FREE AND PERFECTLY ELASTIC MOLECULES IN A STATE OF MOTION. 35 
The actual rate of diminution is very fluctuating and uncertain, varying from 200 
to 500 feet for each degree. The formula of our hypothesis applies only to the con¬ 
dition of an atmosphere resting on a horizontal base ; such, indeed, as may be found 
onlv during; a balloon ascent. 
M. Gay-Lussac, in his celebrated ascent from the neighbourhood of Paris, found 
the depression amount to 72^° Fahr. in 7634 yards. This corresponds to 316 feet of 
elevation to 1 °, if the rate is uniform. We have already determined (§ 27) the value 
of v in air at the temperature of melting ice to be 2244 feet per second; hence 
v 2 /g — 156,593 feet = H, the height of the atmosphere at this temperature (being 
nearly 30 miles). Now, taking Rudberg’s expansion of dry air, the value of v 2 in 
degrees of Fahr. is 493° at this temperature, and = 317*6, which is the 
elevation that ought to correspond to 1 ° by the hypothesis, in which also the rate is 
uniform. If Dalton and Gay-Lussac’s constant of expansion is preferred, the eleva¬ 
tion for 1° is 328 feet. 
The hypothesis requires that the diminution of temperature should be uniform, and 
the best authorities agree that it approximates to uniformity at considerable eleva¬ 
tions. In M. Gay-Lussac’s table of observations taken during his ascent, the indi¬ 
cations of the thermometer are somewhat irregular, as might be expected from the 
manner of making the observations, and the formula (Laplace’s) employed to 
compute the elevations may not, perhaps, answer so well for balloon ascents as it has 
been found to do in mountainous elevations. We have also to keep in view that the 
atmosphere absorbs a large proportion of the Sun’s rays in their passage through, 
besides being supplied with heat from the ground irregularly according to the 
varying characteristics of its surface. Taking all these circumstances into account, 
the accordance between theory and M. Gay-Lussac’s extreme observations is nearer 
than might be expected, and probably will not be found so exact at lesser elevations. 
But the hypothesis admits of being tested without employing any empirical 
barometric formula, because, if it is correct, the tension as shown by the column of 
mercury ought to vary as the sixth power of the absolute temperature (from zero at 
— 461° Fahr.) (XIX). But the observations must be taken at stationary points 
during the ascent, so that time may be allowed for the thermometer to acquire the 
temperature of the stratum of air in which the balloon rests. Let z° be the absolute 
temperature of Fahr. zero; then for any two observations we ought to have 
6 e 
— 7 ; in which t x , e x , are the temperature and tension at any one altitude 
and t 2 , e 2 , the same at any other. The value of z eliminated from this equation ought 
to be 448 or 461. I have tried this with M. Gay-Lussac’s sixteenth and last 
observations, which appear to be the most favourable for accuracy, and z comes out 
equal to 467. The rate of diminution also in the interval of 1800 metres between 
these two observations agrees well with theory, being 310 feet for each degree Fahr.* 
# Note H (Formula for measuring beigbts by thermometer). 
F 2 
( z + 
\z + tj 
