of Edinburgh, Session 1885-86. 
393 
proximate particles, as well as on the average speeds of a P and a 
Q. (This calculation can he effected by a simple method closely 
analogous to that in § 4 above. See the paper immediately following 
this.) But it is very much to my purpose to look back on what 
precedes, so that we may clearly see what assumptions had to be 
made in order that our results might be such as they are. 
And we see at once that the investigation, from the point of view 
taken, would have been barren of interpretable result had it not 
been for the assumptions by which we [so far] * justified (in § 6) the 
statements : — - 
(a) Average value of uv = 0 . 
( b ) Average value of P u 2 - Q« 2 = J(Pp 2 - Qg 2 ) . 
This last may be considered as including — 
(c) Average value of P(u' 2 - u 2 ) = JP(p' 2 -p 2 ) divided by the 
ratio, of the number of Ps which impinged on Qs, to the whole 
number of Ps. 
Now these assumptions were themselves justified solely by the 
Understood “ special ” state of the Ps and Qs separately ; and by the 
u equalising ” property, in virtue of which each system, so far as its 
own internal actions are concerned, tends in the long run to that 
special state. We are not warranted in concluding that either (a) 
Or (b) would hold true unless the separate systems tended by their 
Own internal actions to the “ special ” state. Thus, suppose the Ps 
to impinge on one another, and on the Qs; but the system of Qs to 
have no internal impacts. This would be the case if the Qs were 
mere points ; i.e. particles of diameters infinitely small in com- 
parison with the average distance between two proximate ones. 
Nothing above, so far at least as we have developed it, warrants us 
in concluding that the Qs will tend to a special state, and, therefore, 
acquire the same average energy as the Ps. Obviously, if the Qs 
were in a great majority, they would not only not themselves 
assume a special state, but would also tend to prevent the Ps from 
ever doing so. Think of Le Sage’s ultramundane corpuscles and 
* [Inserted Jan. 8, 1886.] I hope to show at the next meeting of the 
Society that, though neither of these assumptions is correct, Maxwell’s 
Theorem is rigorously true. Neither in Maxwell’s paper nor in this has any 
account been taken of the fact that collisions are more frequent as the relative 
speed is greater. This consideration affects only numerically the results of 
§§ 7, 9, 10 above, and does not interfere with the argument based on them. 
