316 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
y is imaginary if 
| ( - ) n sin e sin 0 + sin a | > cos e cos 0 . 
At summer solstice n is even, and y is imaginary if cos (e + 0)< sin a, 
i.e. if 0 > 90° - € - a , i.e. > 48° 33'. 
At winter solstice n is odd and there are two cases : 
1. sin a> sin e sin 0. 
y is imaginary if cos (0 — e) < sin a, 
i.e. if either 0 >90° + e - a or 0<€ + a -90°. 
2. sin a < sin e sin 0. 
y is imaginary if cos (0 + e) < — sin a, 
i.e. if <£>90° - e + a , i.e. >84° 33'. 
When y = 2mr, cos y = l, cos (0 — ($) = cos (90° + a) ; therefore 
<5 = 0 — 90° — a, which is impossible if 0 < 90° -f a — e, i.e. < 84° 33'. 
When y = (2 n *f l)7r, cos y = — 1, cos (</> + S) = cos (90° — a) ; therefore 
either 8 = 90° -a — cp, which is impossible if 90° — a — (p>e , or <—e, 
i.e. if <p < 90° - e - a , i.e. < 48° 33' or if <f> > 90° + e - a ; 
or S = a — 90° — <p, which is impossible if a — 90° — cp< —e, 
i.e. if </>>e + a - 90°. 
If (p<e + a — 90° both values of S are possible; if (p> 90° + e — « both 
values are impossible. In the latter case cos (0 4- S) is always < sin a and 
y is always imaginary. For such values of 0 the twilight-curve is 
therefore entirely imaginary. 
§ 9. Consider now the slope of the curve. 
dy _ sin e cos x sec 3 S sec <p 
dx sin y 
(sin 0 + sin a sin S). 
In general ^ = oo if y = nir. Exceptions, of course, occur when y cannot 
= njr. Also in general — 0 when x = nir + ^ , and also when sin 0 + sin a 
sin 8 = 0. The greatest value of S is e ; hence if sin 0^> sin e sin a, or 0^>7° 4', 
= 0 when 
dx 
t _ sin 0 
Sill a sin € 
Exceptions may arise if both numerator and denominator vanish 
together. 
