FREE AND PERFECTLY ELASTIC MOLECULES IN A STATE OF MOTION. 33 
than the mean of the lower, the difference being the accelerating effect of gravity in the 
breadth of the stratum. We have also to remember that the nature of the equi¬ 
librium of a medium requires that if the velocities of all the molecules that pass a 
horizontal or other plane in a given time are resolved perpendicular to that plane, the 
sum of the squares of these resolved velocities are equal in opposite directions. In a 
constant infinitesimal time the absolute acceleration of velocity is evidently greater 
with the vertically moving molecules than with those moving obliquely to the vertical; 
thus the aggregate effect of the accelerating force of gravity in increasing the molecular 
velocity must be less than if it acted upon them directly as in the vertical column (§ 30). 
The question is, how much less ? for in such proportion must we consider the force of 
gravity to be reduced, supposing it to act uniformly on all the molecules of the atmo¬ 
sphere. The retardation of the ascending molecules of a stratum is equal to the 
acceleration of the descending molecules. Let us consider the latter. 
We assume from the original hypothesis that in any infinitesimal area the lines of 
molecular motion lie equally in every direction, so that if supposed to issue from one 
point S, they would be directed equally to every point of the surface of a sphere of 
which S is the centre. Let SB represent one of these velocities and B n the vertical 
Fisr. 2. 
5 * ' 
acceleration by gravity in the infinitesimal time dt. With the radius SB describe a 
hemisphere having its base on the horizontal plane PS. It is evident that the locus 
of the point n is the surface of another hemisphere with its base at the distance of 
B n below the plane PS, and that the area of the space between the bases is equal to 
the area between the surfaces of the hemispheres. Now, if gravity acted on each line 
of molecular motion, instead of acting only in the vertical, the common increment of 
velocity that would affect all is B p = Bn = gdt ; and this, when Bn/BS is infinitesimal, 
is equal to the quotient of the area between the concentric hemispheres qpf T, PBAR, 
by the surface of the inner hemisphere. But the actual increase of BS is 
nr = ?iS — BS, and the mean increment is found by adding up all the particular 
values of nr, and dividing the sum by the number of values, or, what amounts to the 
same thing, taking the quotient of the area between the equal hemispheres PBAB, 
cnfb by the surface of one of them. Now the first is equal to the area between the 
bases, which is equal to the product of a great circle by Bn, and the latter is well 
known to be equal to two great circles; therefore the quotient is \Bn. 
MDCCCXCII.—A. F 
