PROPERTIES OF MATTER IN THE GASEOUS STATE. 
813 
the normal) will always be less than the angle of incidence, must cause the molecules 
to assume directions more and more nearly perpendicular to the surface as the tube 
becomes narrower, until some limit is reached. 
The case of a billiard ball started obliquely along the table will serve to illustrate 
this. Each time the ball leaves the side cushions its path will be more nearly perpen¬ 
dicular, and if it could maintain its velocity, and the table was sufficiently long, it 
would eventually be moving directly across the table. This, however, would not be 
the final condition if the cushions were zigzag, for then a number of balls, in whatever 
direction they might be started, w'ould finally attain a certain mean obliquity, depend¬ 
ing on the unevenness of the cushions. And it would seem probable that the latter 
case must be that of the molecules in a tube so narrow that they can range across. 
The ability of the molecules to range across the tube will depend on the value of - ; 
hence it would appear that the most probable assumption with regard to the nature 
of X is that 
+ . ( 91a ) 
where fJ~) and fj~) are functions of some such form as having respectively 
the values unity and zero when - =0, and zero and unity when -=oo ; and k l is a 
coefficient independent of the nature of the gas on which X 3 may depend. 
That there is good reason for making this assumption appears from the comparison 
of the results for hydrogen and air (see result VIII., Art. 10G). 
The equations of motion as a ffected by discontinuity . 
99. Substituting in equation (G6) from equation (54a), and putting — for ^ ct.,.(Mh) 
clx 
as in Art. 92, we have for steady momentum along the tube 
dp ,df spa du .s, — Sr, dd 1 
dx dz [_ fir dz 2 77 - ^dx J 
(92) 
Whence integrating between the limits 0 and 2 
dp spudu — s 2 du 
And substituting for Sj,— s. 2 from equation (83) 
spx du dp 
n t dz dx 
C —2 C + 5 
e «i *—e da. 
“1t (Si_S2 A*; 
1— c fl > s 
(93) 
(94) 
