50 



M. C. Cellerier on the Distribution of 



It will therefore 



passed through will be equal and parallel. 



be the same with the 



be besides in the opposite direction. The vis viva not having 



changed, their value will still be |V; their projection upon the 



common tangent plane, too, not having changed, the old and the 



new velocity will have for their bisectrix the normal to that plane. 



Case I. All the molecules are spheres of the same diameter p. 



Let be the centre of one of the ascending 

 molecules, and let us trace a sphere with 

 centre and radius p. In order that a mo- 

 lecule of the first kind may come into contact 

 with the sphere during the time t, its centre 

 Gr must encounter the surface ; and for this 

 it must be, at the commencement of the time 

 t, interior to the cylinder which would be 

 formed by drawing, through all the points of 

 the upper hemispherical surface, verticals di- 

 rected upwards and of the length Vt, since V 

 is the relative velocity of the two molecules. 

 This cylinder is terminated below by the he- 

 misphere, above by an equal hemisphere ; its volume is that 

 of a cylinder of height Yt and base 7rp 2 ; the total volume of 

 the similar cylinders corresponding to all the molecules of the 

 second kind is H = n'lrp 2 Yt ; the number p. of centres of mole- 

 cules of the first kind which are found there at the commence- 

 ment of the time t is therefore nH, according to the second 

 remark above. Hence, putting 7rp 2 — a } results 



fi=ann f Vt (1) 



This number, then, is also that of the collisions during the 

 time t ; for, according to the third remark, the cases in which 

 two descending molecules would have their centres in one and 

 the same cylinder can be omitted, the collision of the one 

 preventing that of the other. 



Let us employ the above sphere as 

 typical of the velocities, and draw 

 various meridians through the vertical 

 O A. If the point G (the centre of a 

 descending molecule) meets the sur- 

 face on one of them in M', the tangent 

 plane common to the two molecules 

 will be parallel to the tangent plane 

 in M', so that, M being drawn par- 

 allel to the new velocity, its typical 

 point M will be upon the same meri- 



dian, and the 



angle 



AOM double 



