172 Caplain Owen on Circummeridian Altitudes. 
log. of z. A”; find the square of a, as convenient, = a’; add it to 
z.A’, Sum is 7D”, half its log. is log. of ¢ D’, from which deduct 
log. cosine D, leaves the log. of ¢—Q. E. D. 
No. 1. Z 
‘P 
In the figure No. 1, let Z represent the Zenith, P the Pole, and © 
the celestial object: the side L* = Colatitude (L = Latitude). 
A‘= Zenith Distance (A = Altitude). 
D*‘= Polar Distance (D = Declination). 
Now suppose the object moved from © to 8, the latitude and dec- 
lination remaining unchanged (that is, L’ is common to both, and 
PO=Pb=D)), then 20 Zb=z=change of azimuth, and ba 
the corresponding change of altitude, also © Pb = ¢ = the corre- 
sponding change of time. 
Then in the small differential 1@a6, right angled at a. 
ba=difference of altitudes =a. 
©a. secant Az and Od. secant D=t. 
For the Conversion of small Arc of Time into corresponding Arc 
of Azimuth ; or of Azimuth into Time. 
Let us consider their relations in small subsidiary or differential 
rectangular triangles, at the zenith, or at the object itself, in the 
