10 
MR. J. J. WATERSTON ON THE PHYSICS OF MEDIA COMPOSED OF 
direction, and if the number is increased eight times in any one direction it must be 
so in every other. 
This may he viewed in another light. Suppose in both cases, the density being 
1 and 8 respectively, that the molecules are arrested in their motion. It is evident 
that opposite a unit of surface in density 8 there will, in the first row, be four times 
as many molecules as in density 1, and that the average distance between the rows 
is only one-half. Suppose the molecules to resume their motion, and compare density 8 
with density 1, it is obvious that in half the time four times the number will impinge 
on the unit of surface, and in the same time eight times the number. Now it has 
been shown (§ 2) that the elastic force is proportional to the number of molecular 
impacts made with a constant velocity against a unit of surface in a unit of time, 
hence we deduce that the elastic force (e) o f a medium with a constant mean molecular 
velocity ( v ) is proportional to its density (A 3 ) (or e == A 3 , if v or v 3 is constant). . II. 
§ 4. Hitherto the molecular velocity has been supposed constant. We have now 
to enquire how the elasticity of the medium is affected by a change in the velocity 
from v to mv. The intestine action of the medium may be viewed as the traversing 
of a certain mean distance, L, by the molecules in a given time, t ; and in this time a 
certain mean number, A, of impacts take place against a unit of surface. If the 
velocity is increased m times, the distance L is traversed in l/mth the time t, or t/m, and 
in this reduced time the same number of impacts must take place as before took place 
in the time t ; for there is nothing in the change of velocity simply that can alter the 
ratio that subsists between the mean distance traversed and the mean number of 
impacts, unless that ratio were subject to change without any change whatever in the 
medium, which is absurd; hence, in the original time, t, there is m times the original 
number of impacts, A .. 
It was shown in § 2 that if the weight of each of the molecules were represented 
by 1, their mean velocity by v, and weight of plane supported by their impinging 
action n, the number of impacts in a second or unit of time required to support the 
plane is fy = A, or n — yAv, and this equation must evidently be maintained in 
altering the value of the terms. Now, it has been shown that in changing v to mv in 
2 
a medium that does not alter its density we cause A to become mA, and - Av becomes 
2 2 0 0 
-A mvm — -A vm° = nm 3 . Hence n, the weight of the plane, or measure of tension, 
0 9 
must be increased m 3 times so that it may continue to equilibrate the impinging 
action. Thus, we deduce that while the molecular velocity increases from v to mv, 
the elasticity increases from n to nvn or the elasticity of a medium having a constant 
density is proportional to the mean square molecular velocity or vis viva of the medium 
(or e == v 3 , when A 3 is constant)*.III. 
* [II. and III. were given by D. Bernoulli. See 1 PoGG. Ann.,’ vol. 107, p. 490, 1859.—R.] 
