334 
Proceedings of the Royal Society of Edinburgh. [Sess. 
cf> < e + a - 90° 
<f> < 90° - e — a and ^ sin -1 (sin e sin a) 
4>>e -+- a — 90° , > 90° - e - a , < 90° + e - a, < 90° - e-f a, and ^ sin -1 (sin e sin a) 
</> > 90° + e - a 
4> > 90° -- e + a 
The first and the fourth and fifth are each cut in three by <p = 90° — 
a 
2 
and <p = 90° — e. 
The last is cut in two by <p = 90° — ^ , and the sixth is cut 
in three by <p = 90° — e and <p = 90° — ? but the latter partition is nugatory. 
The seven primary regions correspond to the seven different forms of 
the twilight-curve by itself, not taken in conjunction with the sunset- 
curve. 
As one ascends vertically from the plane <p = 0, the regions passed 
through give the sequence of phenomena for varying latitudes, and for the 
values of e and a corresponding to the starting-point. 
(. Issued separately May 14, 1908.) 
