696 Dr. G. Johnstone Stoney on 
one another inversely as the fifth power of the distance. 
The several assumptions which he thus makes are put forward 
not as theory but as hypothesis ; they do not profess to re- 
produce any existing gas, but ‘substitute for the gas an 
artificial model; and Maxwell is careful to keep this 
prominently before the mind of his reader. 
As to his exponential Jaw for the distribution of speeds, 
it is the solution of a functional equation, which in turn is 
the expression of the assumption that the number of molecules. 
whose velocities lie between w, v, w, and w+ du, v+6v, w+ dw 
must be some function of u,v, andw. Now this is true of 
Maxwell’s models, but cannot be the case in any gas in 
which there is an irruption of energy from the internal 
motions to the translational on the occurrence of events. 
which depend either wholly or partly on conditions other 
than the mere translatory speeds of the molecules—such 
conditions for example as the aspects of the two molecules 
to one another when the encounter is about to take place, 
or the phases at which the internal motions had arrived at 
that instant of time, or many other conditions that are 
possible and can be easily conceived. Accordingly, whenever 
a mathematician applies Maxwell’s law under the impression 
that, as regards any particular gas, it is more than an 
appr ‘oximate law, he tacitly assumes either that there are no 
internal events (as in Maxwell’s models), or that if there be 
internal events (as in all real gases) the partition of energy 
between these internal events and the translational motions 
is a transfer taking place at such short intervals that it may 
legitimately be treated by the mathematician as a process 
which goes on continuously and at a constant rate. At the 
bottom of our atmosphere an event that happens once in 
10° encounters occurs to each molecule as often as 7 or 8 
times per second. Even here the assumption that the 
transfer of energy goes on uninterruptedly makes but a 
rough approximation to the truth, and it is utterly remote 
from being an approximation in that penultimate stratum of 
the atmosphere from which nearly the whole escape of 
molecules takes place, and especially in regard to an event 
like the escape of a molecule from the earth, which is mainly 
the outcome of the circumstance that an individual encounter 
has chanced to be very unlike ordinary encounters. Hence, 
in no real gas can the actual law of the distribution of speeds 
be ¢dentical with Maxwell’s exponential law, nor with any of,the 
exponential laws of Maxwell’s successors ; although under 
the conditions which prevail in our laboratories these laws 
may be an approximation sufficient for many useful purposes. 
