FREE AND PERFECTLY ELASTIC MOLECULES IN A STATE OF MOTION. 
27 
plane is therefore u sin 6 4z. To find the increment given to the medium in a unit of 
time we have first to consider the number of impacts in the unit of time if the plane 
were at rest, and then the additional number owing to its motion. Preferring back to 
the reasoning in § 17, it has been shown that the impinging velocity u sin 6 is 
repeated u sin 6 times in a unit of time ; hence the increment of vis viva in the unit 
is u 2 sin 3 -6 4z, and as the mean of all the values of u d sin 3 6 is w 2 = ^v 2 , we have- 
following the reasoning in § 17, and supposing the surface of the plane to be equal to 
the side of the cube that contains A 3 molecules—the increment of vis viva given to the 
medium by the front surface in a unit of time is jr A 3 v 2 4z. By applying the same 
reasoning to the back surface of the plane the same amount of vis viva is found to be 
taken from the medium on that side. 
We have now to consider the additional number of impacts due to the motion of 
the plane. Let us first suppose that no change of density is caused by the motion. 
The action of a medium on a surface at rest is the same as that of a uniform current 
of molecules whose mean distance is \/\ A 3 and velocity w (see § 17), and in the same 
manner the surface meeting this current with the velocity 2 , the effect is the same as if 
the velocity of the current were increased to w -j- z. The additional number of 
impacts due to the motion of the plane is therefore ^ A 3 z, and the mean increment of 
vis viva to each being 4z, the whole increment in the unit of time is 2 A 3 z 2 . The 
same reasoning applied to the action on the back surface shows that the diminution in 
the number of impacts is also ■§• A 3 z, and the mean decrement of vis viva caused by 
each impact being also 4 z, we have the decrement of vis viva in a unit of time also 
2 A 3 z 3 . The sum of these 4 A 3 z 3 is the force required to move the plane with the 
velocity z. The weight n, whose pressure is equal to this force, is found as in the last 
equation of § 17 : there we had w 2 A 3 = ng, and in the same way here we have 
4 
4z 3 A 3 = n 0 g, or n 0 = - z 3 A 3 . 
This result differs very much from the actual resistance of a body moving in air as 
observed by Piobins and Hutton. In Hutton’s Dictionary it is mentioned that the 
resistance to a surface of one square foot, moving 20 feet per second, was found to be 
12 ozs., and that it increased as the square of the velocity. Now if we compute the 
resistance by the formula - z 3 A :J — n Q , which is the common theory at low velocities, 
we shall find n 0 to be 15 ozs. nearly, when z is 20 feet per second, the weight of a 
cubic foot of air being represented by - fe -T () - oz. 
Here then is a notable discrepancy ; the resistance of the medium that represents 
air in specific gravity and tension appears to be four times greater than it ought 
to be. 
§ 27. We have all along assumed, for the sake of simplicity and to avoid any 
addition to the fundamental hypothesis, that the surfaces upon which the medium 
acts are perfectly rigid as well as perfectly elastic, although no such surface, so far as 
