1907-8.] Sunset and Twilight Curves, and Related Phenomena. 317 
§ 10. Let y = 2n7r, 8 = <p — 90° — «; therefore 0<9O°-fa — e i.e. <f84 & 33 / . 
sin <p -f sin a sin 8 cannot vanish, for if 
sin (90* + a - e)^> sin a sin e , 
we get cos a cos e^>0, which is impossible unless a or e = 90°. 
Let cos x = 0, x — mir + ~ , 8 = ( — ) m e. 
Since S is negative, m must be odd and 8 — — e, (p — 90° — e + a. 
To find the limiting value of ^ put x = (2n+l)7r + ^ + /ul, where p is 
ObOC A 
small. 
Then 
cos x = sin p — p , sin x = — cos jx= - ( 1 - ¥l 
sin 
8= - sin e(l - , sec 8= (1 - sin 2 8)" 4 = sec e(l - J/x 2 tan 2 e) , 
y= | sin e sin <f>(l - - sin a j sec <p sec e(l - J/x 2 tan 2 e) , 
ixr 
= 1 - — sec cf) tan e sec 2 e (sin cj> - sin a sin e) , 
A 
since <p + e = 90° + a ; 
sin 2 y = [x 2 sec <p tan e sec 2 e (sin cf> - sin a sin e) , 
, a _ sin *•/*• se ° 3 6 ( s ^ n ^ — s i n a s ^ n € ) 
and — -j-? t — : : : — rn 
ax cos <£.//.. sec e sec <£ tan € (sm </> - sin a sm e ) } 
J • sec e | tan 65° 19'. 
v sm (e - a) 
(e-a) 
§ 11. Let y = (2n+l)ir, S = 90° — a — <p or a — 90° — (p . 
For <p — 90° — a — S, sin <p + sin a sin 8 = cos a cos 8, and for (p = a — 8 — 90° 
it is = — cos a cos 8. 
Hence this expression cannot vanish except in the limiting cases 
a = 90° or <5 = 90° = e. 
Let cos x = 0, x — m7r + ^ , 8 = ( — ) m e. 
If m is even, 8 = e and (p = 90° — e — a. 
Evaluating the limit, we get 
dy _ / sin e. cos a 
dx V sin (e + a) 
• sec e = tan 39° 30'. 
If m is odd, 8 = — e and <p = 90° + e — a or e + a — 90°. 
In the first case the twilight-curve is just vanishing and the points 
x = 2n7r— -T , y = (2m+ \)tt are acnodes. 
A 
