FREE AND PERFECTLY ELASTIC MOLECULES IN A STATE OF MOTION. 29 
equal in mass they will exchange impinging’ velocities ; this interchange modifies the 
vibration and disturbs the harmony between the simultaneous motions of the centres 
of gravity above referred to ; the motion of the centre of gravity of the molecule 
changes suddenly, not so with the centre of gravity of the remainder of the plane. If 
the connection between the plane and the molecules w T ere broken at the instant of 
impact, it is clear that the centre of gravity of the remaining molecules of the plane 
must continue to move in the direction and with the velocity it had at the instant. 
Now as the concurrence may take place at any part of the vibration, either going or 
returning, it is plain that the mean motion of the centre of gravity of the remainder 
of the plane caused by the transference of vis viva from the molecule of the plane to 
that of the medium is zero. But the centre of gravity of the remainder of the plane 
reciprocates the active effect of the molecular force on the new velocity until it is 
destroyed at the end of the first vibration; the molecular force acting as much on the 
remainder of the plane as upon the molecule. The destruction of this by the mutual 
binding force destroys in the opposite direction the same amount of vis insita in the 
plane, or generates it in the same direction, and as we have to attend only to the 
effect upon the centre of gravity of the plane made by the motion transferred from 
the medium to the molecule of the plane, the mean effect must be equal to the mean 
incident vis insita of the molecules of the medium; in short the same as if the striking 
molecule cohered to the plane after impact. This is the case if the plane is at rest 
when struck, but a condition of statical equilibrium requires that the infinitesimal 
descending motion by gravity should be equal to the ascending infinitesimal motion 
given by the impetus of the striking molecules. The upward velocity therefore given 
to the plane by this impetus is only one half what it would be if the plane were at 
. 2iV , V 
rest when struck (see § 2). Thus the expression -~ + -~ (see § 2) becomes ——- , and 
vjn becomes vj2n, and gnj2v = A becomes gn/v = A. 
These alterations make no difference in the subsequent reasoning until we come to 
§ 17 where the equation for A is employed, and in consequence of its change of value 
the terminal equation A 3 v 2 = 3 gn is changed to A 3 id = 6gn. 
This alters the value of v from \/ ~ s , the velocity acquired in falling through one 
and a half uniform atmospheres, to , the velocity acquired in falling through 
three uniform atmospheres, and the numerical value of v in the medium that corre¬ 
sponds with air at the temperature of melting ice is 2244 feet per second. 
§ 28. As this change in the value of v reconciles the discrepancy in the theory of 
resistance, and in the subjects of the two concluding sections, it may be proper to 
illustrate by diagram the general principle that the mean impinging effect of free 
molecules on a cluster of cohering molecules is the same as if the striking molecules 
o o 
cohered at the instant of impact. 
