528 



VISCOSITY. 



[CHAP, xi 



dy 



dz 



dz)' 



. 



dx ] 



(9), 



dx dy J 

 where the harmonics </> n , % n are arbitrary*. 



294. If we neglect the inertia-terms, the equations of motion 

 of a viscous liquid reduce, in the absence of extraneous forces, to 

 the forms 



, ............ (1), 



dy dz 



.,, du dv dw /ox 



with j-+:y- + -7- = ........................ (2). 



dx dy dz 



By differentiation we obtain 



V*p=0 .............................. (3), 



so that p can be expanded in a series of solid harmonics, thus 



P = 2p n .............................. (4). 



The terms of the solution involving harmonics of different alge- 

 braical degrees will be independent. To obtain the terms in p n 

 we assume 



-*+*-;&.' 



(5), 



dz dz 



where r 2 = a? + y* + ^ 2 . The terms multiplied by B are solid 

 harmonics of degree n + 1, by Arts. 82, 84. Now 



L, dpn\ = 



\ dx J 



dx ^ dy dz] dx dx 



* Cf. Borchardt, ' ' Untersuchungen iiber die Elasticitat fester Korper unter 

 Beriicksichtigung der Warme," Berl. Monatsber., Jan. 9, 1873; Gesammelte Werke, 

 Berlin, 1888, p. 245. The investigation in the text is from a paper "On the 

 Oscillations of a Viscous Spheroid," Proc. Lond. Math. Soc., t. xiii., p. 51 (1881). 



