114 



Prof. H. Lamb on the Uniform 



at points distant from the sphere. Since, however, both the 

 constraining forces and the viscous forces are m these regions 

 relatively small, it does not necessarily follow that the 

 character of the motion in the immediate neighbourhood of 

 the sphere will he seriously affected. At points near the 

 sphere the constraining forces tend to vanish, whilst the 

 viscous forces are of the order vUa/r 3 . 



The innovation made by Oseen in the treatment ot the 

 question consists in writing U + u for u, and neglecting terms 

 of the second order in u, v, w only. These symbols now 

 denote the components of the velocity which would remain 

 if a uniform velocitv -U were superposed on the whole 

 system. The hydrodynamical equations accordingly take 

 the forms 





ith 



a# p oy 

 B# p oz 



Si + ^ + aJrV- 



y 



(10) 



(11) 



The inertia terms are thus to some extent taken into 

 account, but it is to be remarked that although the approxi- 

 mation is undoubtedly improved at infinity, where u, v, w = 0, 

 it is in some degree impaired near the surface of the sphere 

 where we now have u= — U. This will be a matter for 

 subsequent examination. 



The solution of the equations (10) and (11) for the purpose 

 in hand can be effected very simply. In the first place we 

 have 



VV = °> ...... (12) 



and a particular solution is therefore obtained if we write 



p=pV 



Be/) 



ft* 



w=- 



B4> 



(13) 

 [14) 



where </> satisfies 



C 2 </> = 0. 



