IN PASSING THROUGH THE ATMOSPHERE. 
239 
Integrating-, 
log C — log sin v' = log r + log \/ 1 + k g, 
C being a constant, 
C , 
• i — V \/ 1 —I— fc P 
smv' v 1 5 
(14.) 
When the ray starts from O (fig. 4.) the angle at which it intersects the inferior stra- 
tum is the apparent zenith distance = 0 ; therefore v' = 0 ; let also r — a the earth’s 
radius, and g = (g) the density at the earth’s surface. Under these circumstances 
sin © — a 1 + ^ (f ) 
(15.) 
Dividing by (14.) 
sin v' a v' 1 + k (g) 
sin© ~ r +kg 
42. viii. Substituting this value of sin v' in eq. (10.) it becomes 
d i _ — k a / 1 + k (g) . 
d . log s 2 0,1 rv (i-j-&g) 3 ' sin *' 
(16.) 
(17.) 
In which, if we consider k (g) and k g each as small compared to unity, and ~ as 
nearly equal to unity, the coefficient of sin 0 on the second side will be constant for 
any value of the zenith distance. Also these approximations will be as exact near 
the horizon as elsewhere. Hence eq. (17.) may be written* 
d - l °s* = -d^e C8-) 
Integrating and supplying the constant E, and calling now § 0 the whole refraction, 
log — = (19.) 
Here E is the value of s when refraction commences at the exterior limit of the at- 
mosphere ; it is therefore the measure of the intensity of solar light there-f-. 
43. ix. From the expression (19.) we may deduce the intensity proper to any alti- 
tude from two observations, as by Bouguer’s method. 
Let 0j 0 2 be the apparent zenith distances ; 
h0 1 the corresponding refractions ; 
s 1 s 2 the corresponding intensities of transmitted light observed. 
By eq. (19.) 
p i p j g 
log — — log — = log J- is known from observation. Let it = R : 
* Mec. Cel., IV. 283. 
f E and e have the same meanings here as V and v in a former part of this paper. In the Mecanique Celeste 
E is supposed = 1, and that letter is used to denote what we have used [s] for, farther on. 
