FREE AND PERFECTLY ELASTIC MOLECULES IN A STATE OF MOTION. 19 
gravitating elastic plane ; and the equilibrium as being maintained when the upward 
velocity given to the plane by the shock of one molecule was equal to the downward 
velocity given to it by gravity acting through half the infinitesimal portion of time 
that elapses between, two successive impacts. During the first half of this time 
gravity acts in destroying the upward velocity : during the second half it acts and 
generates the same velocity downwards, and by applying the equation for the meeting 
impact (§ 2) we found the relation between the pressure and the number of impacts in 
a given time. This relation is expressed by gn^lw = A, in which n is the number of 
molecules whose aggregate weight is the weight of the plane supported by A number 
of molecular impacts in a second of time : the impinging velocity being w, and g the 
accelerating force of gravity. It was also shown that the upward velocity given to 
the plane by one impact is tr/n, and this is likewise the descending velocity with 
which it encounters the molecular shock. 
We have now to examine the case where the encounter takes place with the plane 
at rest. Applying the equation for the meeting impact as in § 2, and putting 
e = /3 — 0 ; n — B; D = 1 ; io = S ; we have 
2w 
e o — Aj = n ^ = velocity upwards of the plane after the shock ; 
w 0 = §o = w 
2 wn 
= — w + 
non 
to 
— w 
n + 1 1 n + 1 7i+l 
= velocity downwards of the molecules after the shock.'"' 
Thus, n being an indefinitely great number, we have the ascending velocity of the 
plane e 0 = 2 iv/n, being double what it was in the former case when the result of 
the impacts was statical equilibrium: and the decrement of molecular velocity 
= w — w 0 = ivfn, which is a new and important feature. In the former case there 
was no decrement of molecular velocity : the molecule and plane continually meeting 
and retreating with velocity of impact and reflexion the same, and inversely 
proportional to their respective weight. 
With its velocity — the gravitating plane ascends to the height (-:) \g — 
9 n 
2w 2 
gn* 
SO 
that the weight of n molecules is raised through this height by the decrement 10 / 1 % of 
the impinging velocity w of one molecule. Employing the differential notation, 
* [It is easy to see that in the case supposed nc 0 = 2 w, when n is great, so that the velocity of the plane 
is 2w/n ; but in the next step there is an unfortunate error which runs through many of the subsequent 
deductions. 
W Ci — Sr, = IV 
Itvn 
= — — w 
Awn 
n + 1 
= — [w — 
n + 1/ ’ 
not 
tv — 
2?y 2 /n.—R.] 
w \ m 
n ,p \ }• The vis viva expended in raising n to the height 2?y 2 /y?i 2 is thus 4nv 2 /n, not 
D 2 
