Solid Bodies through Viscous Liquid. 699 



only a little ; but when b is great enough v may acquire any 

 value up to unity. And since the distinction depends upon 

 what occurs at infinity, it may evidently be extended on the 

 one side to oval surfaces o£ any shape, and on the other to 

 cylinders with any form of cross-section. 



In the analogy already referred to there is correspondence 

 between the displacements (a, /3, 7) in the first case and the 

 velocities (u, i\ 10) which express the motion of the viscous 

 liquid in the second. There is also another analogy which is 

 sometimes useful when the motion of the viscous liquid takes 

 place in two dimensions. The stream- function (yjr) for this 

 motion satisfies the same differential equation as does the 

 transverse displacement («/) of a plane elastic plate. And a 

 surface on which the fluid remains at rest (^ = 0, dyfr/dn = 0) 

 corresponds to a curve along which the elastic plate is 

 clamped. 



In the light of these analogies we may conclude that, pro- 

 vided the square of the motion is neglected absolutely, there 

 exists always a unique steady motion of liquid past a solid 

 obstacle of any form limited in all directions, which satisfies 

 the necessary conditions both at the surface of the obstacle 

 and at infinity, and further that the force required to hold 

 the solid is finite. But if the obstacle be an infinite cylinder 

 of any cross-section, no such steady motion is possible, and 

 the force required to hold the cylinder in position continually 

 diminishes as the motion continues. 



§ 3. For further developments the simplest case is that of 

 a material plane, coinciding with the coordinate plane ^ = 

 and moving parallel to y in a fluid originally at rest. The 

 component velocities u, w are then zero ; and the third velocity 

 v satisfies (even though its square be not neglected) the 

 general equation 



dt 



d 2 v 



in which v. equal to fi/p, represents the kinematic viscosity. 

 In § 7 of his memoir Stokes considers periodic oscillations of 

 the plane. Thus in (I) if v be proportional to e int , we have 

 on the positive side 



v= .^ e int e -x s /(iny) (5) 



When # = 0, (5) must coincide with the velocity (V) of the 

 plane. If this be Y n e int , we have A = V„: so that in real 



2 Z 2 



