Taking q = x^ then q - y, 



3 '""' I'^Tjl^ 



3 "6 



+ np 



'-{'.^.^y-)--X, 



, , 3 a^ 



7'^^" V'T^ "7'^ 



— J', 



whence 



^ = - pa' 



3 



3 a/ 

 1 + 



16 



npU' 



(2 cos^ a - sin a), [130c] 



Y - - — p a 1 + — — yy + — npV sin a. cos a 



3 \ 16 „3 



130d] 



Here x, y are the components of the acceleration of the sphere. The term in U in X repre- 

 sents a repulsion by the wall on a sphere moving toward or away from it, proportional to 

 1/x , or an attraction half as large on a sphere moving parallel to it. 



(See Reference 1, Articles 98, 99, 187, 138; Reference 2, Section 16.30.) 



Two Spheres 



Instead of a wall, there may be a similar sphere centered at O'and moving at speed U 

 in the direction ^' = 0, so as to secure complete symmetry of motion. 



The general motion of two spheres of any size can be treated in terms of series of 

 spherical harmonics^; the motion has also been treated in terms of images by Hicks, ^'^^ 

 otherwise by Bassett,^' ^^^ and in terms of bipolar coordinates by Endo.^^^ 



131. POINT DIPOLES NEAR A SPHERE 



Consider two point dipoles located at {b^, 0, 0) and (6 0, 0), with their axes parallel 

 to the a!-axis but oppositely directed; let their moments be n , fi , where p^ and p have 

 opposite signs. The resulting stream function, if the fluid is at rest at infinity, is, from 

 Equation [119k], 



-A—z^i— -'^2 — 



[131a] 



326 



