A SPHERE IN A VISCOUS LIQUID. 45 



H, 8 be the component velocities of the liquid along and perpendicular to the 

 radius vector ; then, if we assume that no slipping takes place at the surface of the 

 sphere, the surface conditions are .\ 



Also, at infinity II and 8 must both vanish. 

 These equations can be satisfied by putting 



......... (4) 



where i/, and i/ 2 are functions of r and t, which respectively satisfy the equations 



?3h _ 



r* ' ' ' ' ' 



2fc_lrffe 



dr r ' ~ v dt ' 



The proper solution of (5) is 1/1, =f (()/>', which it will be convenient to write in the 

 form 



where x ( a ) IS an arbitrary function, which will hereafter be determined. 



In order to obtain the solution of (6), let us put i/ 2 = re"* 1 '"' dw/dr, where 10 is a 

 function of r alone ; substituting in (6), and integrating, we obtain 



rw = A cos X (r a + a ). 



where a is the radius of the sphere and A and a are the constants of integration. 

 Whence a particular solution of (6) is 



d e" 



e"* 

 ., = Ar - oes X (r a + a). 



Integrating this with respect to X between the limits oo and 0, and then changing A 

 into F (a) and integrating the result with respect to a between the same limits, we 

 obtain 



rj<* d r F() f (r-a + 



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