1907-8. j Sunset and Twilight Curves, and Related Phenomena. 333 
intersection of which give the critical values of e (and also of a). We are 
only concerned with the portions of the curves contained within the square 
e — 0° to 90°, a = 0° to 90°. These curves are represented in the diagram, 
fig. 3. They divide the square into 22 regions, each of which is 
characterised by a different sequence of phenomena for varying latitudes. 
These correspond to the 22 different sequences of phenomena which we 
have already obtained. This gives us a means of tabulating the results 
more succinctly for varying values of a in terms of e. 
a < 
tan-^cote 
tan -1 ^ cot e 
<a< 
90° -e 
90° -e 
<a< 
sin -1 cot e 
sin -1 cot e 
<«< 
180°- 2 e 
180° -2e 
< a < 
2 sin -1 
^ cosec € 
2 sin- 1 
ijr cosec 6 
<a< 
120° e 
a > 
120° e 
a<fe 
I. 1 
Y. 2 
YIII. 3 
IX. 4 
XII. 5 
fe<a<e 
I. 2 
IY. 3 
VI. 4 
VII. 5 
X. 6 
XI. 7 
XIII. 8 
6 < a< 2e 
I. 3 
II. 4 
III. 5 
VI. 6 
VI. 7 
VI. 8 
VI. 9 
a > 2e 
I. 4 
I. 5 
I. 6 
In a similar way the critical values of cp , which are expressed in terms 
of both e and a, determine surfaces, or rather portions of surfaces, contained 
within the cube, e, a, (p from 0° to 90°. The projections upon the plane of 
e, a of the curves of intersection of these surfaces are exactly the curves 
corresponding to the critical values of a in terms of e. These surfaces 
divide the cube into 16 regions, but one of the partitions is nugatory, 
viz. that portion of the plane (p = 90° — | which divides the region bounded 
by 
<p = 90° + e — a, <p = 90° — e , e = 0 and a =90°. 
Hence these regions correspond to the 15 different yearly phenomena, 
and the nugatory partition corresponds to the case in which the critical 
value (p = 90° — ? was found to be nugatory, viz. when a > 2e. 
The regions of the cube can be described without difficulty. The four 
planes 
<p = e + a — 90° , <p = 90° — e - a , <p = 90° + e - a , <p = 90° — e + a 
determine a regular tetrahedron inscribed in the cube. This tetrahedron 
is cut in two by the surface sin <p = sin e sin a. We have then seven 
regions. 
