OF PLANETARY ATMOSPHERES. 
9 
the lower regions of our atmosphere, in consecjnence of convection currents and winds, 
the adiabatic law more nearly represents the true distribution of density. The calcu¬ 
lation of numerical data showing the effect of this correction must be deferred for 
future investigation, but a general discussion of the etfects which may be expected 
to arise from this cause will be given later. 
8. Rate at which the Molecules are escaping across a Concentric Spherical 
Surface. 
Assuming the distribution given by (7) and supposing the planet to be spherical, 
so that the potential at distance r from the centre is M/r, let us calculate the number 
of molecules per unit time which are crossing a concentric spherical shell of radius r 
with sufficient velocity to carry them to infinity if they do not encounter other 
I 
molecules in their subsequent paths. 
For this purpose refer to polar co-ordinates, then the frequency of distribution (7) 
assumes the form 
11 exp [ — hm + %") ~ Httg r sin 6 — M/rj] 
r” sin 0 dr cW clef) cluy du^ du^ .(lO); 
where w,, u.^, are the velocity components relative to axes fixed in space coinci¬ 
ding with the directions of the line elements c/r, reW and r sin i'6(f) at the point (r, 6, ef)). 
Thus Ui, th, are the velocities resolved along the vertical, the meridian and the 
parallel of latitude at the point. 
To find the number of molecules crossing the spherical surface-element sin 6 d6 d(f> 
with velocities within the limits of the multiple differential du^, du-,, du.^, we write 
^q dt for dr in the above expression, and if the number is re(.[uired per unit time we 
divide by dt. 
Of the molecules crossing the surface outwards, those which will escape from the 
planet’s attraction if they do not encounter other molecules, have their residtant 
velocity greater than Q where Q* = 2M/V. Calling these the escaping molecules,” 
the number of escaping molecules across the element r” sin 6 cW d<f) per unit time is 
sin 6 cW d(f> n exp [ — hm (tq'^ -f- up + — fiWg r sin 6 — M/r }] duy du .2 du.^ 
.( 11 )> 
the limits of integration being defined by the relation if + if + ^'3^ > 
Now transform the integral by putting 
tq = q sin a sin /3, ?q = q cos a, Wg = q sin a cos 
and at the same time integrate with respect to 9 and <f). 
Then we obtain for the total number of escaping molecules the expression 
VOL. CXCVI.—A. 
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