360 Proceedings of Royal Society of Edinburgh. [sess. 
we have at once 
tan (azimuth) = 
sin h 
sin X cos h - tan 8 cos X * 
Capt. Weir, in his diagram, virtually puts 
x = sin h sec X 
y = cos h tan X 
so that 
tan (azimuth) = - — , 
(i) 
x and y being found by the intersection of the confoeal conics 
/£#2 
+ , = 1 , the latitude ellipse, 
sec 2 X tan 2 X 
and 
sin 2 /i cos Vi 
= 1 , the hour-angle hyperbola. 
The Amplitude is the value of the azimuth at rising or setting, so 
that the corresponding hour-angle is to he found from 
cos h + tan X tan 8 = 0. 
With this value of h, equations (1) become 
x = sec X ^1 - tan 2 X tan 2 8 
y= - tan 2 X tan 8 
Elimination of 8 gives, of course, the latitude-ellipse as before. 
But elimination of X gives, instead of the confoeal hyperbola, the 
curve 
x 2 + \y - J(tan 8 - cot S)] 2 = ^(tan 8 + cot 8) 2 , 
or 
x 2 + (y + cot 2S) 2 = cosec 2 28 , 
which is a circle passing through the common foci of the ellipses 
and hyperbolas. 
The construction of the “ Diagram” by means of (1) is, theoreti- 
cally, a very simple matter. Thus, take OA as unit length on the 
