10 
Dll. G. H. BRYAN ON THE KINETIC THEOEY 
I dq [ d0\ dj)\ da \ d/S. n exiD [ — hm flrq sin a cos /3 sin 8 — M/rH 
• Q 0 • 0 0 J 0 
q sin a sin /3 . q^ sin a , sin 6 . 
Integrate first with respect to /3 and </>. AVe obtain 
( 12 ). 
2irn f dq [ dO [ da j.in. .m» _ - j-m,-,nn . 
J Q J 0 J 0 L 
Sine I n- sm g 
liniV. 
This expression cannot be integrated very simply wdth respect to a alone or 8 alone. 
dx and 
J rt 
flc sin X c. 
sin X dx. 
as we shonld obtain inteo’rals of the forms 
1 0 
It is therefore not possible to express the number of “escaping molecules” in the 
neighljonrbood of a given parallel of latitude except by integrals of these forms. 
But the double integration can be effected very simply by regarding a, 8 as polar 
co-ordinates of a point on a sphere, and changing the axis of polar co-ordinates to one 
at right angles to the former axis. 
Thus considering the integral 
I" rgtsinasinS 
• J 0 • 0 
v’e choose two new variables, y, xjj, defined by the relations 
sin a sin 8 = cos y, 
sin a cos 8 = sin y cos if/. 
cos a = sin y sin if/. 
and by spherical geometry or otherwise 
sin a da d8 = sin y c/y dijj 
and the integral transforms into 
(13). 
■ i:m>: 
y e sin y cZy = 2?? 
'■’ - 1 
(14). 
Substituting k = limD.rq and k = — hmD,rq in turn in (12), we obtain 
(15). 
The form most convenient for evaluating this integral must depend on the par¬ 
ticular problem considered. If tlie etfects of axial rotation on the rate ot escape are 
small, then 2 cosh hmP.rq may be expajided in powers of limD.rq, and we thus obtain 
in the limiting case when H = 0 
