FREE AND PERFECTLY ELASTIC MOLECULES IN A STATE OF MOTION. 59 
Suppose 
T = (461 + 60), t — (461 + 80-55), and r=(461+ 70); 
the formula computed gives li = 3127, which, divided by t — r = 10’55, gives 
296^ feet as the value of 1° in such an instrument. This amount varies but little, 
through a considerable range of temperature and pressure. 
This value may be obtained by observing the temperature at the bottom and at the 
summit of a known height, and dividing the elevation in feet between the two 
stations by the difference. Neither the law of the vapour nor the amount of air 
enclosed with it is required. 
What if we dismiss the vapour altogether and enclose dry air only ? It is evident 
that the line CS will then become parallel to the axis and distant from it F, the 
sixth root of F G , the density, which is constant. The element G becomes infinitely 
negative, and -— = 1, thus simplifying considerably the expression for h. which 
\/1 Ct 
is now converted into h = 317‘6 jl — (^''j j T. 
Let t — b 6 , and r = (6 — /3) 6 '= b 6 — 6b 5 (3, when (3jb is a small fraction. By division 
we have on this hypothesis jjt — 1 — 6 fi/b, and {r/t'f =1 — /3/b, which converts the 
ft ft . 
equation for h into h = 317’6 y T. To express j in terms of t and r, we have 
t —- t — 6//’/3, and 
t — T 
P 
b 
Hence, so long as this fraction is small in comparison 
to unity, we have the following simple expression for the height in terms of the indi 
cations of the thermometer :— 
This gives the nearly constant value, 53 feet, for each degree of Fahr. thermometer, 
at moderate elevations and ordinary temperatures. 
This is the lowest possible value for difference of temperature that can be obtained. 
In ascending through an increment of the height of the atmosphere, we experience 
one decrement of temperature, and five decrements of density, which, together, make 
six decrements of tension. These six decrements of tension must be effected in the 
enclosed air of the instrument before an equilibrium is established, and as the density 
is a constant quantity they must be produced by means of a lowering of temperature 
to the amount of six decrements. Thus, six decrements of temperature in the instru¬ 
ment correspond to the same differential height as one decrement of temperature in 
the atmosphere, or six degrees correspond to 317‘6 feet, the difference of height that 
causes a difference of 1° while in its natural condition of vertical equilibrium. 
It appears, therefore, that dry air is in every respect the best in theory for measuring 
heights with the thermometer by means of such an instrument as is referred to in 
Note H. The theory upon which its theory rests has been shown to agree with 
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