4 
Transactions of the Royal Society of South Africa. 
the right-hand number of this equation is the correction for atmospheric 
absorption, and includes as unknown quantities the height of the atmo- 
sphere = d, and the coefficient of absorption = y. 
This final equation can be put in the form — 
(m^ - nioY + 2(77^, - mo) cos 0 ^ - y^^l + 1^ = 0. 
Eeturniag to the value of t, viz. — 
t'd + 2t Gose-d-2 = 0, 
we may remark that when d is very small, that is when the ratio of the 
height of the air to the radius of the earth is practically zero, the above 
expression becomes — 
2t cos 0 = 2, 
t = sec 0, 
and — 
(mi - m^) = y sec 0. 
This limiting value holds good for zenith distances up to 70°, but after 
that it ceases to hold good. 
The rigorous equation of condition — 
{m^ - m^Y + - cos 0 ^ - y^^l + 1^ =0, 
well represents the relation between atmospheric absorption and zenith 
distance at any altitude, always premising that the atmosphere consists 
of homogeneous concentric spherical shells round the earth's surface, and 
that it is free of foreign bodies such as dust or masses of vapour. 
The variations in the barometer for different regions of the earth's 
surface show that neither of these considerations is admissible. 
2. Consideration of Winterberg Observations. 
Over five hundred observations were made from the summit of a hill, 
whose altitude would be about 4,000 feet. 
A number of stars were selected, and their decrease in brightness 
noted as they passed from the zenith to the horizon. 
These quantities were plotted down and the interpolating curve drawn. 
The following are measures on the curve : — 
Altitude 79° Decrease = 0-64 m. 
81 0-75 
83 0-95 
. 85 1-28 
87 1-82 
89 2-70 
