320 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
The curve is always a vertical sinuous curve. 
For the twilight-curve, if a is positive, the critical values of a, 90° — e — 0 
and 90° + e — 0, become impossible, and the others all become 0. 
When a = 0 the curve assumes a limiting form, 
cos y = 
- (1 + sin x) sec x tan a = — tan a cot 
if a? <90°. 
y is imaginary if a > 45° — 
x 
2 
i.e. x > 90° — 2a, and when x = — 90°„ 
y = 90°, so that the curve consists of closed ovals as in fig. 4. 
The curve then starts as fig. 14, and when a = 0 passes instantaneously 
through the two nodal forms and this limiting form into 6. Actually at 
the winter solstice in latitude 0 = a the time at which twilight ends is quite 
indeterminate, just as at the equator the time of sunset is indeterminate,, 
the sun being just on the horizon. 
III. Yearly Phenomena of Light and Darkness. 
§ 15. Let daylight, twilight, and true night be denoted by D, T, 1ST 
respectively, and let a combination of these, such as DTN, denote the 
phenomena which occur from noon till midnight. There are six possible 
kinds of days : D, T, N, DT, TN, and DTN ; DN being impossible unless. 
a = 0. 
These are marked by the following critical phenomena : — 
1. The sun does not rise if $<0 — 90°. This is only possible if 
0 >90° — e. 
2. The sun does not set if <5 >90° — 0. This is only possible if 
0> 90° — 6. 
3. There is no true night if 8<a — 0 — 90° or >90° — a — 0. The 
former is possible only if 0<e + a — 90°, the latter only if 
0>9O° — e — a. 
4. There is no twilight if <5<0 — a — 90°. This is only possible if 
0> 90° — 6 + a. 
5. There is perpetual twilight while 90° — a — 0 < £ < 0 — 90°. This is 
only possible if 0 > 90° — ^ . Previous to attaining this value, 
Lu 
0 must have become > 90° — e, hence this is only possible if a < 2e. 
6. The twilight - curve has two minima instead of one if 
sin 0 < sin e sin a. 
7. The twilight-curve becomes imaginary if 0>9O° + e — a. 
