32 
MR. J. J. WATERSTON ON THE PHYSICS OF MEDIA COMPOSED OF 
Section V.— On the Vertical Equilibrium of a Medium, Supposing it to 
form the Atmosphere op a Planet. # 
§ 30. Suppose the height of the atmosphere AB to form the axis of a parabola of 
which the vertex A is at the summit. If a body begins to fall from A it is evident 
from the law of falling bodies that its acquired velocity at any point is porportional to 
the ordinate of the parabola at that point. 
Divide AB into an infinite number of parts so that the length of each shall be 
proportional to the ordinate of the parabola at that part. Suppose that in each of 
these parts one molecule is vibrating upwards and downwards, striking against the 
upper and lower molecules of the adjacent parts with a velocity proportional to the 
ordinate of the parabola and equal therefore to what a body would acquire in falling 
from the vertex. It is evident that each of the parts or infinitesimal divisions will be 
traversed in the same time dt by its molecule, and that the impinging velocities of 
each pair are equal, so that there is a perfect equilibrium and. constancy of phenomena ; 
but the upper impact of a molecule against the one above it is made with less velocity 
than the lower impact against the one below it, because the accelerating force of 
gravity increases the velocit}? during the interval of descent, and the acceleration is 
represented by the increment of the parabola’s ordinate in that interval. If g be the 
accelerating force of gravity, or velocity bestowed on a falling body every unit of time, 
the acceleration in each interval of descent, or infinitesimal division of the height AB, is 
evidently gdt. If this constant increment of velocity should by any cause be reduced 
in any given proportion, the aggregate effect must evidently be the same as if the 
force of gravity g were reduced in the same proportion. 
In such a vertical column of single molecules it is apparent that the equilibrium 
acquires! a continually increasing velocity in the molecular motion from the summit to 
the base ; and since the vis viva of a molecule is measured by the square of its velocity, 
it is also obvious that the molecular vis viva increases in the simple proportion of the 
distance from the summit. And knowing v 2 the amount of vis viva in the molecules at 
the base, we also know the height of the column v 2 /2g, which is simply the height due 
to the molecular velocity. 
§ 31. In a medium the nature of the action that sustains the upper molecules must be 
the same. The mean of the upper molecular impacts of a stratum must have less force 
* [This section attempts to deal with one of the most difficult points in the theory. That the loss of 
velocity suffered by every ascending molecule will lead to a smaller mean velocity above than below 
seems, at first, sight, inevitable. This consideration was urged by Guthrie (‘ Nature,’ vol. 8, p. 67,1873) ; 
and, in his reply (p. 85), Maxwell narrates that a similar argument, which occurred to him in 1866, 
nearly upset his belief in calculation. Waterston’s result really depends upon an assumption that, at a 
given height, the molecular velocities are all the same; whereas, according to the true Maxwellian law, 
all velocities are to be found at all heights. The force of this consideration will be appreciated when it 
is remembered that those molecules which at any time move at a low level with low velocities, would not 
of themselves reach a high level at all.—R.] 
t [? requires.—R.] 
