3 
7 . Now if we take for n the value = -° 9 > corresponding to the multiplier 45, employed by Mr. Les- 
lie, the refractions in the immediate neighbourhood of the horizon will become too great by about i' ; 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 suface. Professor Bessel’s approximation is also found to make the 
horizontal refraction too great. Mr. Laplace’s formula, which 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 de- 
pression 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 varying 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. The theory will however afford us some general indications for this 
purpose ; showing, for example, that the coefficient of the second term cannot be smaller than ^ s, 
whatever positive value we may attribute to n ; and if we adjust the second and fourth coefficients, so as to re- 
present the refractions near the zenith and at the horizon, without regarding the value of the subsequent terms, 
we shall obtain the third, by dividing the fourth by half of the second ; since that part of the fourth coefficient, 
which occurs in the case of horizontal refraction, is always derived from the third by taking the fluxion with 
respect to v only, and is therefore found by multiplying the third by whatever the relations of the 
other quantities concerned may be. 
8. On every supposition, the coefficient of the first term must be — , and that of the second must not 
s 
greatly differ from . The third coefficient, on the hypothesis of a law analogous to Mr. Leslie’s, 
S 2 
will be 1500 ; if W e suppose the temperature to vary more uniformly, and make z — y ( 1 + 1 x — t), the 
number will become 1900 ; or, taking z =. yx 1 , 2200, m being 766, and t 176: and Mr. Laplace’s formula 
will of course give a value still larger. In fact the result of observation is represented with sufficient accu- 
r r 2 r 3 r* 
racy by the equation .0002825 — v — + (2.5+ .$v z ) — + 3400 v — + 3400 ( 1. 25 -f- . 25 v z ) — , the 
s s s a 4 
barometer standing at 30 inches, and the thermometer at 50° : and this formula appears to be at least as 
accurate as the French tables. We have, for example : 
Altitude 
Refr. 
Conn. d. T. 
Altitude 
Refr. 
C onn. d. T, 
O 1 
/ » 
/ n 
O 1 
1 a 
/ / 
O O 
33 5 2 
33 52 
20 O 
2 39 
2 40 
5 0 
9 57 
9 5 6 
0 
0 
1 41 
1 41 
10 0 
5 21 
5 21 
45 0 
58.15 
58.3 
The difference is somewhat greater a few degrees above the horizon ; thus at 2° 17' 50", this formula makes 
the refraction 17' 16", the French tables 17' 4", Bradlkv’s 17' 30", and Dr. Brinkley’s observations 
reduced, 17' 9" : but in such cases we can scarcely expect a greater degree of accuracy. 
9. The terrestrial refraction may be most easily determined by an immediate comparison with the angle 
subtended at the earth’s centre, the fluxion of which is t and j s initially the first part of the coef- 
'OX DXOiT 
ficient of the second term of the series already obtained, and is equal to 6 ; so that this angle, while it remains 
small, is six times the refraction : commonly, however, the refraction in the neighbourhood of the earth's 
surface is somewhat less than in this proportion. 
10. The effects of barometrical and thermometrical changes may be deduced from the fluxion of the equa- 
tion, if we make m, p, and n, or rather t, vary : and for this purpose it will be convenient to employ the form 
