OF PLANETARY ATMOSPHERES. 
13 
—=i== + AhV = const. 
+ r- 
This equation may be written 
^ \ = C (a constant).(20), 
Avhere • 
At the point of minimum density in the equatorial plane, 2 = 0, r = a, and the 
curve of equal density through this point is the one for which C = 1‘5. This curve 
has a double point at the point in question. 
The curves for which C > 1'5, and for which the density is therefore greater than 
the equatorial minimum, each consist of a closed oval and an infinite brancli having 
the line ^(r/«)" = C as asymptote, while the curves for which C < 1‘5 consist of 
infinite branches only. It is clear then that if any gas of appreciable density were 
to pass by diffusion beyond the closed portion of the surface C = 1‘5, it would diffuse 
itself indefinitely over the unclosed surfaces of equal density, and its molecules would 
not be able, by their collisions, to maintain the equilibrium of the distribution of 
molecules within the surface. 
Calling the surface C= 1'5 the critical surface, one condition for permanence is 
that the molecules which reach this surface must be so few and far between, that 
collisions rarely take place between them. 
Such molecules will then describe free trajectories under the planet’s attraction. If 
their velocity be greater than that required to cany them to infinity, they will leave 
the planet, describing parabolic or hyperbolic orbits. If the velocity be less tlian 
that amount, they will describe ellipses, and return to the planet’s atmosphere. 
Now at the singular point of the critical surface, the velocity due to axial rotation 
is just sufficient to make a particle, if projected with it, describe a circular orbit. 
For a parabolic orbit the velocity is ^2 times as great. Hence a molecule at the 
singular point moving tangentially in the direction of rotation, will just leave the 
planet if its relative translatory velocity, due to the temperature of the atmosphere 
which it has left, be ^"1 — 1 times the velocity due to rotation. If it be moving in 
any other direction, or it be situated at any other point of the critical surface, its 
velocity will have to be correspondingly greater. 
10. The Critical Density-ratio. 
It thus becomes important before proceeding further to calculate and tabulate, for 
different gases at different temperatures, the ratio of their densities at the surfaces 
of different members of the Solar System to their densities at the corresponding 
