52 M. C. Cellerier on the Distribution of 



volumes or of their sections j, irp 2 , whence results 



*'=i^" (2) 



In particular, if it be wished that the acute angle of the new 

 velocity with a given fixed direction LOI/ shall be greater 

 than a given acute angle a, it will be necessary that M be 

 within a zone bounded by two circles whose poles are L, I/, 

 and having an angular distance « from those two points. 

 The height of the zone being 2p cos a, its surface will be 

 co = 47T/3 2 cos a, and the number of collisions satisfying the pre- 

 ceding condition will be 



(3) 



Prof. Clausius has remarked that for the new velocities all 

 directions are equally probable. This is what equation (2) 

 expresses, since the number of collisions yJ is independent of 

 the position of co. It is this fundamental property that we 

 have now to demonstrate for molecules which are not spherical. 



Case II. All the molecules of the first kind have any surface 

 S, those of the second kind any surface S'; those of the same 

 sort are not only equal, but similarly oriented ; that is to say, 

 the homologous straight lines joining two- points of them are 

 parallel. 



Let us trace apart a typical sphere with centre 0, radius 1, 

 for which we will again use the same figure, A being its high- 

 est point. If we wish the new velocity of a descending mole- 

 cule to have M for its typical point, the normal to the plane 

 of the collision must be M', the bisectrix of OMA. On an 

 ascending molecule S' we shall find the point M", at which 

 the interior normal is paralled to M'O; there is only one such 

 point, because we shall suppose S, S' convex, without points 

 or sharp edges. We shall displace S by a motion of transla- 

 tion only, so that this surface shall touch S' in M". 



On repeating the operation for all the points M within an 

 infinitely small circle <u, the locus of the points W will be an 

 element co' of the sphere, that of the points M" an element a>" 

 of S', that of the corresponding positions of G a surface- 

 element 7 in the space. Carrying through each of these points 

 a vertical of the length Yt, we shall have formed a thin prism 

 of the volume h = Vty cos i, i being the angle made by the 

 plane of y with the horizontal. We will name it the efficient 

 volume of the region «, because it is that in which Gr must be 

 at the commencement of the time t in order that the new 



