54 G. Jounstone Stonry On Polarization Stress in Gases. 
_ In these A, stands for the expression (13) ; and introducing the following values 
which are given by Clausius as correct to the second order of small quantities (loc: 
cit : p. 526, footnote.) , | 
g=9 
Nee 
50 
oa aoe ¢ 
We find— 
A,=13°8 
From this and (14) | 
« =1°8xXpq*e? + terms of the fourth and higher orders. (15). 
But by Clausius’s theory (loc: cit : p. 516) 
G=4Pou?ge+ terms of the third and higher orders. (16). 
Whence, approximately, omitting the fourth and higher orders of small quan- 
tities, and writing v for wu, since they are nearly equal, and then putting P for its 
equivalent 45, 
12 al 
Now, by Boyle’s law PT where P., 9,, and T, have reference to standard 
temperature and pressure. Whence, finally 
PAL aie ; alk G? 18 
=| (18) S58 pea ceo eae (18) 
An equation which assigns the same law as we obtained above in equ. (B) by the 
wholly different method of direct mechanical considerations. 
23. Equation (18) appears to give also the amount of the polarization stress. 
But this is illusory. The hypothesis upon which it restsis adequate as regards 
the conduction of heat, but is insufficient for a quantitative investigation of the 
stress, as I will now proceed to show. 
The general formule for the conduction of heat and for the polarization stress 
are the following— 
1 
G=180 f IV? udp 
a 
r=H0 IV2.(3u2—1)dp 
=—l 
(See Clausius’s memoir p. 514, and equation (G) above). | Now, - and 3y,—I1, 
which occur as factors in these integrals are the first and second terms of a series of 
spherical harmonics (Laplace’s co-efficients) of the simple kind that are functions of 
» only, and which therefore represent the radii of solids of revolution from points 
on their axes. Itis moreover obvious that we can expand IV? and IV’ in series 
