364 Lord Rayleigh on the 



stream-function subject to the conditions that both ty and 

 dtyjdr shall have values arbitrarily given at all points of 

 the circumference of the circle. It is not necessary actually 

 to write down the formulae ; but it may be well to notice that 

 the same solution applies to the question of determining the 

 transverse displacement w of a thin circular plate when w and 

 dw/dr have arbitrarily prescribed values on the boundary. 



As a preliminary to further questions it will be desirable to 

 consider for a moment the form of the general equations of 

 viscous motion. In the usual notation, 



du , du , du du v 1 dp , _ " . 



dt . dx dy dz p dx v ' 



with two similar equations. Further, if (f denote the re- 

 sultant velocity, and f , r), J be the component rotations, 



du , du du , da* ~ ^ , n /orN 



In steady motion du/dt = ; and if the terms of the second 

 order in velocity (35) be omitted and there be no impressed 

 forces except such as have a potential, the equations reduce 

 to the form already considered. A solution thus obtained for 

 small velocities will fail to satisfy the conditions when the 

 velocities are increased ; but the equations lead readily to 

 an instructive expression for the forces X, Y, Z, which must 

 be introduced in order that the solution applicable without 

 impressed forces to small velocities may still continue to hold 

 good. From (35) we see that the necessary forces are 



^ = i^-^H2w V , .... (36) 



with two similar equations. In this the term ^ dq^Jdx need 

 not be regarded, as it tells only upon the pressure and does 

 not influence the motion. We may therefore write 



X=2w V -2v£, Y=2uZ-2u£, Z = 2v%-2u V . . (37) 



These equations show that 



uX + «Y + wZ = (n 



fX + ^Y+SZ-OJ 7 * * ' * ( 38 ) 



signifying that the force whose components are X_, Y, Z, acts at 

 every point in a direction perpendicular both to the velocity 

 and to the axis of rotation. As regards its magnitude, 



