100 REPORT — 1894. 



And it can now be shown precisely in the same way as before that 

 the mean values of 



A2(.i,2, B^u-j^, CW 



are equal to one another. 



Therefore the mean values of 



^A«.,S iBa.2^ iCV 



are now inversely projiortional to A, B, C, a^id Maxwell's law of partition 

 of energy does not hold good. 



It is only when the distribution of Tslocities or momenta is deter- 

 mined by a function ,/ of the energy alone that we can assert that the 

 mean values of the different squares forming the kinetic energy are 

 equal. 



APPENDIX B. 



On the Laiv of Molecular Distribution in the Atmospliere of a 

 Rotating Planet. 



Suppose a mass of gas-molecules to be situated in a field of force that 

 is symmetrical about a fixed axis — for example, the field due to the 

 attraction of a spheroidal planet. 



Let this axis be chosen as the axis of ;:;;, and, in the first place, let the 

 molecules be monatomic, so that they may be regarded as smooth homo- 

 geneous spheres or material points. 



Let m, M be the masses of two molecules, E„, Ejj their total energies, 

 J), P their angular momenta about the axis of z, so that 



p=m{vx-uy) P=M(VX-UY) . . (72) 



Then, since the field is symmetrical, the angular momenta p, P are constant 

 in the absence of collisions, and at a collision the total angular momentum 

 is unaltered, so that 



p^-V=p' + V' 



Also similar relations are satisfied by the energies E„, Eji- 

 Therefore, if the distributions of co-ordinates and velocities be given 

 -by the expressions 



fdx dy dz du dv dw and F dX dY dZ dJJ dV dW, 



the conditions of permanence between collisions, and the functional equation 

 for collisions 



are satisfied by 



f=n exp —(h'E„,-kp) F=N exp -(/iEu-itP) 



where h, k are any constants whatever. Putting k=^hQ, we have, if x he 

 the potential energy of m, 



f=nexY> -h(T-Qp + x) 

 ==01 exp— A {im (?*^ -|- v"^-^ w^) — mQ (vx- uy) -f y] 



=nexp-A{im[(M-f-fi2/)' + (v-fia;)2+w2]-|-X-imn2(a;2 + y')} (73) 



