A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

Geodesy. 145
Let A be the place of the observer, i, N the objects above
the horizon; let a sphere of radius unity be described, touching
the horizon at A, and intersecting the lines AM, AN in the
points B, C. Draw the gréat circles ABO, ACO; then if
Z, H' denote the elevations of A-M, AN above the horizon, we
have Rf= - arc AB, H' = ar A C; that is, considering the
spherical triangle AB C, if= c, H' = % b. Now the angle A
of the spherical triangle ABC is the horizontal angle which
corresponds to the oblique angle MXAN. Hence, if 0 denote
the difference, we have (459)
0 = tan 'A sin2 (b + c) - cot A- sin2 ~ (b - c);
0
A
Fig. 59..-. 0 = tan }i sins a (H + H') - cot -4 sin2 j (H - ( l). (460)
In practice H, I' and 0 are very small. Hence this formula
may be replaced by the following, which is Legendre's:
0= {i(f+ r')}2 tan J fMAN- ( (Hf- H')} cot IMAVN,
(461)
an approximate value of the difference between the circular
measures of the oblique and horizontal angles, which must be
added to the former to obtain the latter.
L