﻿Equations of Motion of a Viscous Fluid. 453 



unci we Lave 



5(f,n)__/ ,, a 



B(«. *) 



("*-i)-°- • • • a*> 



Eliminate ■% between (11) and (13) and after some reduction 

 we obtain 



;(»S + &«*)) + s $3f a --(--s)«<« 



. . . (15) 



Then (14) and (15) are the equations for this type o£ motion. 

 One interesting result which flows from these equations is 

 that the only possible motion, which consists of pure circula- 

 tion about the axis without any accompanying motion in the 

 meridian plane, is that generated by the rotation of two 

 infinite coaxial circular cylinders about their common axis. 

 Put ty = const, and (15) becomes 



and hence from (11), taking the case of steady motion, 

 dm _ 1 dQ _ 



dijr 2 -57 dm 

 the solution of which is 



n = Abt 2 + b, 



or . B 



V = Al37 H — , 



which is the solution referred to. Hence if any body of 

 revolution other than a cylinder be rotated about its axis in 

 a viscous fluid, the consequent motion of the fluid about the 

 axis must in all cases be accompanied by a certain motion 

 in the meridian plane. For the special case of a sphere 

 this was pointed out by Stokes*". 



When the motion is entirely in the meridian plane, we 

 have il — 0. Equation (14) is identically satisfied and (15) 

 becomes 



d(w, s) V ot -or d: / 



* ' Mathematical and Physical Papers/ vol. i. p. L03, 



