THEORY OF VISCOSITY AND THERMAL CONDUCTION, IN A MONATOMIC CAS. 285 
this being the only way in which the first-order derivatives can occur in F, in order 
that F may be an invariant. 
Here P^C 2 ), P 2 (C 2 ), P 3 (C 2 ) denote certain undetermined functions of U, V, in 
which these variables appear only in the form U 3 + V 3 +W 2 or C 2 . The first term of F 
is evidently of odd degree in U, V, W combined, and the second and third are of even 
degree; it is convenient to denote them by 0(U, V, W) and E(U, V, W), when we 
wish to refer to the odd or even part of F separately.] 
It is easy to see that, in a uniform gas, f, satisfies the necessary conditions (3), (4). 
In the non-uniform case these conditions require F to satisfy the equations 
(17) 
/ 0 F dll dV dW = 0, 
(18) ||| U/ 0 F dll dV dW = 111 V/ 0 F dU dV dW = J j W/„F dU dV dW 
Clearly the odd part 0(U, V, W) of F satisfies (17), and the even part E(U, V, W) 
satisfies (18), but not vice versa, so that these equations place certain restrictions on 
0 and E. 
3. The Equation of Transfer of Molecular Properties. 
§ 3 (A) The rate of change of rQ, the aggregate value of Q (u, v, w) per unit volume, 
may be analysed into three parts, viz., that due to molecular encounters (which we 
denote by AQ), that due to the passage of molecules in or out of the volume-element 
considered, and that due to the action of the external forces. The equation 
expressing this analysis may readily be shown* to be 
(19) 
|(,Q) = AQ- 2 
x,y, z 
A(mQ)-^X(5y 
ox m \ du 
We may define AQ by the statement that (AQ )dxdydzdt is the change produced 
by molecular encounters during time dt in the sum AQ taken over all the molecules 
in the volume-element dx dy dz : evidently AQ = rQ dx dy dz. 
If in (19) we make Q equal to unity, in which case AQ is clearly zero, the equation 
becomes 
dv _ fdvll 0 dvV 0 
dt \ dx ' dy dz J 
( du 0 dv 0 8w 0 \ 
\dx dy dz) 
dx 
dv 
+ V) ^ t w o 
dy 
which is the equation of continuity. 
* Cf. Jeans’ ‘ Dynamical Theory of Gases.’ 
