14 
DE. G. H. BRYAN ON THE KINETIC THEORY 
critical surfaces, assuming the law of permanent distribution of § 6. This can easily be 
done from the formula, according to which the density at any point is proportional to 
g — 7i/K{V — 
Calling the ratio in question the critical density-ratio, the condition for a permanent 
atmosphere requires this ratio to be very large. It must in fact be the ratio of a 
density of molecular distribution at the surface of the })lanet comparable with that 
of the gases in our atmosphere to one at the critical surface in which the molecides 
are very few and far l)etween. For most purposes it will be sufficient to knov' the 
value of this ratio coi-rect to the nearest power of 10, and this is shovui quite as 
clearly l^y tahidating the logarithm of the ratio to base 10, for which purpose it is 
only necessary to multiply the difference of the potentials of gravitation and centrifugal 
force at the planet’s surface and at the critical surface hylimy,, where g is the modulus 
of common logarithms. The tables are tlius easy enough to calculate, but it is necessary 
to have them before one in order to arrive at any definite conclusions. 
The logarithm to base 10 of the critical density-ratio, as thus defined, is equal to 
(Vo + iu— VO.(21), 
where Vq is the gravitation potential at tlie j^lanet’s surface, 
u is the velocity at the planet’s equator due to axial rotation, 
Vj is the coml)ined potential of gravitation and centrifugal force at the critical 
surface. Since this is constant over the critical surface, we may take V^ to be the 
potential, due to gravitation alone, at the extremity of the polar radius of the ciitical 
surface. Tliis polar radius is | of tlie equatorial radius, and the latter is obtained by 
e(piating tlie planet’s attraction to centrifugal force. 
hm is equal to 3/Q'q wliere is the mean of the squares of the velocities of the 
molecules. This quantity is proportional to the molecular weight and inversely 
proportional to the absolute temjjerature. 
Starting with the case of the Earth and hydrogen, I have used the data given in 
Dr. JoHNSTONi'] Stoxey’s paper, pp. 310, 312, viz.. 
Earth’s equatorial radius = 6378 kiloms. 
Value of p' at equator due to Eartli’s attraction at equator = 081'5 centims./sec.^. 
V, the equatorial velocity due to rotation = 464 metre/sec. 
= (11140)' T, in C.G.S. units at absolute temperature T. 
It will be convenieiiL co assume a temperature of 100° absolute (—173° C.) in tlie 
first calculations, since tlie values for other temperatures can be more easily calculated 
by taking a round number to start with. 
We thus obtain the followine- values fin' the various teinis :— 
O 
pAmVo the term due to tlie gravitation potential at the Earfli’s surfiice = 65’718 
Call this term A. 
