304 Mr. stokes, OX THE FRICTION OF FLUIDS IN MOTION, 



that in this case a permanent motion in annuli is impossible, and that, whatever may be the 

 law of friction between the solid sphere and the fluid. Hence it appears that it is necessary to 

 suppose that the particle.s move in planes passing through the axis of rotation, while they at 

 the same time move round it. In fact, it is easy to see that from the excess of centrifugal 

 force in the neighbourhood of the equator of the revolving sphere the particles in that part 

 will recede from the sphere, and approach it again in the neighbourhood of the poles, and this 

 circulating motion will be combined with a motion about the axis. If however we leave the 

 centrifugal force out of consideration, as Newton has done, the motion in annuli becomes 

 possible, but the solution is different from Newton's, as might have been expected. 



The case of motion considered in this article may perhaps admit of being compared with 

 experiment, without knowing the conditions which must be satisfied at the surface of a solid. 

 A hollow, and a solid cylinder might be so mounted as to admit of being turned with different 

 uniform angular velocities round their common axis, which is supposed to be vertical. If both 

 cylinders are turned, they ought to be turned in opposite directions, if only one, it ought to 

 be the outer one ; for if the inner were made to revolve too fast, the fluid near it would have 

 a tendency to fly outwards in consequence of the centrifugal force, and eddies would be produced. 

 As long as the angular velocities are not great, so that the surface of the liquid is very nearly 

 plane, it is not of much importance that the fluid is there terminated; for the conditions which 

 must be satisfied at a free surface are satisfied for any section of the fluid made by a horizontal 

 plane, so long as the motion about that section is supposed to be the same as it would be 

 if the cylinders were infinite. The principal difficulty would probably be to measure accurately 

 the time of revolution, and distance from the axis, of the different annuli. This would probably 

 be best done by observing motes in the fluid. It might be possible also to discover in this 

 way the conditions to be satisfied at the surface of the cylinders ; or at least a law might be 

 suggested, which could be afterwards compared more accurately with experiment by means of 

 the discharge of pipes and canals. 



If the rotations of the cylinders are in opposite directions, there will be a certain distance from 

 the axis at which the fluid will not revolve at all. Writing - 9, for 9, in equation (23), we have 



for this distance 



/ab{bq, + aq.^) 



bqi +aq, 



9. Although the discharge of a liquid through a long straight pijje or canal, under given 

 circumstances, cannot be calculated without knowing the conditions to be satisfied at the surface of 

 contact of the fluid and solid, it may be well to go a certain way towards the solution. 



Let the axis of x be parallel to the generating lines of the pipe or canal, and inclined at 

 an angle a to the horizon ; let the plane yz be vertical, and let y and z be measured downwards. 

 The motion being uniform, we shall have ti = 0, v = 0, w =f(x,y), and we have from equations (13) 



dp dp dp . id" to d^it)\ 



dm 



dp dp . Id'w d'w\ 



In the case of a canal --- = ; and the calculation of the motion in a pipe may always be reduced 



d D 

 to that of the motion in the same pipe when — is supposed to be zero, as may be shown by 



dx 



reasoning similar to Dubuat's. Moreover the motion in a canal is a particular case of the motion 



in a pipe. For consider a pipe for which — = 0, and which is divided symmetrically by the 



dw 

 plane xx. From the symmetry of the motion, it is clear that we must have — = when x = -, 



