Stoner—Of Atmospheres upon Planets and Satellites. 311 
Cuapter Il.—Dynamical Equations. 
In making our calculations with reference to the planets and satellites of the 
Solar system, it will simplify the work, and be sufficient for our purpose, to treat 
them as spherical bodies, consisting of layers each of which is a spherical shell 
of uniform density. In that case, if B be one of these bodies 
‘ ; M 
a (the acceleration at the surface of B, due to attraction) = Ri (2) 
and 
K (the potential of gravitation at its surface) : - 5s, (3) 
where JM is the mass of B, and F# the radius of its spherical surface. 
Now 4, the potential, as we learn in the science of Dynamics, expresses 
the kinetic energy stored up per unit of mass by a small* body in falling upon 
the surface of B from infinity. Hence, 
vy 
ie Qo ’ (4) 
where v is the velocity which would be acquired by a small mass in falling from 
infinity. Ifa missile were projected from B with this speed, it would just be able 
to reach infinity, 7.e. this speed is the least which would enable a molecule to 
get completely away from B. _We may, therefore, call it the minimum speed of 
* By a small body is to be understood one whose mass bears to the mass of B, a ratio so small 
that, from the physical standpoint, it may legitimately be regarded as a small quantity of at least the 
first order. For this purpose, a ratio of a tenthet, that is, of a unit in the tenth place of decimals, 
is sufficiently small in almost every branch of physical inquiry. If If be the mass of B, and m the 
mass of the body falling upon it, then the energy changed from potential into kinetic energy, by 
allowing them to fall together from infinity, 
mm? MV 
ak ore 
if we suppose them to have started from rest, and if on coming together they have acquired the 
velocities V and — v. Now, by the Principle of the Centre of Mass, ZV + mv = 0. Therefore the 
acquired kinetic energy may be written 
_ me ee 
int 1g i) 
mov 
=>) 
which differs from being 
by an insensible quantity if the ratio m/f is sufficiently small. And it is much more than suffi- 
ciently small from the physical standpoint, in the cases we are concerned with, where m is the mass 
of a gaseous molecule, and If the mass of a planet or satellite. In fact m is here of about the fifth 
order of small quantities compared with J, if we take a tenthet (10~) as about the ratio between 
quantities of two consecutive orders. 
