20 
MR. J. J. WATERSTON ON THE PHYSICS OP MEDIA COMPOSED OF 
dw = ivjn ; and 2 dw w — 2nr/n = did 1 , the vis viva expended in raising n to the height 
2 vPjgn % . 
§ 17. If we recur to § 4 we may remark the necessity of considering molecular 
velocity in two points of view when applying the arguments of §§ 2 and 16 to an 
enclosed volume of a medium. The first point is that upon the molecular velocity of 
impact depends the intensity of the shock on the plane, the ascending velocity given 
to it, and therefore also the time between the impacts, if the weight of the plane is 
considered constant. Thus as any one velocity is to the time of ascent and descent 
of the plane caused by that velocity, so is the mean of the impinging velocities to the 
mean of the time intervals, or inversely as the number of impacts in a unit of time ; 
and the equilibrium does not require that the succession of impacts should be 
regular; the rapidity of the succession may fluctuate, but the average time and 
velocity must be constant. The second point is that any augmentation of velocity 
causes an increase in the frequency of the encounters (§ 4). In the equation ngj2w — A, 
if A were not a function of iv it would remain unchanged, if n and w increased or 
diminished in the same proportion ; but it was shown in § 4 that it was proportional 
to w, when A 3 , the density, is constant, and to A 3 or n or c when w is constant; hence 
A — wc A 3 , in which c is a constant factor that has to be determined. We have also 
to determine the ratio between iv 2 , the mean square impinging velocity, and v : , the 
mean square absolute molecular velocity, in the equation ngj2w = A = wc A 3 , or 
2 
n = - wc A 3 . 
9 
Suppose the unit of volume in which the medium is confined to be a cube, the 
upper side of which is the plane n, and let v 3 be the mean square velocity of the 
molecules, so that if the squares of the respective velocities of all the molecules be 
added together, the sum will at all times be equal to A 3 t> 3 . Resolve the motion of 
each molecule at any instant into the six rectangular directions parallel to the side of 
the cube and add up the squares of the resolved velocities that are perpendicular to 
one side ; it is evident that the sum must be -jj A 3 r 3 , as the force is equally distributed 
in every direction, and in the stratum of the medium next the plane n one-sixth of 
the force of the molecules that happen to be in the stratum at any given instant is 
directed perpendicularly upon the plane. Suppose the breadth of the stratum 
is 1/A, the number of molecules that at all times are to be found moving in it is A 3 , and 
half of these are diminishing their distance from the plane, and half increasing their 
distance with the mean square velocity g- u 3 . 
The molecules moving equally in every direction must necessarily impinge equally 
in every possible direction on the plane, so that if their lines of motion were brought 
from every point of the surface of the plane where they impinge and made to issue 
from one central point, they would radiate equally to every part of the hemisphere; 
and as soon as those belonging to any one direction have impinged and thus with¬ 
drawn from forming part of the constant aggregate force g v 3 A 3 , their place is 
