THEORY OF VISCOSITY AND THERMAL CONDUCTION, IN A MONATOMIC GAS. 295 
components is reciprocal, so that J = + I. It may readily be seen that tire positive sign 
is the correct one, by considering a particular case of variation, say dlh = dU 2 = d U, 
d \/1 = dV 2 — d\J , dWi = dW 2 = dW. This is equivalent to the addition of a small 
velocity (dU, dV, dw) to the whole system ; obviously this will reappear in the final 
velocities, so that also dufi = d\j' 2 — dU, dM\ =.dw' a = dV, dWfi = dWb = dW. In 
this case, and therefore always, we have J = 1, so that 
(55) dUh dV'i dW\ dU' 2 dVb dWh = dlh dVi dWi dU 2 dV 2 dW 2 . 
With a little more trouble this might also be proved analytically from the equations 
of this section. 
From the component equations corresponding to (42), (43), i.e., from 
(56) 
(57) 
U i — X 0 + fx 2 j 2 x 
K) 
Vi = Yo + ^Y 
E, Wl — Z 0 + /X 31 Z R , 
U 2 = Xn-M^Xp, Vo = Y 0 - Mi2 1/2 Ye, Wo = Z 0 -/xi 2 1/2 Z r , 
Ml2 A B> 
C 0 2 = x 0 2 +y 0 2 +z 
2 
0 ? 
b'jf — x r 2 + y r * + z r 2 , 
it is easy to prove that 
(58) 
3 (Ui, V t , W 1; U 2 , V 8 , W 8 ) 
0 (X 0 , Y 0 , Z 0 , X R , Y r , Z r ) 
Hence, by further transformation to polar co-ordinates, we have 
(59) riUj dV l dWj dU 2 dV 2 dW 2 = - (miM 2 )" 3/2 dX 0 dY 0 dZ 0 dX R dY R dZ R 
= — (miM 2 ) _i/2 C 0 2 C k 2 dC 0 dC R d cos @ 0 d cos @ R d<£ 0 d^ R . 
Since dUidVidWidll 2 dV 2 dW 2 is essentially positive, the negative sign on the right 
of (59) must be made positive, if the limits of C, cos 9, and <p in each case are taken 
as 0 to + oo, — ] to +1, and 0 to 2?r respectively; it may readily be seen that the 
negative sign corresponds to reversed limits of integration of one of the variables cos 9. 
§ 5. The General Expression for AQj. 
Definition of AnQi and A 12 Qj. 
§ 5 (A) The rate of change of ^Qx due to molecular encounters, i.e., AQi, may be 
divided into the two parts A u Q l5 A 12 Qx due respectively to the encounters of the 
molecules ni x among themselves, and those with molecules m 2 . Thus 
(60) AQi = A n Qi + Aj 2 Qi. 
We shall chiefly consider A 13 Q 1} whence A U Q X may be obtained by changing the suffix 
2 into 1 throughout. 
YOL. ccxyi.—a. 2 s 
