526 VISCOSITY. [CHAP, xi 



o&amp;gt; a 2 /i&amp;gt; is small. If we put v = -018 (water), and a = 10, we find that the 

 equatorial velocity co a must be small compared with 0018 (c. s.)*. 



When the terms of the second order are sensible, no steady motion of 

 this kind is possible. The sphere then acts like a centrifugal fan, the motion 

 at a distance from the sphere consisting of a flow outwards from the equator 

 and inwards towards the poles, superposed on a motion of rotation f. 



It appears from Art. 286 that the equations of motion may be written 



= ^- + ^%, &c., &c 

 where 



Hence a steady motion which satisfies the conditions of any given problem, 

 when the terms of the second order are neglected, will hold when these are 

 retained, provided we introduce the constraining forces 



X=2(wr)-v\ Y=2(u-w), Z=2(v-ur)) ......... (iii)t 



The only change is that the pressure p is diminished by ^pg 2 . These forces 

 are everywhere perpendicular to the stream-lines and to the vortex-lines, and 

 their intensity is given by the product 2^co sin ^, where o&amp;gt; is the angular 

 velocity of the fluid element, and ^ is the angle between the direction of q 

 and the axis of to. 



In the problem investigated in this Art. it is evident d priori that the 

 constraining forces 



X=-A, F=-fy, Z=0 ............... . ........ (iv). 



would make the solution rigorous. It may easily be verified that these 

 expressions differ from (iii) by terms of the forms -dQ/dx, dQ/dy, -dQ/dz, 

 respectively, which will only modify the pressure. 



293. The motion of a viscous incompressible fluid, when the 

 effects of inertia are insensible, can be treated in a very general 

 manner, in terms of spherical harmonic functions. 



It will be convenient, in the first place, to investigate the 

 general solution of the following system of equations : 



VV = 0, VV = 0, W = ............... (1), 



* Cf. Lord Rayleigh, &quot; On the Flow of Viscous Liquids, especially in Two 

 Dimensions,&quot; Phil. Mag., Oct. 1893. 

 t Stokes, 1. c. ante, p. 515. 

 Lord Rayleigh, L c. 



