1907-8.] Sunset and Twilight Curves, and Related Phenomena. 323 
§19. The critical values of a are therefore 
i. |e. y. 2e. Impossible if e>45°. 
ii. e. vi. 120° -§e. 
iii. 90° vii. 180° - 2e. 
iv. tan -1 J cot c. viii. sin -1 cot e. 
ix. 2 sin -1 \ cosec €. 
;» Impossible if e<45°. 
) 
Consider the relative magnitudes of these values. 
i< ii, 
i< iii if c < 54°, 
i< iv if e<42° 41' approx., 
i< v, 
i< vi, 
i< vii if e< 67° 30', 
i < viii if e<58° approx., 
i< ix, 
ii< iii if e<45°, 
ii< iv if e<35° 16' approx., 
ii< v, 
ii< vi if e< 72°, 
ii< vii if €<60°, 
ii < viii if e<51° 50' approx., 
ii< ix if e<66° 14' approx., 
iii > iv, 
iii > v if e<30°. 
iii < vi. 
iii < 
vii. 
iii < 
viii. 
iii < 
ix. 
iv> 
v if e < 24° 6' approx, 
iv< 
vi. 
iv< 
vii. 
iv< 
viii. 
iv< 
ix. 
vi> 
vii. 
vi> 
viii. 
vi> 
ix. 
vii > 
viii. 
vii< 
ix. 
viii < 
ix. 
The critical values of e are therefore 
30°, 45°, 54°, 60°, 67° 30', 72°, 
and the solutions of the equations 
tan §e = \ cot e, 
sin = cot e. 
= 1 
tan e = J cot e, i.e. e = tan 
-i i 
n/2’ 
sin e = cot e, i.e. e = cos 1 J(^/5 - 1), 
tan 2e = J cot e, i.e. e = tan -1 -!- ; 
v 5 
or arranged in order 
24° 6', 30°, 35° 16', 42° 41', 45°, 51° 50', 54°, 58°, 60°, 66° 14', 67° 30', 72°. 
§ 20. The critical values of a therefore assume the following orders for 
different values of e : 
I. 
e< 24° 
6' 
i 
ii 
iii 
vii 
iv 
II. 
24° 6'< e< 30° 
i 
ii 
vii 
iii 
iv 
III. 
30°<e<35° 
16' 
i 
ii 
vii 
iv 
iii 
IY. 
35° 16'<e<42° 
41' 
i 
vii 
ii 
iv 
iii 
