568 VISCOSITY. [CHAP, xi 



with the condition Cf-^ (ha) = o&amp;gt; ................................. (ii), 



a being the radius, and o&amp;gt; the angular velocity of the sphere, which we suppose 

 given by the formula 



Putting h = (l i)$, where /3=(&amp;lt;r/2r)*, we find that the particles on a 

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



. ......... 



/j(Aa) r 3 l + iha 



where the values of / x (hr\ / : (ha) have been substituted from (3). The real 

 part of (iv) is 



a 



-|8(r- a) sin {erf- (*-) + }] ...... (v), 



corresponding to an angular velocity 



(vi) 



of the sphere. 



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



03 



l+iha 

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



This is equivalent to 



, r 

 ~ 



dt ~ 

 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 



w=u, = 0, w .............................. (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 

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



(hr)&amp;lt;l&amp;gt; n ...... (xi) ; 



~_ n 

 * Another solution of this problem is given by Kirchhofif, Mechanik, o. xxvi. 



