528 VISCOSITY. [CHAP, xi 



dy ^ dz J 



(9), 



+ _ 



dz + y dx * dy) 



where the harmonics (f) 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), 



.,, du dv dw /ox 



with ;j- + :r 4 -:r- :=0 ........................ ( 2 )- 



dx dy dz 



By differentiation we obtain 



.............................. (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 



*f&amp;gt; + r^ ^,\ 

 da; dx r 2 ^ 1 



dy dy 



dz dz 



where r 2 = x 2 + y 2 + z*. The terms multiplied by B are solid 

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



dx J dx \ dx y dy dz/ dx dx 



-. , 



* Cf. Borchardt, &quot; Untersuchungen uber die Elasticitat fester Korper unter 

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

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

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



