1895-96.] Prof. Tait on Clerk-Maxwells Law of Distribution. 125 
Is it not obvious, at once, that such a group must present at all 
times , and from all sides , precisely the same features ? In other 
words : — that the solution of the problem is UNIQUE. (This 
word practically contains the whole point of the question). If not, 
the higher part of the Theory of Probabilities (in which M. Ber- 
trand himself is one of the prominent authorities) is a mere useless 
outcome of analytical dexterity ; and even common-sense, with 
consistent experience to guide it, is of no value whatever. 
A first consequence of this perfect community of interests is that 
(on the average, of course) the fraction of the whole particles, 
whose component speeds in any assigned direction lie between 
x and x + Sx is expressed by 
f(x)Sx 
where f is a perfectly definite (and obviously even) function. 
It is clear from this that the density of ends in the velocity space- 
diagram depends on r only ; but we require further information before 
we can find hoiv. (M. Bertrand seems to admit the first statement ; 
but he insists that, otherwise, the solution is wholly arbitrary.) 
3. [But, before seeking this, we may take another mode of view- 
ing the situation : — as follows. It is, of course, nothing more than 
an illustration of the argument just given. 
Suppose, merely for the purpose of examining the condition of 
the gas, and therefore without any inquiry into other physical 
possibilities, which have nothing to do with the argument : — 
That (a) each particle of the group is self-luminous, and all 
give out, with equal intensity, light of one definite period. (To 
illustrate the remark just made, note that this luminosity is not 
attributed to collisions, nor to any assigned physical causes). 
( b ) The wave-length of light reaching the eye from a moving 
source is altered by an amount proportional to the speed with which 
its distance from the eye alters. 
(c) The displacement of light by a grating on which it falls 
normally is proportional to the wave-length. 
(d) An ideal grating may be assumed, of any requisite regularity 
and fineness ; and, again for the sake of argument only, it may be 
supposed to act, however fine it be, in the same manner as do 
ordinary gratings. 
