FREE AND PERFECTLY ELASTIC MOLECULES IN A STATE OF MOTION. 31 
is V - , and thus the mean result of these two equally probable impacts is the trans- 
n 
mission of the vis insita of the impinging molecule to the plane as if it cohered to it 
after impact. 
(4.) At either of the extremities of the vibration the same law obviously applies, but 
it is the two impacts where vis viva is taken from and again returned to Q whose effects 
ought to be viewed together. In the meeting impact let the velocity of Q from v 
change to u with a different direction ; then according to the second case of impact the 
upward effect on the plane is to give it the ascending velocity v + ^ - u , and the loss 
of velocity is v — u. Let this loss be returned in an overtaking impact so that v — u 
shall become v ; then according to the first case of impact the upward effect on the 
plane is ujn. In these two impacts Q returns to its original condition of motion, 
and the mean effect is v/n. A continual and equal interchange of vis viva being 
necessary to the persistent molecular condition of the plane and of the medium, the 
same is effected by means of impacts which take place equally in the ascending 
and descending vibration. This equality seems to be a necessary condition because 
the motions that are taken account of are the velocities of impact resolved in a vertical 
direction only, and the plane of impact cannot now be assumed always to be horizontal 
as in the case of the rigid plane; hence the absolute velocity in the vibration and the 
resolved impinging velocity are independent variables. 
§ 29. Such is the view of the phenomena which seems to authorise the change that 
has been imposed on the value of the mean square molecular velocity. It has no 
pretension to be considered as a demonstration, and we are therefore not permitted to 
make use of it as a synthetical deduction from the hypothesis. 
Nevertheless, if it is admitted as being probable, the probability is increased if it 
reconciles at once all the discrepancies that have been met with, and at the same 
time neither affects any one of the preceding deductions where the analogy to the 
properties of gases is perfect nor introduces any other point of discordance. 
If we now revise the mode of estimating the law of resistance in § 26 it is obvious 
that the mean increment of velocity communicated by the plane now considered as 
molecular to the free molecules of the medium is not 2z, but 2 , and hence the mean 
increment of vis viva in each incident molecule is not 4z, but 2z, and the increment in 
a unit of time not 2A 3 z 2 , but A 3 z 2 . The sum of the front increment and back decre¬ 
ment is not 4A 3 z 2 , but 2A 3 z 2 ; and as w 2 A 3 is no longer equal to ng, but to 2 ng, we have 
2A 3 z' 2 — 2 ng, or n = - , A 3 z 2 , which is the equation derived from the common theory of 
the resistance of the atmosphere at low velocities. 
It will be remarked that the resistance is as much derived from the minus pressure 
behind as from the resistance in front, whereas the common theory only takes account 
of the inertia of the front which is assumed at low velocities as constituting the whole 
© 
of the resistance.'" 
* Note G- (objection to undulatory theory of heat). 
