52, G. JouHNsToNE Stoney On Polarization Stress in Gases. 
V being their velocity. Hence the number traversing the slice in the specified 
direction within a unit of time is 
ey . NIV cosy. do. 
Aor 
Multiplying this by mVcosy we get the resolved part of their momenta along r. - 
The sum of all such components of the momenta, all estimated as positive, is P,, 
the stress in the direction of r, Whence, and writing » for m N, we find 
P= ff f TV? cos*do 
the integration being extended over the unit sphere. 
Hence the stresses in the directions of three rectangular axes are 
Raff, TV? co2ade 
Tv 
P= = fj jive co2Bdae 
Pat {five co*ydo 
a [3 y being the director angles of the element of solid angle d-. Introducing polar 
co-ordinates, we have 
do=sin 6 dé dé 
cos a=cos 8 
cos B=sin 0 cos ¢ 
cos y=sin 6 sin 
by which the expressions for the stresses become 
2 — } 
re ef " {" V7 cos26 sin 6dédo 
TH o ° 
2 — 
Paz ff © IVF sin%e cos*adadg | (E) 
o o 
ey) 2 ——— 
P= f f TV? sin 20 sin 20d0d¢ 
TT o 0 
These are the most general expressions for the stresses in three rectangular 
directions within gas polarized in any way; and they will be the only stresses 
between portions of the gas separated by planes parallel to the planes yz, za, ay, if 
the axes are so chosen that there are no moments round them arising from the 
molecular encounters”. 
21. This condition is easily secured in the case which we are investigating, viz., 
when heat is making its way between a heater and cooler that are paralled to one 
another, and of large extent compared with the interval between them ; since the 
* Equations (E) cannot be integrated unless [V? is given as a function of 6 and 4, 7.c., unless the law 
of polarization in the gas is known. But they show that in general the stresses in different directions are 
unequal, which is here what is chiefly insisted on. aan 
When the gas is unpolarized, I becomes equal to unity, and V? is independent of the direction, and 
may therefore be put outside the integrals. In this case all three equations concur in giving the well 
known expression for the stress in unpolarized gas, viz., 4 9 V’. 
