A SPHERE IN A VISCOUS LIQUID. 61 



also be zero, and the liquid would be frictionless. We shall therefore assume that the 

 roots of (59) are real. 



The constants A and B must be determined from the condition F (0) = F 7 (0) = 0, 

 whence 



^-fri 



p-y J 



also this value of F satisfies the conditions that F (a) e~* , and F" (a) c~*' should 

 vanish when a = oo : whence the value of v is 



8inf f-r_0_ / 



~ V(7rvO J [r (3k + a) \ 



p - 



13. We shall lastly consider the motion of liquid contained within a sphere, which 

 is rotating about a fixed diameter, when there is no slipping, and when the angular 

 velocity is uniform. 



In this case v must satisfy the differential equation (43), and also the condition (i.) 

 of 10 ; but (ii.) becomes v = when t = for all values of r < a : also we have a 

 third condition, viz., that the velocity must be finite at the centre of the sphere. 



A particular solution of (43), subject to the condition of finiteness at the origin, is 



/T d If \ ( 



V = * A V * dr r e *V' { ~ 



(r 



whence if p and <j are any quantities which are independent of r and t, a solution of 

 (43) is 



AT d 1 p.,. r \ (r-) 3 l (r + ) 9 1 1 7 



V = * V ,7 dr r\ q F W L eX P' { - } - eX P' { - M \ \ <*" 



= " l (d\ f F (a) e- (cos X (r - a) - cos X (r + a)} da. 



tiT T 1 Q J q 



If we put p = a, q = 0, F (a) = a, the double integral when t = is equal to r by 

 FOURIER'S theorem, for all values of r between a and 0. If we put p = oo, q = a, 

 the integral when t = is zero for all values of r which do not lie between o and oo . 

 The solution of the problem is therefore contained in the formula 



