254 
PROFESSOR CLERK MAXWELL OX STRESSES IX RARIFIED 
It is easy to write down the surface conditions for a surface of any form. 
Let the direction-cosines of the normal v be l, m, n, and let us write 
d p 7 cl d , cl 
T for 1 T +m Tj +n T 
We then find as the surface conditions 
d r 
u 
-G-R1-F) «- Imv-lnwl~)(6+iG^)=0 
0 J- 
0 
d 
d 
4 p6\dx dv 
je 
dv 
4 p9\dy civ 
W 
— Gry-[(1 —-w 2 ) w— nlu—nmv\\-\ ~ n< T 
dv L ' ’ J 4 pd\clz 
dv 
dv 
*+<)= 
pi) 
In each of these equations the first term is one of the velocity-components of the gas 
in contact with the surface, which is supposed fixed; the second term depends on 
the slipping of the gas over the surface, and the third term indicates the effect of 
inequalities of temperature of the gas close to the surface, and shows that in general 
there will he a force urging the gas from colder to hotter parts of the surface. 
Let us take as an illustration the case of a capillary tube of circular section, and for 
the sake of easy calculation we shall suppose that the motion is so slow, and the 
temperature varies so gradually along the tube that we may suppose the temperature 
uniform throughout any one section of the tube. 
Taking the axis of the tube for that of z, we have for the condition of steady motion 
parallel to the axis 
dp 
!d?w d 2 w 
clz ^{w + df 
(72) 
Since everything is symmetrical about the axis, if we write r~ for ar+y 2 we find as 
the solution of this equation 
(73) 
A , 1 (l P 0 
W=A+ jS*'" 
If Q denotes the quantity of gas which passes through a section of the tube in unit 
of time 
Q = 2ir\pivrdr 
J A i 1 d P « 
= ^(A + --a- 
At the inner surface of the tube we have r=a, and 
i l 1 d P 2 
—or 
4 pb clz 
■ (Ci) 
. Q . 1 T 
jrpcr Sp, dz 
a* 
■ • (75) 
