570 VISCOSITY. [CHAP, xi 



terms in p rx , since the surface-harmonics of other than zero order will 

 disappear when integrated over the sphere, we find 



z + C l Ba .................. (xx), 



where B_^-\a\ &amp;lt;7 X = 2/iAa/ (ha) ..................... (xxi), 



by Art. 306 (19). Hence, by (xii) and (3), 



{2/0/ (ha} Wa * h (ha}} 



This is equivalent to 



X- - 1 ^ *+ -3, P V + u ...... (xxiii). 



The first term gives the correction to the inertia of the sphere. This 

 amounts to the fraction 



^ + 4^ 



of the mass of fluid displaced, instead of ^ as in the case of a frictionless 

 liquid (Art. 91). The second term gives a frictional force varying as the 

 velocity*. 



310. We may next briefly notice the effect of viscosity on 

 waves of expansion in gases, although, for a reason to be given, the 

 results cannot be regarded as more than illustrative. 



In the case of plane waves f in a laterally unlimited medium, 

 we have, if we take the axis of as in the direction of propagation, 

 and neglect terms of the second order in the velocity, 



du _ I dp d?u m 



Tt~~^Tx + * v dx* .................. ! 



by Art. 286 (2), (3). If s denote the condensation, the equation 

 of continuity is, as in Art. 255, 



ds _ du 



dt~ dx ...... 



* This problem was first solved, in a different manner, by Stokes, 1. c. ante 

 p. 518. For other methods of treatment see 0. E. Meyer, &quot;Ueber die pendelnde 

 Bewegung einer Kugel unter dem Einflusse der inneren Keibung des umgebenden 

 mediums,&quot; Crelle, t. Ixxiii. (1871) ; Kirchhoff, Mechanik, c. xxvi. The variable 

 motion of a sphere in a liquid has been discussed by Basset, Phil. Trans., 1888; 

 Hydrodynamics, c. xxii. 



t Discussed by Stokes, I. c. ante p. 518. 



