322 Proceedings of the Koyal Society of Edinburgh. [Sess. 
They are further distinguished as follows : 
1 
a < tan 1 J cot e, 
2 
a< 90° — e. 
5 
tan -1 J cot e<a< 180° - 2t, 
6 
a > 90° — e. 
12 
a>180°-2e. 
7 
a < 2 sin -1 \ cosec 
e or 
<2e, 
3 
10 
a>2e, 
2e>a>fe, 
according as 
45°. 
14 
a<|e. 
11 
a > 2 sin 1 J cosec 
€. 
4 
a< sin -1 cot e, 
9 
a >e. 
8 
sin -1 cot e < a < 1 20° - §e, 
15 
a< e. 
13 
a> 120° — |e. 
Also for 3, e<45°, and for 8 , 11, 12, 13, e>45°, the others being possible 
for all values of e. 
§ 18. The critical values of <p , obtained from § 15, are as follows : 
(a) 
c/><sin 1 (sin e sin a). 
Two minima in the twilight-curve. 
<*> 
<f> < e 4- a — 90°. 
Twilight-curve open at winter solstice. 
(Impossible if a< 90° - e.) 
(c) 
</> > 90° — e — a. 
Twilight-curve open at summer solstice. 
(Always occurs if a >90° - e.) 
(d) 
> 90° — e. 
Perpetual day. 
(e) 
+ > 90”-^. 
Perpetual twilight. 
(Nugatory if a>2e.) 
(/) 
</> > 90° — e + a. 
Perpetual night. 
(Impossible if a>e.) 
(?) 
</> > 90° + e — a. 
Twilight-curve imaginary. 
(Impossible if a<e.) 
Consider the relative magnitudes of these values 
a>b , 
a<c if tan e tan a<J, 
a<d if sin a<cot e, 
a<e if sin A <4 cosee e, 
2 2 
a<f, 
a<g, 
b<d 
b<e 
b<f, 
t<g, 
if a< 180° — 2e, 
if a<120°-fe, 
c<d. 
c<e. 
c<f. 
c<g. 
d<e ifa<2e. If d>e, 
e is nugatory. 
d<f. 
d<g if a< 2e. 
e<f if a>|e. 
e <g if a< 2e. 
