564 VISCOSITY. [CHAP, xi 



and therefore, for the angular velocity (xiv), 





 the real part of which is 



(xxi). 



As in the case of laminar motion (Art. 298), this represents a system of waves 

 travelling inwards from the surface with rapidly diminishing amplitude. 



When, on the other hand, the viscosity is very great, pa is small, and the 

 formula (xiv) reduces to 



nearly, when the imaginary part is rejected. This shews that the fluid now 

 moves almost bodily with the sphere. 



The stress-components at the surface of the sphere are given by (13). In 

 the present case the formulae reduce to 



Prx= - 



a j 



(xxiii). 



sr, a r 

 If 8$ denote an element of the surface, these give a couple 



>'(A; 



(XX1V) ' 



by (xiii) and Art. 305 (7). 



In the case of small viscosity, where /3a is large, we find, on reference to 

 Art. 267, putting ha = (1 - 1) a, that 



~ ..................... (xxv), 



approximately, where = (1 - 1) pa. This leads to 



N=-%7rij.a 3 (l+i)pa> ......................... (xxvi). 



If we restore the time-factor, this is equivalent to 



7r/ia 3 (/3a)6) ............ ( xxvii ). 



The first term has the effect of a slight addition to the inertia of the sphere ; 

 the second gives a frictional force varying as the velocity. 



308. The general formulae of Arts. 305, 306 may be further 

 applied to discuss the effect of viscosity on the oscillations of a 



