1907-8.] Sunset and Twilight Curves, and Related Phenomena. 311 
XVIII. — Sunset and Twilight Curves, and Related Phenomena. By 
D. M. Y. Sommerville. Communicated by Professor Dyson. 
(With Three Plates.) 
(MS. received February 22, 1908. Read May 4, 1908.) 
§ 1. The objects of the present paper are, first, to describe certain 
curves which approximate to the graphs of the time of sunset (or sunrise) 
and end of twilight (or daybreak) all the year round for various latitudes ; 
second, to tabulate the yearly phenomena of light and darkness for 
different latitudes under various conditions. 
The time of sunset depends upon the latitude and the sun’s declination, 
z 
and the extent of its variations depends upon the obliquity of the ecliptic. 
The extent of the variations of twilight depends also upon a certain angle 
a, which is the maximum depression of the sun’s centre below the horizon 
for which the light reflected by the upper strata of the atmosphere is 
sensible. The investigations will be conducted, in the first place, on the 
assumption of the present value of the obliquity e = 23° 27', roughly, and 
the observed value of a = 18°. We shall then extend the investigations by 
assuming both e and a as arbitrary within the range 0° to 90°. 
I. The Sunset-Curves. 
§ 2. Let Z be the zenith, P the North Pole, S the sun on the horizon, 
and N the north point. Then NP is the latitude = <p, PS is the sun’s north 
