102 REPORT— 1894. 



Here the potential energy of centrifugal force is 



and therefore the law of distribution 



/=«exp-A{T, + x + Vo} .... (77) 



where T,. is the kinetic energy of the relative motion ; 



■X is the potential energy due to the field ; 

 Vq is the potential energy due to centrifugal force. 



Hence, as before, the Boltzmann- Maxwell Law holds for the system 

 obtained by applying the reversed angular velocity — fiand the centrifugal 

 force whose potential is — ^fi- (aS^ + y^) at every point of the gas. 



It would not be difficult to extend the proof to the case of a rotating 

 ellipsoidal planet with three unequal axes, where the field of force is not 

 symmetrical about the axes of rotation, but the investigation would 

 hardly be sufficiently interesting to be worth giving in detail. It will also 

 be admitted, without difficulty, that similar conclusions must hold good 

 when the planet and atmosphere besides rotating have a common motion 

 of simple translation. 



In a communication read at the Nottingham meeting of the Associa- 

 tion ' I worked out certain results of applying the Boltzmann- Maxwell 

 Law to the atmospheres of planets ; but in these calculations no account 

 was taken of axial rotation, as I did not at that time see how the effect of 

 tliis rotation could be determined. The numerical results there obtained 

 hold good, without modification, at points alone/ the polar axes of the 

 various bodies considered. The effects of centrifugal force on the dis- 

 tributions now furnish a promising subject for future investigation, about 

 which I hope to say more shortly. 



APPENDIX C. 



On the Application of the Determinantal Relation to the Kinetic Theory of 

 Polyatomic G'ases. By Professor Ludwig Boltzmann. 



We shall consider a gas whose molecules are compound (or poly- 

 atomic), but are all similarly constituted. Let a, b, c, . . . be the co-or- 

 dinates which determine the position and configuration of a molecule of 

 such a gas ; and let p, q, r, . . . be the corresponding momenta. Let us 

 suppose that the time during which any one molecule acts upon or is acted 

 upon by other molecules is short in comparison with the whole time of its 

 motion. Let the gas be contained in a vessel of invariable form. After 

 a certain time the state of the gas will become stationary, aiid the ques- 

 tion is, what is then the probability that the co-ordinates and momenta of 

 any one molecule lie between certain limits 1 To express the probability 

 by means of a number let us suppose the stationary state to last for a long 

 time, ©. Divide this time into n infinitely small parts, 5. We shall call 



' ' The Moon's Atmosphere and the Kinetic Theory of Gases,' Nottingham Rejjorty 

 p. G82. 



