Laws of Irreversible Phenomena. 391 



dynamical phenomena, except that, being irreversible, they 

 do not satisfy the condition d'Q, = 0. Lord Rayleigh has shown 

 how in many cases we can put 



d'Q=-2F<fc, (1) 



employing F to indicate a function of the variables q { (sup- 

 posed to be " normal "ones) and s fj homogeneous of the second 

 degree with respect to the s., which he calls the Dissipation 

 Function. The assumption we make is therefore that 



d'q=-dtts i ^ = -^d (/i ; ... (2) 

 and that 



following the rule laid down in § 1 we put 



S'Q=-5^%, (4) 



and from (III.) we obtain 



•A{8T-2g8 ?j +SP j 8 ?i -2||8< ? J=0; . (5) 

 hence 



j" 



d .(VL)-VL+* 



dt\ds./ d& d^ 0*. ' 



These are Lord Rayleigh's equations, with V written in the 

 place of U. 



§ 10. Irreversible Hydrodynamics. — Let us now proceed to 

 consider a viscous fluid ; we shall call fx its coefficient of 

 viscosity. Owing to the viscosity of the fluid its motion is 

 accompanied by irreversible production of heat ; owing to its 

 compressibility, there is reversible production or destruction 

 of heat. We shall suppose that every such loss or gain in 

 every element of the fluid is being immediately and exactly 

 compensated, so that the temperature of the element remains 

 constant. At the interior of a large quantity of fluid we 

 take a portion, of mass \\\ dxdydzp, p being the density at the 

 point (x,y, z). Let p be the ordinary mean pressure ; u 1 v, w 

 the components of the velocity, X, Y, Z the components of 

 the extraneous acceleration, at the point [x, y, z) and time t. 

 The equations of motion, as given by Navier, Poisson, Stokes, 

 and Maxwell, are as follows : — 



