318 Proceedings of the Royal Society of Edinburgh. [Sess. 
Evaluating the limit when (p = e+a — 90°, we get 
sm e. cos a 
dy = 
dx V sin (e + a) 
sec e = tan 39° 30'. 
§12. The limiting case given by sin 0 = sin e sin a deserves special 
investigation. Here the pairs of minima at sinx = - — Sin .- < fi- (7r>y>0) 
s r sm e sm ct 
coalesce with the interjacent maximum at sinx=— 1 to form a point of 
undulation which is an apparent minimum. It is easy to show that at 
this point the differential coefficients up to the fourth all vanish, while 
^ = 3 tan a tan 2 e sec e . 
dx^ 
§ 13. Let us now collect our results. Plate I. gives, within the limits 
7 37T 
x = — df to ~ , y = 0 to 7r, the different forms of the twilight-curve in com- 
Z z 
bination with the sunset-curve, the upper one being the twilight-curve. 
There are seven forms, not including limiting forms. Their evolution 
is easiest to follow by considering <p and e as constant and a as variable. 
The critical values of a are 
sin -/sm_^\ 90° — e + <4, 90° -€-d>, e + <A - 90°, 90° + <■-<#>. 
\sin e J 
90° - e - <f> is only possible if </> < 90° - e , 
e + <£-90° „ „ cf> > 90° — e , 
90 ° + e-<£ „ „ <A>e, 
E + <f> and sin -1 ! sin ^ ) are only possible if d> 
\sin e / 
90 c 
< e . 
Also 
;in ^ ^ < 90° - e - <f> if tan <£< - 
Vsin e / 1 
tan e 
+ 2 tan 2 
while 
90° — € + cf) > 90° -€ — <]), 
J sin <f> 
>sm 
V; 
vsm e 
>€ + cf) — 90°, 
The critical values of cp are 
e, 90° - e and tan -1 
90° + e — cf> > 90° - e — <f> , 
>€ + </>- 90° , 
sin 
. 1 /sui_^\ >e + ^_ 90 . 
\sin e / 
tan e 
and we have 
tan 1 
1 + 2 tan 2 e 
tan e 
1 + 2 tan 2 e 
<€, 
< 90° — e 
e<90° — e if e<45°. 
