[100] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION 



which in fact is the same as the second of the equations (8). The solution thus obtained is as 

 we have seen 



«' = wf(z, t), (181) 



/denoting a function the form of which there is no need to write down, which satisfies (180) 

 when written for v. Now it will be seen at once that the expression (181) satisfies the exact 

 equation (179), and therefore the approximate solution obtained by the method of Art. 8 is in 

 fact exact, except so far as regards the termination of the disk at its edge, which is what it was 

 required to prove. 



Passing from semi-polar to polar co-ordinates, by putting z = r cos 6, -sr = r sin 8, we get 

 from (179), after writing /x'p for fx, 



dV 2 dv l d I . dv'\ v 1 dv 



l? + r'd^ + 7^8 dlV me de) ' ?~^e "f/lF (182) 



Suppose now the solid to be a sphere, having its centre at the origin. Let a be its 

 radius, a its angular velocity, and suppose the fluid initially at rest. Then v' is to be deter- 

 mined from the general equation (182) and the equations of condition 



«'= when t = 0, v' = as sin 9 when r = a, v = when r • «o . 

 All these equations are satisfied by supposing 



v = v sin Q, 

 v" being a function of r and t only. We get from (182) 



dV' 2 dv" 2w" _ l dv" 



17 + r "d7 ^~^~dJ (,83) 



If we suppose 8 constant, v" will tend indefinitely to become constant as t increases inde- 



dv" 

 finitely, and in the limit — = 0, whence we get from (188) and the equations of condition 



at 



«"= aH when r = a, v"= when r = <» , 



v = ~zr> v = ii~ sin0. 



This is the solution alluded to in Art. 8 of my paper On the Theories of the Internal 

 Friction of Fluids in motion, fyc. 



Note B, Article 65. 



Let us resume the problem of Art. 7, but instead of the motion of the plane being 

 periodic, let us suppose that the plane and fluid are initially at rest, and that the plane is 

 then moved with a constant velocity V, and let the notation be the same as in Art. 7. 



