312 Proceedings of the Royal Society of Edinburgh. [Sess. 
polar distance = 90° — S(S being the declination, north), and ZPS = h is the 
sun’s hour-angle. 
Then, in the right-angled triangle SNP, 
cos h = — tan S tan <£ . 
h, expressed in hours, is the local apparent time of sunset, i.e. the 
number of hours that have elapsed when the sun’s centre is on the horizon 
since its passage across the meridian. If E is the equation of time, i.e. 
the mean time of meridian transit, h + E is the local mean time of sunset. 
Finally, if X is the west longitude of the place, expressed in hours, A-f-E-f X 
is the Greenwich mean time of sunset. A correction is usually made for 
refraction, r. At the equator r is about 2 minutes, and at latitude 60° it 
varies from 4 minutes at the equinoxes to 7 minutes at the solstices. The 
G.M.T. of apparent sunset is then h + E + X + r, and the G.M.T. of 
apparent sunrise is 12 — /i + E-f-X — r. The time of sunset corrected for 
refraction may also be calculated by the equation for twilight putting 
a = 34', the amount of mean horizontal refraction. Parallax may be 
neglected (though it would require to be taken into account in the case of 
the moon). 
§ 3. In order to discuss the curves which represent the graph of the 
time of sunset for any place all the year round, it is desirable to get an 
approximate equation. For this we require an expression for the sun’s 
declination at any time. 
Let S (fig. 2) be the sun’s position at any time, and project S upon the 
equator. Then NS = 8, T S is the sun’s longitude = X, and the angle NTS = e. 
We have then 
sin 8 = sin € sin X . 
If we neglect the eccentricity of the earth’s orbit the sun’s motion in 
longitude will be uniform and X will be proportional to the “ equinoctial 
time.” 
