fixed diameter, and similar results 

 are obtained. To a first approximation the angular velocity is equal 

 to Qe -Ai , where X is a positive constant, which shows that the motion 

 ultimately dies away. 



