56 MR A. B. BASSET ON THE MOTION OF 



since this component does not produce any effect on the motion of the sphere, which it 

 is our object to determine, we may confine our attention solely to the calculation of v. 

 In addition to (43), v must satisfy the conditions : 



(i.) At the surface of the sphere v = aa> for all values of t. 



(ii.) When t = 0, v = for all values of r greater than a, the radius of the sphere. 



Let v = Rc~ AV< where R is a function of E, alone ; substituting in (43), we obtain 



dr* r dr 

 the solution of which is 



whence 



R = A j - cos X (r a + a) I . 



d f e~ xly( 1 



v = A Tr \~r cos x < r ~~ a + a) r 



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 



/* d I f" _,, x j (r-a + ) 



v = * v rtSr - 



Performing the differentiation and then integrating by parts, we shall obtain 

 1 /TT f JF(a) I J (r - a + 



- ~ ~ 



provided F(0) = and F(a)c- ' = when a = oo . 

 The surface condition (i.) will be satisfied if 



F() + aF'() = - - -, 



7T 



whence 



F()=-^(i--n 





the constant of integration being determined so that F(0) = ; this value of F(a) 

 also satisfies the condition that F(a)e~ a> = when a = oo . We therefore obtain 



a?<osme [ \a I a\ , 1 f (r - a + *) 3 ] 



v = .-.- l^-f-ll -- 1 ***> exp. \ - -\ da.. . (44) 



ri/(int) Jo b' r/ 4 V < J 



