ON OUR KNOWLEDGE OF THERMODYNAMICS 101 



Take axes of ^, v, K rotating about the axis of ;:; with angular velocity I? 

 and instantaneously coinciding with the axes of x, y, z. 



Then the relative velocities of a molecule referred to these moving 

 axes are 



l=u + Qy, 17= V — fix, ^=tv 



and we at once find 



9 (u, V, w) 

 Hence the distribution may be written 



neK^-h{km{i? + ii' + ^)-irX-h'^^\-- + ^'')] •'i-'^nd^didi^dt . (74) 



Therefore the velocities relative to the moving axes follow the Boltz- 

 mann-Maxwell distribution, and in addition to this the molecules have a 

 superposed motion of rigid-body-rotation with angular velocity fi. And 

 the density at any point is the same as if the gas were acted on by ' centri- 

 fugal force 'having a potential —\p'^{^^ + r)^), and the reversed angular 

 velocity — fi were applied to every molecule. 



Hence the Kinetic Theory may he applied to the atmospheres of planets by 

 reducing the jylanets to rest and applying centrifugal force to the atmospheres 

 in the usual way. 



It is interesting to notice that the temperature 3/27i is the mean 

 kinetic energy of the translational motion relative to the jilanet and not the 

 total mean translational enei'gy. 



The results can evidently be generalised for the case when the mole- 

 cules are rigid bodies of any kind. Let u„ o)„, w. be the angular velocities 

 of such a molecule about axes through its cm. parallel to the axes of 

 a-., y, z, and let wi, w,, i.'a be its angular velocities about its principal axes. 

 By Appendix I., duji djj.^ djjs is independent of the time, and e^'idently 



O {ti>„ o)y, M.) I determinant of the direction cosines | -i 



577i 7. 77 \ 1 between the two sets of axes J 



Hence the permanent distribution is of the form 



n exp — h (T — fi;; -f- x) • dx dy dz da dv dw du>^ dio^ duj^ 



Now if A, B, C, D, E, F denote the moments and products of inertia, 

 the kinetic energy of rotation of the molecule is 



T„=|(A, B, C.-D, -E,-F3[..,, a,,, c,)^ 



=^^^° p=m{vx-uy) + -^f- .... (75) 



ciut 



therefore 



T-np + x=hn [{u + nyY- + {v- nxy- + w']-hnn^- (x-^ + 2/^) 



-f-i (A, B, C,-D,-E,-F3[a,„ (.,, a,,-fi)2-iCfi2-Hx 



and since evidently 



tlierefore 



f=n exp -h {hn {k^ + f,' + 1") -F i ( A, B, C, - D, - E, -F ][ w„ o,„ a,,)^ 



-imn^^-W)-icn-'+x} .... (76) 



