the Molecular Velocities in Gases. 49 



numerous groups, each of them will contain an immense num- 

 ber. The molecules are classed in groups according to the 

 nature of their velocities ; the molecules of one and the same 

 group are supposed to be distributed in a sensibly uniform 

 manner throughout the space. 



(2) If n is the number of molecules of a group, the number 

 of those of -which the centre of gravity is comprised in a de- 

 termined volume H will be nil, with an imperceptible relative 

 error. Indeed this is equivalent to saving that it is propor- 

 tional to the volume. Inow it is evident that, when the total 

 volume is divided into a great number of equal parts, the num- 

 ber of centres contained in each will be very great, and the 

 ratios between them sensibly equal to unity. There would 

 only be a difference for the proportions contiguous to the side, 

 it not being possible for the centre of a molecule to exist at a 

 distance from this which falls below a certain limit ; but this 

 limit being infinitely small in comparison with the total dimen- 

 sions, the result is not thereby altered. 



(3) We shall have to value the total number of collisions, 

 or that of certain collisions, during a period constantly desig- 

 nated by t ; this is supposed such that there is an immense 

 number of collisions, and yet they affect only an imperceptible 

 fraction of the total number of molecules of one and the same 

 group ; also the cases in which during that time one and the same 

 molecule suffers two collisions can be completely neglected, 

 their number being infinitesimal in proportion to the former. 



(4) The directions of the velocities will be indicated by 

 drawing parallel to each of them a radius of a fixed sphere, 

 which will be named the typical sphere: the point of the sur- 

 face at which it ends will be the typical point of the velocity ; 

 any portion of the surface will be called the typical region of 

 the velocities whose typical point is on that region — which will 

 serve for distributing them in groups. 



Fiest Disposition. 



All the centres of gravity of the molecules have one and the 

 same velocity JY, in parallel directions, wliich ice figure vertical; 

 hut n of these molecules are descending, and the others, in number 

 n', are ascending. 



Let G, G 7 be the centres of gravity of two molecules which 

 collide, and their common centre of gravity; this is in the 

 middle of GG', and consequently immovable before the colli- 

 sion; it will therefore continue to be so after it in the absence 

 of any extraneous force : consequently, if after the collision 

 G, G' are in P. P / and an instant afterwards in Q, Q', the tri- 

 angles POQ, P'OQ' being equal, the small spaces PQ, P'Q' 



Phil. Mag. S. 5. Vol. 13. No. 78. Jan. 1882. E 



