240 
PROFESSOR FORBES ON THE EXTINCTION OF THE SOLAR RAYS 
hence 
II = 11 
J_5j_ 
sin ©j 
l- 1 
sin © 2 j 
( 20 .) 
Also by eq. (19.) 
logE = log Sl + H4^ i (21.) 
whence the intensity without the atmosphere becomes known. 
For any other zenith distance S n , 
l°ge B = logE- H^|- (22.) 
44. x. This expression fails at the zenith ; but for all considerable elevations 
above the horizon the thicknesses of air traversed are as the secants of the zenith di- 
stances, and consequently the logarithm of the loss of light is in the same proportion. 
(See eq. (1.) p. 227). If [s] represent the intensity after a vertical transmission, and 
i x that at an elevation of 45°, 
_E 
M 
log rk : Iop” 
E 
8 S, 
1 : sec 45°, 
= 1:^/2, 
i E /I . E . „ . 
Io sh = v v Io sir ( 
E 
For instance, Bouguer’s result for a vertical transmission, is *8123. Substituting 
this in eq. (23.) the intensity at alt. 45° would be found ; and for any other elevation 
the logarithm of the intensity compared to E would be found to be proportional to 
the refraction divided by the sine of the apparent zenith distance. From what has 
just been said, the atmospheric masses intercepted are in the same proportion as the 
logarithms of the intensities. Thus, let ^ and /a 2 be the masses of air traversed at 
45° of elevation and at any other angle : 
H _ sa 2 
• ^2 s i n 450 • s i n © 2 
Taking the value of the refraction from Ivory’s latest Table*, 
_ 58''-36 S$ 3 
l Jj l • 1^2 ^/x * gi n © 2 
For example, at the horizon, where % Q = 2072", the thickness at the zenith being 
unity, and therefore that at 45° = a/ 2, 
2072" 
58"-36 — 35’5034. 
45. The other thicknesses are computed (always with reference to the perpendicular 
mass of the atmosphere) in the following Table: 1. by the approximate law of the 
secants ; 2. by Laplace’s analogy ; 3. from Bouguer’s Table contained in his ‘Optics,’ 
which it will be seen scarcely differs from Laplace’s result at the horizon. The first 
* Philosophical Transactions, 1838. 
