1907-8.] Sunset and Twilight Curves, and Related Phenomena. 319 
Hence we have the following cases : — 
( 1 ) 
cf> < tan -1 
tan e 
1 + 2 tan 2 e ’ 
The curve starts as fig. 1 and passes through the form with the points of 
undulation as a passes through the value sin 1 into fig. 2. When a 
assumes the value 90 ° — e—(p the curve becomes nodal and passes into 
fig. 5. When a = 90° — e-f<^> it breaks up into figures of eight, and then 
becomes closed ovals as in fig. 6. When a = 90° the two values of S, viz. 
90° — a — (p and a — <fi — 90°, both become — <p and the ovals reduce to points. 
These variations are illustrated in Plate II. for values of a positive and 
negative at intervals of 10° from —90° to +90°, and for 0 = 10°, e = 30°. 
The curves for negative values of a are entirely similar to those for 
corresponding positive values, only displaced through i r along both axes. 
(2) d> >tan - ’ 1 — ^ an e , and < e and < 90° - e . 
v ^ 1 + 2 tan 2 e 
The variations are similar to the preceding case with fig. 4 substituted for 
2, when 90° — e — < a < sin -1 S * n ^ . 
sin e 
(3) €<cf)<90° - e. 
Here the curve, after assuming the form of fig. 4, vanishes when 
a = 90° + e — <p. 
(4) 90° -e<<£<e. 
The curve starts as fig. 14, and passes through a nodal form when 
a = <p + e — 90° into 4, thence through 5 and 6. 
(5) </>>eand >90° -e. 
Here the curve starts with 14, passes through 4, and vanishes when 
a = 90° + e — <p. 
These variations are illustrated in Plate III. for values of a positive and 
negative at intervals of 10° from —50° to +50° and for 0 = 70°, e = 30°. 
§ 14. The case for e = 90° requires special treatment. Here sin S = sin x ; 
8 = x from T to S.S., 
= 7 t — x from S.S. to W.S., 
= x -% t from W.S. to T- 
The sunset-curve becomes 
cos y=— tan c/> tan x when x lies between 2mr ± — ; 
A 
cos y = tan tan x when x lies between (2n + l) 7 r ± - . 
