G. Jounstone Stonny Cn Polarization Stress in Gases. 53 
polarization of the intervening gas will evidently be disposed symmetrically round 
the direction in which the heat is travelling. Hence, taking this direction as our 
axis of x, there can be no moments round this axis, or round any axis at right 
angles to it. The stresses (EK), therefore, are the only ones to be taken into account. 
Moreover, we can integrate equations (E) at once by ¢, since IV° is, in this simple 
ease, a function of 6 only. Doing this, and writing » for cos 6, we find 
nee 
P= . fs LV? pedp | 
yet a 0 etl == 
P=P=f fl" TV? (1—p2)dp.| 
Whence, since «, the polarization stress, =P,— P,, we have finally 
pte 
tf” TV (Bu l)du -... (G) 
This then 2s the complete mathematical expression for Crookes’s stress. It could be 
(F) 
integrated if we knew the law of the polarization of the gas, for then IV? would be 
a known function of p. 
22. Clausius, in investigating the diffusion of heat across the layer of gas, makes 
the assumption (Phil. Mag. Vol. 23, pp. 425 and 524) that the numbers and velocities 
of the molecules “ emitted” by a thin stratum of the gas (i.e. that have passed out 
of the stratum after having encountered other molecules within it) may be ade- 
quately represented “by assuming at first motions taking place equally in all 
directions, and then supposing a small additional component velocity in the direc- 
tion of positive « to be imparted to all the molecules.” In other words, it is assumed 
that the motions of these molecules may be represented by radii vectores from a 
shghtly excentric origin to points equally distributed over the surtace of a sphere. 
Tt will be instructive to trace the consequences of this hypothesis, both because of 
what it will do and what it will not do. 
Upon this hypothesis Clausius finds the following convergent series for V? and I 
(loc: cit: pp. 434 and 516) 
V2 =? Ququet (Qur-+gq,2)u2e2-+ Spntok: 
i (le ie2- eee )—f. pet wet .. 
where 4g. (p. 525) is the small component velocity spoken of above, u is the mean 
velocity of molecules moving in the plane yz, and the other letters have the 
meanings assigned to them by Clausius. Multiplying these together, going to the 
second order of small quantities, and arranging by powers of #, we find 
TV2=u?(1—40'e2)-+ Ae} Ayp? .... (12) 
where 
A,=—29?+ 2urtqitur ..... (13) 
Introducing the expression (12) into equations (F) and (G) we find 
Pr=Low(1—tr'e?) 4+ 1pAye2+ 
P,=P,=19u2(1—49'e2)4+ JepAye2t ... ; (14) 
k= spAge*t ... 
