126 
Proceedings of Boyal Society of Edinburgh. [sess. 
These premised, the spectrum of the gas will be a band, whose 
visible breadth depends only on the fineness of the grating and 
the luminosity of a particle. But this band will present, at all 
times and from all sides of the group, exactly the same appearance. 
Its brightness, therefore, at any given distance from its central 
line, will be constant. But this means that the fraction of the 
whole number of particles which have any given speed in the line of 
sight, depends on that speed alone. The utmost speed of a gaseous 
particle is exceedingly small compared with that of light, and the 
alteration of wave-length is not affected by the part of the motion 
of the luminous particle which is transverse to the line of sight.] 
4. We have not yet exhausted the consequences of absolutely 
perfect (average) community. For every particle, in virtue of 
citizenship, has a right to, and obtains, its due quota of whatever is 
shared among the group. Its tenure of any one value of x ceases 
(usually in a most abrupt way) some 10 10 times per second, but 
leaves it absolutely free to have, during each of these brief periods, 
any values of y and 2 which may fall to it. There are, in fact, 
definite specifications of a*, y, z speeds ; but they are distributed 
among the particles with absolute independence of one another, in 
a manner which is perpetually changing at an exceptionally rapid 
rate. And the entire independence of x , y, and 2 speeds is shown 
by the fact that, in a collision, there is a mere interchange of speeds 
along the line of centres at impact : — whatever be the speeds of the 
impinging particles in other directions. 
Thus the assumption, which Maxwell allowed “might appear 
precarious ” (it is carefully to be observed that he did not say it 
appeared so to himself) is fully justified. In any element of volume 
of the space diagram of velocities, the density is proportional to 
whatever rectangular axes be employed. This, of course, gives at 
once Maxwell’s result, viz. : — 
To any one who is doubtful about the accuracy, or the cogency, 
of the reasoning just sketched, we may put the matter in another 
form. The solution, we saw, is unique. But this is obviously a 
/(*) Ay) f( z ) 
