34 
ME. J. J. WATERSTON OH THE PHYSICS OF MEDIA COMPOSED OF 
Thus, it seems to be clear that the molecules of a medium are collectively only half 
as much affected by gravity as if they all moved in vertical lines; but it has been 
shown (§ 30) that if they moved in vertical lines the height of the atmosphere would 
be the height due to the molecular velocity; but as the increments of their velocity by 
gravity is only one-half what they would receive if their motion were vertical, that 
height must be computed as if the force of gravity were only one-half the actual 
amount. Thus, if v~ be the mean square velocity at a depth H below the summit of 
the atmosphere, the height due to this with the full effect of gravity is v 2 j2g, and with 
half effect it is v^/g = H. Thus we arrive at the following deduction. The molecular 
vis viva increases simply as the depth below the summit of the atmosphere, and the 
height of the summit above any stratum is equal to the quotient of the mean square 
molecular velocity at that point by the accelerating force of gravity, or to double the 
height that a free and unresisted projectile would ascend if projected vertically with 
an initial velocity equal to the square root of the mean square molecular velocity in the 
stratum . .XVII. 
§ 32. To ascertain the law of density we have the equation in § 17, modified as in 
§27 to -g-A s u 3 = gn, for the reasons given in last section. By this we have 
v^jg — Qn/ A 3 = H, which applies to any part of the atmosphere at all heights. 
Differentiating the equation = H, we have — cZH. But ~ is 
. , . , 7IT , 6n3dA 6n 3d A TT 3dA . „ 6dn Qn3dA 
evidently equal to did, and —= — X = Jd —therefore —-—— 
= dR = 6c/H — H ; or 
A 
3dA 
A 
5dE _ 
H ’ 
and by integration we have A 3 = H 5 . 
Thus we deduce that the density of the medium at any depth below the summit of an 
atmosphere is proportioned to the fifth power of that depth .XVIII. 
§ 33. As we had u 2 = II we may further deduce that u 3 A 3 = H 6 , or that the elastic 
force of the atmospheric medium at any point is proportioned to the sixth power of the 
depth of that point below the summit and to the sixth pouter of the mean square 
molecular velocity .XIX. 
These deductions are all embraced by the equations v^jg — H = A% and 
riA 3 _ H 6 = 
§ 34. To compare these results with what is known of the physical condition of 
our atmosphere, we have first the obvious correspondence between the diminution of 
molecular vis viva and of temperature in ascending. No sufficient explanation of this 
has, I believe, been yet offered, for it is needless to attempt to do so by supposing the 
specific heat of air to increase as its density diminishes, as no difference of specific heat 
disturbs the equilibrium of the temperature of bodies placed in horizontal contact. 
The very fact of a gaseous atmosphere presenting a constant inequality of temperature 
at different elevations seems to prove that the law of the vertical equilibrium of tem¬ 
perature is essentially different from the law of horizontal equilibrium. 
