§24 On the almospherkal Refraction. 



zontal refraction is 1824". All these quantities are very different 

 from the mean horizontal refraction determined by observation, 

 vvhich is 2100" according to Laplace. But there must be one 

 particular atmosphere in the series, which, while it possesses the 

 general property of representing the refractions near the zenith, 

 will likewise coincide with observation at the horizon. Now I 

 have found that this takes place in the atmosphere that has its 

 total height equal to 1 7372 fathoms, or four times that of Cassini ; 

 and the formula I have sent you was obtained by integrating the 

 differential expression of the refraction in this hypothesis. The 

 character of the formula may therefore be described by saying, 

 that in all probability it will be found to coincide with observa- 

 tion better than any other founded on the supposition of a uni- 

 form decrease of heat. 



In the atmosphere to which my formula belongs, the elevation 

 necessary for depressing the centigrade thermometer one degree 

 is 70 fathoms, considerably short of the observed quantity, which 

 is about 90 fathoms. As in the series of atmospheres, there is 

 one agreeing with observation in the quantity of the horizontal 

 refraction, so there is also one that will agree with observation 

 in the elevation for one degree of depression. In the atmosphere 

 the height of which is five times that of Cassini, the elevation 

 for one degree of depression is 87 \ fathoms, nearly equal to the 

 observed quantity ; and in this case the horizontal refraction is 

 2164", or 58" more than according to observation. In reality, 

 neither the horizontal refraction, nor the height necessary for one 

 degree of depression, is determined with great precision; but 

 it is certain that on the one hand 70 fathoms is too little, and 

 on the other 2106" is as great a quantity as can be admitted: 

 and hence we may infer that the supposition of a uniform de- 

 crease of heat in the atmosphere, caimot be reconciled with the 

 astronomical refractions. But, although this be strictly true, yet 

 the refractions are so nearly represented by tl;e law mentioned, 

 that the actual deviation from it must be very inconsiderable. 



It would be superfluous to say any thing here of the solution 

 of this problem contained in the Mtcanique Celeste, the merit 

 of which is so well known, and so justly appreciated. Both the 

 solution now mentioned, and the one given above, seek to ap- 

 proach the truth by means of probable conjectures; and the ul- 

 timate results come nearer one another than was to be expected 

 in two methods employing very di^erent processes of investiga- 

 tion, and leading to formulae of calculation that have nothing 

 in common ; the atmosphere in the one case being of indefinite 

 extent, while in the other the total height does not exceed twenty 

 miles. 



The elevation for one degree of depression, which in my for- 

 mula 



