360 Astronomical and Nautical Collections. 



affords a very correct determination of the refraction, is said to 

 agree sufficiently well with direct observation also; but, in fact, 

 this formula gives a depression considerably greater than was 

 observed by Gay Lussac, in the only case which is adduced in 

 its support ; and the progressive depression follows a law which 

 appears to be opposite to that of nature, the temperature vary- 

 ing less rapidly at greater than at smaller heights, while the 

 observations of Humboldt and others seem to prove that in 

 nature they vary more rapidly. Notwithstanding, therefore, the 

 ingenuity, and even utility of Mr. Laplace's formula, it can only 

 be considered as an optical hypothesis, and we are equally at 

 liberty to employ any other hypothesis which represents the 

 results with equal accuracy ; or even to correct our formulas 

 by comparison with astronomical observations only, without 

 assigning the precise law of temperature implied by them. 



[D. We may compute the effect of a temperature supposed to 

 vary uniformly with the height, by making z=y (l + tx—t), 



, , , z , . , , z dz zdy 

 or ~yx', we have then — —\ + tx—t, or x', and d — = 



y y y yy 



= fdz, or =:te'~'dx, which are initially the same. But tdiX=i 



— <dw , dz zdy Mm , dz z ty mzz — tyy 

 -, and — =— ^ i, whence 3-=: = ^, and 



mz y yy mz Ay y mz wyz 



dw ^ myz , ,^ ^nwdz + wizdi/ 



3^=C — ; consequently df— — ^ - — 2myz. 



dz 7nzz — tyy jnzz — tyy ' ^ 



mzdz — fydy . . .. „ ^ m , . , ,dt Im + m^ 



-z ^-^; and mitially t= , and f '=d ~=z — ■ — ~— 



(mzz—tyi/f •' ' m—t dr \m — t 



(m — ty)ps y ^ ^ m — tj ps Y ^ ^ ^ 



^^7)77= {?-f+^«- «-i>}7f = {(2f'-« «- »} 



;=: ^ (^—1)- (2?— 1) . Now, if we suppose the tem- 



p s p s 



perature to vary 1° in 300 feet, we have 5^ • 3^ = f^"^. 



for the variation of density depending on temperature in 



^nQnnnnn °^*^^^ earth's radius x; hence t should be 139, and f 



