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

92 Spherical Excess.
Hence, from (392), we get
A 2B
cos2 — smn - sin2 -COS s = 2 si n 2 si n 2 (NEUBERG.) (393)
2 sin 1 A sin - - sin ' C'
This theorem is the correlative of 357. We can get, in the same manner,
cos (s-a), cos(s-b), cos(s-c); cos, sin tan tan2 &c.
SECTION II. —LEXELL'S THEOREEM.
88. If the base BC of a spherical triangle ABC be given in
magnitude and position, and the spherical excess in magnitude, the
locus of the vertex is a small circle of the sphere.
STEINER'S PROOF. Lemma.-If upon the base BC a spherical
triangle be constructed, such that A - E is given, the locus of
A is a small circle, namely, the circumcircle of the triangle.
For if O be the pole of the circumcircle (see fig., ~ 75), the
angle OBC= OCB = (A - E). Hence O is a given point, and
the circle is given in position.
I,; LEXELL'S THEOREM.-Let AB C be one position of the triangle,
2E the spherical excess constant. Let the points B', C' be the
P,
Fig. 32.
antipodes: of B, C; let P be the pole of the circle AB'C';
then we have 2 E=A + B + C- 7r = B'AC'+ r - AB'C' + wr