Motion of a Sphere through a Viscous Fluid. 115 



The solution is completed if we write 



u ** +< w== -5*w, ^=-|* + < (i5) 



o.B oy o~ 



where u\ v', w are solutions o£ the equations 



(v 2 -2^)«'=0, f- .... (16) 



and 



9w' ~bv r ~dw' 



We have here written, for shortness, 



£=U/2y. 



(IT) 



(18) 



Since the vortex-lines must be circles having the axis of x 

 as a common axis, we may assume 



fc_0 „- 3 X y_"^% 



(19) 



where % is a function of x and -sr (the distance from the axis 

 of x) only. It follows from (16) that we must have 



(v 2 -2*!-)x = °. • •-,... (20) 

 an additive function of x being obviously irrelevant. Hence 



2k 



3* 





B# 3 T'"3.*' 



'dx 'dz^'dx^i/' 

 dy 3* = 9;t'9 





>- . (21) 



I 2 



