OF PLANETAEY AT^rOSPfTERFS. 
11 
I e 7'~(fdq = — e 
^ o + 2 , 
- p 2" 
7 0 O 
Il-ITI'^ 
^ 0 1 + 
= 47r"n?-- 
h-mr 
(since = 2M/'r) . (IG), 
a result more easily obtainal)le independently of the present method. If, however, 
the expansion of 2 cosh hniflrq in a series becomes inapplicable, we must express the 
integral by means of error functions. It becomes 
47^2.„ytMm/',- 
hh)d9J 
Now 
f gV'mnV I Q-lhraiq + nrf _ 2 1" 1 
j Q J Q J Q j 
f = 
•'q 
"^[q — 9.r)dq = 
y — ih7n{q—ilr 
JAmQ2 
hm 
- nr)r 
km 
[ e ^''^'Qrdq = , yy . Erfc [(Q — flr)^{^hm)], 
jq y/{Z/im) 
and similarly for the reduction of the remaining integral. Hence we obtain 
finally 
hVDJ 
+ 
_|_ ^-7i?)!nrQ _ 2^ 
VLry/{^hn) ^Erfc [(Q — 9r)y/— Erfc [(Q + 9r)y/[^hmy] 
(17), 
and when tlie argument is large we may expand the error-function complements by 
means of the descending series referred to in § 3, equation (4). Remembering, too, 
that Q" = 2Mb-, Ave obtain 
477-71 r 
hhrdD. I 
,hritnr(i 
1 + 
Hr 
+ e 
1 
4(Q - VLr) 
VLr 
1 — 
xO + 
1 . 3 
4(Q + n?-) 
1 
hmdd — 9.r)~ ]ihii-{Q, — n?-)- 
1 1.3 
2 ]ihn?(Q 
km{Q. + n?’) 
(Ci 4 9rf 
This formula might be applied to form an estimate of the rate at Avhich a planet is 
losing its atmosphere. We shall noAv, however, show that the law of distribution, 
and therefore the above formula, cease to hold good beyond a certain distance from 
the planet. 
9. Limit to the Height of a Planet's jitmos'phere. 
It is obvious, on the n.07r-kinetic vleAv, that no matter could be retained in relati^^e 
equilibrium beyond the distance at which a planet’s attraction is just Italanced by 
C 2 
