THEORY OF ELECTROMAGNETISM. 
715 
Thus we get for the expression on the left of eq. (38) 
(||{a; + £ + S© 0 Va; + SC c V;r + Sc c V,x + SH h V^c -f S'FG<M} ch 
- J| {Sd2 {0 & Vx + YaH/47r - YC) + S pV dt] 
of which the volume integral is zero [equations (18), (21)], and the surface integral is 
the expression on the right of equation (38). 
To get the ordinary expression for the flux of heat due to conduction we have merely 
to suppose x to contain the term — S©y©/2$, where y is a self-conjugate linear vector 
function of Class I., of § 9 above. The heat flux referred to the standard position of 
matter due to this term 
= 9f7 (S©y©/20) = - y©, 
and, therefore, by Prop. VI., § 10, the actual flux of heat is — y'©'. 
43. It is known (Tait’s ‘Heat,’ 1st ed., §412) that if 9 0 be the lowest available 
temperature, 0 O F is the rate of dissipation or degradation of energy in Sir William 
Thomson’s sense. Now by equations (35), (36), 
F = J||(* + {)/$& .(39), 
so that (x + £) 9J0 may be called the rate of dissipation of energy per unit volume. 
There seems very good reason then to call X the dissipation function. It only differs 
from Lord Rayleigh’s function in the terms that lead to the conduction of heat. 
If, as will usually be the case, x is quadratic in © and the velocities, £ = x, and the 
rate of dissipation per unit of volume will be 2x0J6. For instance, the rate of dissipa¬ 
tion per unit volume of the standard position due to conduction = — S©y©# 0 /# 3 , and, 
therefore, per unit volume of the present position it is — S©'y'©'^/^. * 
III. Establishment of General Results. 
A. Value of SL for a Finite ‘portion of Matter. 
44. As already remarked (§ 34) the 8 in equation (1) § 1 3 implies variation in every¬ 
thing but the temperature. This will be assumed for the present. Thus 81 depends 
[§ 27 (25)] on the variations of 
p> p, 'k ; d, D, C, H. 
* I suppose this result has been noticed before, though I do not know by whom. 
4 y 2 
