1907-8.] Sunset and Twilight Curves, and Related Phenomena. 315 
a straight line, 2 / = 6 hours; for the North Pole we have two vertical 
straight lines, x = 0 and x = \ year, corresponding to the equinoxes. Out- 
side the Arctic circle 0<9O° — e and we have a horizontal sinuous curve 
with maximum at the summer solstice and minimum at the winter solstice. 
Within the Arctic circle the sinuous curve is vertical, cutting y = 0 and 
y = 12 hours at right angles, leaving open gaps at the solstices. These gaps 
correspond to a period of perpetual day in summer which continues so long 
as S > 90° — (f>, and a period of perpetual night in winter which continues 
so long as $<0 — 90°. In every case the ordinate at the equinoxes is 
six hours. 
For southern latitudes the phenomena are, of course, similar but 
reversed. 
II. The Twilight-Curves. 
§ 7. The twilight-curves are somewhat more complicated. It is found 
by observation that twilight lasts so long as the sun is not more than 
18° below the horizon. 
Here 
ZS = 108°, PS = 90° - 8, ZP = 90° - <p , and ^ZPS = y. 
Hence 
cos 108° = sin 8 sin (f> + cos 8 cos <p cos y , 
or cos y = - (sin 8 sin <f> + sin 18°) sec 8 sec cp , 
where S has to be expressed in terms of x by an equation such as 
sin 8 = sin e sin x . 
We shall assume throughout that <p is positive. For southern latitudes 
the seasons are simply reversed, for the equation is unaltered if the signs 
of both <p and <5 be reversed. We shall also write the general angle a 
instead of 18°, only putting in its value as it is required. 
§ 8. We shall now determine some critical points in the curves, and in 
this way we shall obtain the critical values of (p which mark a change in 
the character of the curve. Some of these values of (p will be impossible 
for the actual values of a and e. As they will afterwards be required in 
tabulating the general yearly phenomena of light and darkness under 
various conditions, it is convenient to notice them all here as they come 
out. The possible critical values of (p for a = 18° and e = 23° 27' will be 
given in numbers. 
When x — mr+^r, S = ( — ) n e, and 
cos y — -{(-)” sin e sin <j> + sin a} sec e sec </> . 
