178180.] WORK DONE IN CHANGE OF VOLUME. 221 



180. If we substitute from (11) in the equations of motion (2), 

 we find that the latter may be written &quot; 



d (du dv dw\ d?u d?u d?u 



&c., &c. 



When the fluid is incompressible these become 

 du v dp 



f^ = P X -Tx + ^ 



dv - d 



where y 2 has its usual meaning. 



In the case of compressible fluids (gases) we may by writing 



dv dw 



reduce (14) to the form (15), but since in all probability the laws 

 of Boyle and Charles hold with regard to p but not to p, there is 

 no real gain of simplicity. 



The equations (14) or (15) have been obtained by Navier, 

 Cauchy, Poisson*, and others, on various considerations as to the 

 nature and mutual action of the ultimate molecules of fluids. The 

 method adopted above, which seems due in principle to de Saint- 

 Venant and Stokes (, is independent of all hypotheses of this kind, 

 but it must be remembered that it involves the assumption that 

 p xx +P&amp;gt; Pxv&amp;gt; &c. are linear functions of the coefficients of distortion. 

 Hence although (14) and (15) may apply with great accuracy to 

 cases of slow motion J, we have no assurance of their validity in 

 other cases. 



It may be remarked, however, that the calculations of Max- 

 well, who has investigated the viscosity of gases on the assump- 



* For references see Stokes, B. A. Reports, 1846, and 0. E. Meyer, Crelle, t. 59. 

 t Camb. Trans. Vol. YIII. 1845. 



J That they do so is in fact shewn by the experiments of Maxwell and of Helm- 

 holtz and Piotrowski. 

 Phil. Tram. 1867. 



