FREE AND PERFECTLY ELASTIC MOLECULES IN A STATE OF MOTION. 15 
is the same is best seen by the reductio ad absurdum method of reasoning. If the 
ratio is different, the motions of the heavier molecules must be resolved in a different 
way from those of the lighter, or the plane of concurrence must incline to one set of 
molecules in a different manner to that of the other set. Now any such effect is quite 
inconsistent with the fundamental hypothesis, and would require us to admit the 
influence of a modifying power whose nature and mode of bringing about the effect in 
question is unknown. 
The ratio p of the resolved to the absolute vis viva is actually one-third, and will 
become obvious in the next section, but it seems needless to require the demonstration 
in this place as all that we have to be assured of is the constancy of the ratio, whatever 
its actual value may be. 
In seeking to demonstrate the nature of the vis viva equilibrium, the solitary 
condition that we have to reason from is that the mean value of (/3 y 2 -f- fif) is equal 
to 2/3 2 (§ 2) and the mean value of (S 0 2 + Sp) = 2S 3 That this is a necessary 
condition is obvious, because if either were less there would be a continual transfer 
from the molecules B to the molecules D, or from D to B and vice versd. 
By squaring the equations in § 2 and adding, we have the following :— 
/V + A 2 = 2{/3 3 - . p + _Ai_. (S n + py 
B + D 
4D 
(B + D) 3 
4B 3 
v + V = 2{S 2 - ^ • s* + (TT5? • (8* + /»•)}. 
If in any case it happens that /3 0 2 + /3p = /3' 3 , we shall have 
4D 2 
(B + D) 3 
(§ 3 + n 
or /3 2 B = S 2 D, and 8 0 2 + Sp — 2S 2 . Hence, if the squares of the impinging velocities 
happen to be in the inverse ratio of the molecular weights, then in either molecule the 
sum of the vis viva of the twofold encounter (one meeting, the other overtaking with 
the same impinging velocities) before impact, or 2/3 3 , is equal to the sum after impact, 
or to /3 0 2 + 
But this is only one case out of an infinite number where the ratio is different. 
Generally, we may express the equation thus : 
/V + Pi = 2/3 3 + p, and 8 0 2 + S x 3 = 2S 3 + q. 
Now, suppose that in an indefinitely great multitude of impacts the sum of all the 
individual values of ftp -f- /3p and S 0 2 -j- Sp are taken, we shall have the mean of the 
values of the first equal to in accordance with the necessary condition of per¬ 
manence noticed above (by (3 m z we mean to denote the mean molecular vis viva or 
