68 PROFESSOR TAIT ON THE 



Some of the immediate consequences of this principle are obvious without 

 calculation: such as 



(a) Even distribution, at any moment, of all the particles throughout the 

 space in which they move. 



(/;) Even distribution of direction of motion among all particles having any 

 one speed, and therefore among all the particles. 



(<•) Definite percentage of the whole for speed lying between definite 

 limits. 



These apply, not only to the whole group of particles but, to those in any 

 portion of space sufficiently large to contain a very great number of particles. 



(d) When there are two or more sets of mutually colliding spheres, no one 

 of which is overwhelmingly more numerous than another, nor in a hopeless 

 minority as regards the sum of the others, similar assertions may be made as 

 to each set separately. 



2. But calculation is required in order to determine the law of grouping 

 as to speeds, in (c) above. It is quite clear that the spheres, even if they once 

 had equal speed, could not possibly maintain such a state. [I except, of 

 course, such merely artificial distributions as those in which the spheres are 

 supposed to move in groups in various non -intersecting sets of parallel lines, 

 and to have none but direct impacts. For such distributions are thoroughly 

 unstable ; the very slightest transverse impact, on any one sphere, would at 

 once upset the arrangement.] For, when equal smooth spheres impinge, 

 they exchange their velocities along the line of centres at impact, the other 

 components being unchanged ; so that, only when that line is equally inclined 

 to their original directions of motion, do their speeds, if originally equal, remain 

 equal after the completion of the impact. And, as an extreme case, when two 

 spheres impinge so that the velocity of one is wholly in the line of centres at 

 impact, and that of the other wholly perpendicular to it, the first is brought to 

 rest and the second takes the whole kinetic energy of the pair. Still, what- 

 ever be the final distribution of speeds, it is obvious that it must be in- 

 dependent of any special system of axes which we may use for its computation. 

 This consideration, taken along with (b) above, suffices to enable us to find this 

 final distribution. 



3. For we may imagine a space-diagram to be constructed, in which lines 

 arc laid off from an origin so as to represent the simultaneous velocities of all 

 the spheres in a portion of space large enough to contain a very great number 

 of them. Then (b) shows that these lines are to be drawn evenly in all direc- 

 tions in space, and (c) that their ends are evenly distributed throughout the 

 space between any two nearly equal concentric spheres, whose centres are at 

 the common origin. The density of distribution of the ends (i.e., the number 

 in unit volume of the space-diagram) is therefore a function of r, that is, of 



