48 G. JoHNsToNE Stoney On Polarization Stress in Gases. 
of unpolarized gas travelling simultaneously in opposite directions, which the en- 
counters within each stream tend to develop, and the condition of stationary 
unpolarized gas, towards which the mutual interference of the two streams modifies 
the structure. Hence there is some polarization stress and some flow of heat all 
along the tube, though of less amount than in the case considered in section 10. 
We may still employ equation (7) as the expression for the polarization stress, 
if we use for p’ and ’’ the densities of the streams at some particular cross sec- 
tion of the tube, andif uw’ and w’’ are modified into what they become as the 
interference of the two streams with one another is increased. It is not necessary 
to ascertain what this modification will be; it is enough for our purpose to know 
that uw’ and w’’ will be some functions of V’—V’’ (where Vand V’” are the 
averages of the cubes of the velocities of the molecules that pass forwards and 
backwards respectively through the cross section), and that they will be propor- 
tional to this quantity when all three are small. 
15. We may base upon this circumstance an investigation of the laws of the 
phenomenon when the difference between the temperatures of the heater and 
cooler is small compared with their absolute temperatures. This case is of im- 
portance because it is that which most frequently occurs, and is the only one in 
reference to which accurate experiments have been made. In this case 9’ and 0’ 
will each be nearly 4p using » for the density of the gas at the position in the 
tube which we are considering; and V’—V’’ being small may be appropriately 
represented by 5V. Then, remembering that u’ and w’’ are proportional to 8V, 
we obtain from equ. (7) the following expression for the polarization stress— 
ESOP oo « oo 6 (8) 
Where the symbol « means approximately varies as. Moreover, it can be shown* 
* One of the ways in which this may be proved is the following :— 
Clausius has shown (Phil. Mag., vol. 23, p. 514) that 
— 
G=7ho J IV%pdp 
—1 
whence 
G= {0p VU V2—1/V2) 
where I’ and V’ are the average values of I and V* under the integral for positive values of p, i.c., for 
molecules traversing the section of the tube towards the cooler; and I’’ and V’ are the corresponding 
averages for negative values of y, 7.¢., for molecules traversing the section of the tube in the opposite 
direction. 
Now, it is evident that these quantities are capable of expansion in the following form :— 
Vv, , (ev) 
rapa +a0%) Se 
me OV 
Ve=Wa(l ECS +...) 
VRAVAI +d +...) 
in which V® is the average of the values of V® for all directions. Whence 
G=%p(A,—B, +C,—D,) V26V : 
-}- terms containing higher powers of dV. 
