OF PLANETAEY ATMOSPHERES. 
7 
equations of motion of the molecules. In the case commonly considered tlie only 
known integral is the energy, and hence and Fo are taken to he functions of the 
energies E, and Eo. 
Condition (ii.), combined with the fact that the sum of the energies is unaltered by 
an encounter, then leads us to the well-known forms 
Fj OC c 
-/■Ej 
F .1 oc e 
But if the field is symmetrical about the axis of 2 , and F,, are the angular 
momenta of the molecules about this axis, we also know that— 
(o) In the absence of an encounter F^ and Fg are constant; 
{Jj) The total angular momentum is unaltered by encounters, so that the values 
before and after an encounter satisfy the relation fi- F^ = F/ -F F./. 
We thus see that a more general solution satisfying conditions (i.) and (ii.) now 
exists, of the form 
F, 
rqe' 
/tEj d* A:F 
F.t = 71,e 
( 7 ), 
the frequency functions F^ and Fo now containing exponential functions of the 
angular momentum as well as of tlie total energy. Take the case in which the 
molecules are regarded as material points of mass m, and siq)pose in the fii'st 
instance tliat x, ?/, 2 , and a, c, tv represent co-ordinates and velocities referred to 
fixed axes, the axis of 2 coinciding witli the axis of rotation. Also let V he the 
gravitation potential of the field of force, and siq)pose /.' = IiD.. Tlien the above 
expressions for F^ and Fo show that a permanent distribution can exist, in which the 
number of molecules whose co-ordinates and velocities lie witliin the limits dx dy cA 
and du dv dw respectively, is of the form 
7% exp — hi7i{^[u~ + ~~ ^ — uy) — Vj dxdydz dudvdtv . (8), 
where A, n, may he any constants iqj to tliis point, and ti is proportional to the 
total number of molecules. 
Now take axes of y, rotating about the axis of 2 or { with angular velocity Q 
and instantaneously coinciding with the axes of x, y, 2 . Then the velocities of a 
molecule relative to these axes are , y', C respectively, where 
= t = “ + 
f dt] 
r= 
^ ,u 
w. 
and the expression determining the frequency of distribution becomes 
71 exp - hm {i (P -f y'-^ + _ V - i n-^ {e T y'^)] d^dy dCd^dydC (9). 
The distribution of co-ordinates and relative velocities is thns the same as if the 
axes were fixed, and the potential of the field of force had the term — ^12 + y~) 
added to it. This term represents the potential of centrifugal force due to rotation 
with angular velocity H. 
