in the Kinetic Theory of Gases, 299 



from the origin to points within the element of volume dQ at 

 P shall be said to have velocities OP (dQ) . Or if we give to 

 dQ any particular form, as co^dojdS, co being OP and dS 

 being a small solid angle, we may speak of the velocities OP 

 (co 2 dco d$) . If X, /jb, v be direction-cosines of the axis of a 

 cone containing the elementary solid angle dS, we may speak 

 of the direction A, /u v d$ as comprising all lines drawn from 

 the vertex of that cone and falling within it ; and \/ulvco 2 day d$ 

 as comprising all velocities between co and dco in directions 

 within that cone. 



4. In Maxwell's distribution, if ice consider all pairs of 

 molecules t M and m, having common velocity Y , and relative 

 velocity R + r ; for given Y all directions o/Rorr are equally 

 probable. 



Let OC = V, POp = tt + r, 



PO_™ _R 

 pO ~M~ r 



If OC be given, the number of the 

 pairs in question for which the angle 

 POO lies between 6 and 6 + dO is pro- 

 portional to 



Now 



M . PC 2 + m .pC*= MR 2 + mr* + W+n~i V 2 



-2(M.R-m.r)Vcos0, 



which is independent of 6 because MR — mr = 0. The number 

 is therefore proportional to sin d6, and this proves the pro- 

 position. 



5. If the molecules behave in their mutual encounters as elastic 

 spheres, then for^ given direction of the relative velocity before 

 encounter, all directions after encounter are equally probable. 



I think it unnecessary to give a proof of this proposition. 



6. (a) Every distribution of velocities among the molecules 

 which satisfies the condition that for given Y all directions ofR 

 are equally probable, is undisturbed by encounters, or by the 

 mutual action of the molecules, and is therefore, in the absence 

 of external forces, stationary. 



(b) No distribution whatever of velocities among the molecules 

 is undisturbed by encounters, or by the mutual action of the 

 molecules, unless it satisfies the condition that for given Y all 

 directions of R are equally probable. 



Of these theorems (a) corresponds to the well-known pro- 



