an Incompressible Viscous Fluid. 349 



For the determination of the constants a and /3 we have only 

 one condition ; so that at most we can only find one of them in 

 terms of the other. This condition is obtained by equating 

 to zero the normal velocity at the surface of the sphere ; if the 

 radius of the sphere is a, we must have 



-~ = for r = a. 

 dr 



Differentiating (/> with respect to r gives 



equating this to zero and making r=a, we find 



from which 



AfOT^' ( 22 ) 



Substituting this value of & in the last form of (j> } we have 



For the determination of the functions U, V, W, we have 

 V 2 U=-2f, V 2 V=-2^ v 2 W=-2f, 



with the condition 



dXJ dV dW = Q 

 dx dy dz 



Equations (10) hold for the particular case of motion that we 

 are studying ; and therefore 



The functions U, V, W can be expressed in a series the 

 terms of which depend upon the general spherical harmonics. 

 The following solution is merely a generalization of one given 

 by Mr. J. Gr. Butcher in the ' Proceedings ' of the London Ma- 

 thematical Society. Mr. Butcher says, in his article " On 

 Viscous Fluids in Motion," that the method is due to Stokes, 

 to whose article, however, I am not able to refer. 



The equations to be solved are of the form 



V 2 S=-0 (24) 



V 2 5 =0. J 



In the problem solved by Mr. Butcher the solution is made to 

 Phil. Mag. S. 5. Vol." 10. No. 63. Nov. 1880. 2C 



