Oh Atmospherical Refraction. 369 



We have then, for the case of horizontal refraction, 



dw „ ^ d'u .9352 



;t- = 4.453 = 2 X 2.2265 ; r^ = x 4.453 = 681 12 =: 



24 X 2838, and^, = (4.453)^ X 5790732O x 5.7162 + 4.4.53 



X 15296*= 7657200 000 — 720 x 1O635000 : consequently, 

 .0002825 = 2.2265 r^ + 2838 r* + 10635000 r" ; now if r"" = 

 .0001,wehave .0002825=: .00022265 + .00002838 + . 000010635 

 [+ .000020935]: consequently, .0001 is too little for r-, and we 

 may try .00011, giving .0002825 = .00024491 + .00003434 + 

 .00001420 [— .00001095]. But in order to keep up the proba- 

 ble sequence of the progression, the remainder should be about 

 equal to the last term, or about .000011, and .0000209 should 

 have been diminished by about .00001 instead of .0000318 ; so 

 that we must take ,000103 as the true value of r-^ on this hypo- 

 thesis, and r=:34'53", which is exactly^ too great by about 

 r ; a difference by far too considerable to be attributed to the 

 errors of observation only ; and we must infer, that the law of 

 temperature, obtained from the height of the line of congelation, 

 is not correctly true, if applied to elevations remote from the 

 earth's surface. [If indeed this law were fully established, and 

 capable of being applied, with any little modification, to the 

 exact computation of the refraction, it would be necessary, for 

 the lowest altitudes, either to compute a greater number of the 

 fluxional coefficients, or to divide the refraction into two or 

 more parts, and determine the successive changes of density 

 required for each of them. We should also have] for finding, on 

 this hypothesis, the height x, corresponding to the pressure y 



y ** 1 1 

 and the density z, the expression mx — m=.\ — 1 hi 



!" + -^\^;~^; ; y being = —II , and ^^ = 1 + 4,.^ : 



2:: —qy{\ — z) z -\- n — nzz ' 



and the actual state of the atmosphere would probably be very 



well represented by this formula, taking n = .1 or. 11, rather 



than .09. 



(C) ]Professor Bessel's hypothesis is also found to make the 



horizontal refraction too great. Mr. Laplace's formula, which 



2 B2 



