295] STEADY MOTION OF A SPHERE. 531 



The components of stress across the surface of a sphere of radius r are, 

 by Art. 283, 



If we substitute the values ofp xx ,p xv , p xt , ..., from Art. 284, we find 



d ' * L...(viii) 



+ypzy +zpzz^-zp+pr-l\w+iji(xu +yv + zw) 



In the present case we have 



........................... (ix). 



We thus obtain, for the component tractions on the sphere r = , 



x , o/nU 11 z 



-Po + $^-, Pry=--Po, Prz=-- 



If dS denote an element of the surface, we find 



ISp n dS=fyrp'aa, flp r ,dS=0, Hp n dS=0 ............... (xi). 



The resultant force on the sphere is therefore parallel to #, and equal to 



The character of the motion may be most concisely expressed by means of 

 the stream-function of Art. 93. If we put x=r cos 0, the flux (27r>^) through a 

 circle with Ox as axis, whose radius subtends an angle 6 at is given by 



(xii), 

 as is evident at once from (v). 



If we impress on everything a velocity - u in the direction of x, we get 

 the case of a sphere moving steadily through a viscous fluid which is at rest 

 at infinity. The stream-function is then 



(xiii)*. 



The diagram on p. 532, shews the stream-lines \}r = const., in this case, for a 

 series of equidistant values of ^. The contrast with the case of a frictionless 

 liquid, depicted on p, 137, is remarkable, but it must be remembered that the 



* This problem was first solved by Stokes, in terms of the stream -function, 

 I.e. ante p. 518. 



342 



