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 wliose 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, a«d lastly to the decrease of temperature 

 obserred near the earth's surface, we obtain precisely Mr. Ivory's at- 

 mospheric 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 j(j^QQ 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 not 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- 

 •ure is expressed by any number of terms, containing any powers whatever of 

 (lie densities. 



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