170 Captain Owen on Circummeridian Altitudes. 
To convert an Arc of Azimuth into corresponding Arc of Time at 
any Altitude of the Object. 
To the logarithm of the arc (z) in azimuth expressed in degrees 
and decimals, add the log. cosine of the middle altitude ; the sum will 
be the log. of z X cosine of declination ; this log. doubled will give 
the log. of its square (or 2”. A*”); to 2°. A‘? add a’ (twice log. a 
= log. of a’); the sum will be #?. D”; the half its log. will be the log. 
t. D’, to which add the log. secant of the declination; the sum will 
be the log. of ¢, the interval of time in degrees and decimals ;— 
A‘ expressing the log. cosine of middle altitude. 
D* the log. cosine of the declination. 
a expresses the difference of the altitude, noted in degrees and 
decimals. 
t, an arc of time noted in degrees and decimals. 
2, a corresponding arc of azimuth, also noted in degrees and 
decimals. 
Both these rules result from the foregoing equation, or, ?’?. D” 
=a’?+ 27. A’*; and #. D’?—a’= sum X difference = z*. A**. Or, 
(t.D'+a) X (¢.D’—a)=2. A”. 
But perhaps the simplest and most convenient practical method 
of converting an arc. of time into its corresponding arc of azimuth, 
is to assume a proximate latitude, and calculate them trigonomet- 
rically, by finding both the times and the azimuths for each alti- 
tude, and taking their differences as calculated. 
If, however, it be convenient, let the observer with the theodolite 
note the times by a good common watch at each observation also; 
then will the calculation be unnecessary, and some curious and 
useful problems be deduced from such simultaneous observations. 
