832 
PROFESSOR 0. REYNOLDS ON CERTAIN DIMENSIONAL 
(which for simplicity may be taken concentric and of radius cl), then in order that 
Pc, Pc 
c _ c c 
may be small, we must have J —large as well as - 
2^c,+s Pc,—s S S 
Our equations, therefore, may be seen to hold good when the radius of the solid 
surface is large compared with s and the distance between the opposite surfaces is also 
large. 
c c _ c 
On the other hand, in the limit when either - or- 1 —- are very small p c —p\ and 
s s 
p c ,— pc will depend entirely on the action of the gas within the layer of thickness s 
immediately adjacent to the surface. In these cases, however, when - oi 
are 
c-, 
or - 
s s 
small, the action within this layer may be easily expressed. 
114. Let the temperature of the internal surface (sphere or cylinder) be such that 
the mean value of a for the molecules which rebound from this surface (considered as a 
group in a uniform gas) is a Cx ; while the temperature of the external surface is such 
that the mean value of a for the molecules which rebound is a. 
c ! c 1 — q. 
The condition that - or 
s 
are small necessitates that the molecules which come 
up to the inner surface arrive as from a uniform gas such that a — a. That is to say, 
none of the molecules which rebound from the inner surface can return until their 
C — Q 
characteristics have been completely modified by the external surface. For if ——- is 
small the molecules will cross the interval between the surfaces without encounter, 
• 11 I I 1 Cl Co 
S w S 
while if - is small, although - 1 - g may be large, the characteristics of the gas wall be 
but slightly affected by the internal layer at a distance s from that surface, and, by 
theorem II., the approaching molecules will arrive as from a uniform gas in the mean 
condition of the gas at a distance s. 
I shall first consider the case in which 
is 
small. 
The number of molecules 'which arrive at the inner surface is proportional to p'a.', and 
the number which rebound is proportional to p,a r , and since the numbers must be the 
same we have 
P 
a =n.a,. 
The momentum imparted to the surface by the incident molecules is ^ —p , and that 
o 
imparted by the rebounding molecules is therefore 
pc = ip 
(136) 
Since the molecules which rebound from the internal surface all proceed to the 
external surface, and the surfaces are concentric, we have 
