SUPERIOR REGIONS OF THE EARTH’S ATMOSPHERE. 397 
with sufficient exactness the lower part of the atmosphere, which 
is within reach of observation and experiment, and whose physical 
bases are such that this law may be mathematically extended, as an 
approximation, to about two-fifths of its whole height. Now this is 
easily accomplished, and we are led to it by the refractions them- 
selves. For, if they are calculated on the supposition that the pressures 
are proportional to the first power of the densities, the value obtained 
is too great; but if the second power of the densities be employed, 
the value obtained will be too small. The true law lies, therefore, be- 
tween these two limits, and an approximation to it may be obtained by 
taking an indeterminate expression consisting of two terms affected to 
each of those two powers. If this expression be subjected, as it should 
be, to the general conditions of equilibrium in the layers, as well as to 
the particular circumstances of pressure and temperature which take 
place in the inferior layer, and lastly to the decrease of temperature 
observed near the earth’s surface, we obtain precisely Mr. Ivory’s at- 
mospherie constitution, with all its numerical constants determined, and 
identical with those which he obtained by means of other considera- 
tions*. Now Mr. Ivory has proved that this law being applied to the 
inferior layers perceptibly agrees with the barometrical formula which, 
with respect to these layers, is the faithful expression of facts imme- 
diately resulting from the decrease of the densities. We are therefore 
justified by this combination of identical results, in extending its appli- 
cation to the limits of elevation already indicated as being equal to 
about +;%55 of the earth’s semidiameter. The rest of the refraction is 
then obtained independently of every hypothesis respecting the consti- 
tution of the superior layers, with a limit of error less than 0" 15 even 
for the horizontal refraction, as has been already asserted. 
And it is net only advantageous, but theoretically necessary, that we 
should have to form no hypothesis in respect to the state of these last 
layers, which are unknown to us. For the law of decrease founded on 
the two first powers of the densities, though it adapts itself to all the 
* The expressions thus obtained differ from those of Mr. Ivory only in their 
including the decrease of the weight which Mr. Ivory has neglected in con- 
sideration of the small height of the earth's atmosphere. But theoretically 
speaking, this consideration is no longer applicable to the atmospherical system 
which he employs in the integration of the differential element of the refraction, 
because it still gives an infinite extent to the atmosphere. Mr. Ivory, no 
doubt, found it necessary to proceed in that manner, in order to render the 
analytical integrations, on which the refraction depends, practicable: but this 
restriction is no longer necessary when we employ the numerical quadratures, 
and then any expression whatsoever that represents the real state of the atmo- 
sphere may be employed without limitation. As to the equations of equi- 
librium which determine the ratios which the pressures and the densities bear to 
the height, they are always integrable, and with the same facility when the pres- 
sure is expressed by any number of terms, containing any powers whatever of 
the densities. 
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