1875]. The Atmospheres of the Planets. 451 
and (1) becomes— 
ap=—g,adds oy he) |~6 (3) 
From the theory of the expansion of gases 
_ pho thet 
isa hae aacaapar aD. ieee (4) 
and dividing (3) by this value of s— 
ape ee. 5, Itet 
, eee ange) 4S ei (5) 
This is the differential equation between the pressure and 
height above the surface, and when integrated by substitu- 
tion in (4) gives the law of decrease of density, with increase 
of height above the surface. This equation cannot, how- 
ever, be integrated unless the manner in which the tempera- 
ture varies with the altitude is known. For the present this 
may be assumed to be given by— 
; It+e tf 
= ° ° ° . ° 6 
¢'(s) Saas (6) 
and then very conveniently it may be considered that— 
OS): = a Tae A ay eee cards) 
Substituting these values in (5) after integrating and 
determining the constant by the condition that p=, when 
s=o; whilst replacing for brevity the ratio © by e and (5) 
a) 
becomes— 
p=po e- 8°24) a ee (8) 
and by substitution in (4)— 
d=6, $s). e789 7919) oa pee (9) 
These two equations, once the form of ¢’(s) is known, are 
sufficient to entirely determine the normal physical condition 
of the atmosphere of any planet, in so far at least as de- 
pends on variations indensity, pressure, and temperature. 
As the expansion of gases for a small increase of heat is 
very small, for conditions when the atmosphere undergoes 
only slow and slight variations in temperature as the height 
above the surface increases, the law of decrease of tempera- 
ture possesses only secondary importance, and the condition 
of the atmosphere of the planet approximates to that it 
would possess were the temperature throughout uniform. 
For many purposes it may, indeed, be considered that the 
temperature of the atmosphere is constant without intro- 
ducing any very material error, especially in the higher regions, 
