FREE AND PERFECTLY ELASTIC MOLECULES IN A STATE OF MOTION. 
37 
the same height. All this is an obvious corollary of what precedes. The vertical 
condition of equilibrium of an atmosphere is the same whether that atmosphere consists 
of one homogeneous medium or of a mixture of different media having the same specific 
gravity .XX. 
§ 36. The relation between the total height of an atmosphere of the medium to the 
height due to the molecular square velocity (XVII.) enables us easily to estimate the 
effect of diminished gravity, and assign the limits of temperature at the base, beyond 
which an atmosphere cannot be retained. 
If the force of gravity is constant, it has been shown that the height of the 
atmosphere is equal to the height a body would ascend with the molecular velocity, 
and supposed to be acted upon by half the force of gravity; but as it really 
R 
diminishes as we ascend, according to the ratio 
R + H 
(in which R is the radius of 
a l R 
with the variable half-force ^ 
the planet and H the height above its surface), the true height must be computed 
2 g 
n , r , TT ) , instead of with the constant half-force T 
2 \R + H/ 2 
Let v be the initial velocity of the vertical projectile at the surface of the planet, w 
its velocity at the height h ; then shall — — dt, and dt f— = — dw — the 
retardation in the differential time. Substituting the value of dt, the differential of 
the time in this equation, we have — dw — 
have — 2 w die = — dw 2 = dhg 
dh g 
w 2 \R + h 
, and multiplying by 2 iv we 
R + h 
Integrating this expression gives 
v 3 — w 2 = R</ (1 — 
R + h 
, so that when vr = 0, we have by eliminating h (which 
then represents the total height of the atmosphere) h = 
RA 
% - v 
-; being its value in 
terms of the radius of the planet, of the mean square molecular velocity at its 
surface, and of the force of gravity at its surface. In the former expression for H, 
where the force of gravity was supposed constant, we had H = v 2 jg, or v 3 = </H. 
Substituting this value of v 2 in the equation for h, we have h = 
H 
if H 2 /R is infinitesimal in respect to unity. 
Thus the correction to be applied for the diminishing power of gravity in ascending- 
increases as the square of the height, and employing the preceding data, the total 
height of the earth’s equilibrated dry atmosphere, considered as a medium at the 
temperature of melting ice, is by this theory 157,776 feet, being 1183 feet more than 
the last determination, with constant force of gravity ; and the correction to be 
added to the height computed with constant force of gravity is in feet 1‘2 X H 2 , the 
square of the height in miles. 
§ 37. We may express the last equation in a more general form with the molecular 
vis viva as the constant instead of the mean square molecular velocity. If the 
