314 
Proceedings of the Royal Society of Edinburgh. [Sess.. 
x= ‘Amr - ^ , j — (2m + l)7r ± cos 1 h ; 
y cannot lie between mir ± cos -1 1 k |. 
The curve therefore consists of an infinite number of repetitions of 
sinuous curves lying between the pairs of parallels y = mir — cos -1 1 k \ and 
y = (m— l) 7 T + cos _1 1 k |, and the concavities and convexities of consecutive 
branches are opposed. 
Let | k |>1. 
When x = mr , y = 7mr+ 
y = Zmir , X = nir + ( - ) n sin 
in-i * 
y = (2m + l)7r , x=mr - (-) r 
-i 1 . 
x cannot lie between ^ 7 r + sin 1 ~i- and {n-\-l)Tr — sin 1 -?~ . 
\k\ \k\ 
The curve therefore consists of an infinite number of repetitions of 
sinuous curves lying between the pairs of parallels x = nir =h sin -1 ^ , with the 
concavities and convexities of consecutive branches opposed. 
Another special case occurs when 0 = 0; then 
Also if 0 = 90°, 
cos y = 0 or y = mr + - . 
sin x = 0 or x = mr . 
§ 5. With the other equation the only essential difference is in the form 
of the curve in the limiting case | k \ — 1. 
We have 
dy tan 0 sec 2 8 cos x 
dx ~ cosec e cos 8 sin y 
Putting 0 = 90° — e, S = e, si = — , y — ir, and evaluating the limit, we get 
V^ = sec e = tan 47° 28'. The same value is obtained when S = — e, x = — ^ , 
dx 2 
y = 0 ; while when 8 = 0, x = 0, y = ~, we get ^ = cos e = tan 42° 32'. So 
JL OiX 
that the curve does not differ very greatly from a straight line. 
§ 6. Corresponding to these complete curves, we have the curves of 
sunset. The limiting case for north latitudes is k — — 1, or tan 0 tan e= 1, 
which gives 0 = 90° — e, — i.e. the Arctic circle. For the equator 0 = 0, giving 
