568 VISCOSITY. [CHAP, xi 



with the condition Cf 1 (ka)=-a> (ii), 



a being the radius, and o> the angular velocity of the sphere, which we suppose 

 given by the formula 



o> = a e ^ +) (iii). 



Putting A = (l i) ft, where /3 = (<r/2i/)*, we find that the particles on 

 concentric sphere of radius r are rotating together with the angular velocity 



1* 1+ika 



where the values of/! (hr), /j (ha} have been substituted from (3). The real 

 part of (iv) is 



i^+g/^ 3 *"*^^ 



-/3(r-a)sin{<r*-/3(r-a) + e}] ...... (v), 



corresponding to an angular velocity 



o> = acos (otf + e) ................................. (vi) 



of the sphere. 



The couple on the sphere is found in the same way as in Art. 307 to be 



v 



Putting ha=(l - i) /3a, and separating the real and imaginary parts we find 

 This is equivalent to 



The interpretation is similar to that of Art. 307 (xxvii) *. 



2. In the case of a ball pendulum oscillating in an infinite mass of fluid, 

 which we treat as incompressible, we take the origin at the mean position of 

 the centre, and the axis of x in the direction of the oscillation. 



The conditions to be satisfied at the surface are then 



M = u, v = 0, ? = .............................. (x), 



for r=a (the radius), where u denotes the velocity of the sphere. It is evident 

 that we are concerned only with a solution of the Second Class. Again, the 

 formulae (10) of Art. 306, when modified as aforesaid, make 



r)(t> n ...... (xi) ; 



* Another solution of this problem is given by Kirchhoff, Mechunik, c. xxvi. 



