234 PROFESSOR FORBES ON THE EXTINCTION OF THE SOLAR RAYS 
for all low elevations. This infers a knowledge of the height of the atmosphere ; but 
more than this, it requires that we should be able to ascertain the density of each par- 
ticular stratum of air, for each stratum is traversed at a different obliquity, and the 
quantity of matter which each stratum opposes to the passage of the ray is evidently 
proportional to the density of the stratum, and the length of path in the stratum. 
The mass of air penetrated will be the integral of all such elementary parts. 
26. Supposing the atmosphere divided into three strata of equal thickness but 
differing in density, it is evident that the horizontal ray H A will traverse the coin- 
Fig. 3. 
ID 
paratively short space H a of the highest, the longer space a b of the middle stratum, 
and by far the longest course b A in the lowest (the curvature of the path being 
small). Now as the lowest stratum is also the densest, it follows that on both these 
accounts the stratum in which we are placed acts most energetically, and hence the 
thickness traversed by a horizontal ray may be represented by a highly converging 
series upon almost every physically-possible hypothesis of density. 
27- This figure also shows that, supposing the strata numerous and thin, the 
thickness of each stratum traversed will be equal to its vertical thickness multiplied 
into the secant of the angle of incidence of the transmitted ray. 
28. Lambert contented himself with finding in the first place in fig. 2. the length 
of the line A H or A B from simple geometrical considerations, which he gives in 
these terms*, 
cos y -f x = \/ cos 2 y + 2 y -f- y 2 \ 
where y is the zenith distance, y is the height of the particular stratum DBH (the 
earth’s radius being =1), and x is the length of the path A B or A H. He then dif- 
ferentiates the expression in respect of x and y, and multiplying the element of the 
path thus found by the density of the stratum (variable with y according to some law 
to be assumed), he expands the quantity to be integrated in a series, of which, how- 
ever, he has not attempted to find the exact value, but stops at the first term which 
is proportional to the secant of the zenith distance, as we have seen. The only person 
* Photometria, p. 393. 
