456 The Atmospheres of the Planets. (October, 
Differentiate this with respect to w, and after multiplying 
(12) by integrate when— 
u 
$()=5 —_{ u—log (1—f-+fe-"")} 
oP ; 
substituting these values f 9’ (s) and @ (s) in the two equa- 
tions (8) and (9), and they become— 
eI L=f+pe- hie Stare (14) 
and— 
sae" a at ee eae (15) 
Upon the values given to the two constants f and 7 will 
the rapidity and degree of decrease of heat depend, the for- 
mer influencing mainly the amount and the latter more 
immediately the quickness of the decrease in temperature 
with the variation in density. To determine f, the condi- 
tion exists that f must be such a value that at the summit 
of the atmosphere the temperature will be the same as the 
temperature of space; or if this be supposed extremely 
low, that at which the elasticity of the atmosphere is equalled 
by the attraction of gravity. Accordingly if ¢’ be this tem- 
perature,— 
apuit! ol pee 
I+ét, 
The other constant 7 can be determined if the temperature 
at any given great height were known; but this not being 
so, it must be arbitrarily fixed. It is noticeable that when 7 
is unity, its influence vanishes, but as it approaches zero its 
influence is to make the condition of the atmosphere 
approach those of a uniform temperature; whilst as it 
increases from unity and approaches infinity, it gradu- 
ally approaches the condition of an atmosphere of the uni- 
form temperature given by— 
I+et 
I+e ft, =(I sie) 
on a surface heated to ¢,. In both cases it may be held to 
approach from the value unity the condition of a uniform 
temperature. 
Consequently, as it is apparent for beyond any but small 
values of #, this variable is greater than 8s; for any admis- 
sible assumed law of decrease of temperature, the density of 
an atmosphere at a given height is less than the condition 
