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

126 Inversions.
3. Prove Delambre's Analogies.
From ab e = (ac + ab) sin. a (ac - ab) cos a
From abc we have bc = =
cos ~ (j - y) sin ~ (8 - y)
and substitute as before.
4. Being given three circles on the sphere; there are eight points on the
sphere, any one of which, if taken as the pole of the primitive, the three
circles will be projected into three equal circles.-(STEINER.)
5. tan 2 = Vtan 2 s. tan ~ (s- a) tan (s - b) tan (s - c).
Express tan ~ the angle ab'c, in terms of the sides of ab'c.
1 + cos a + cos b + cos c
6. Prove that cos E =
4 cos 2 a cos ~ b cos ~ c
In the triangle ab'c, we have ab' = ac cos cab + cb' cos cb'a.
7. To express the spherical excess of a spherical quadrilateral in terms of
its sides and diagonals.
c'
u
t\\ /
c
Fig. 49.
Let ABC:D be the quadrilateral; and denoting the sides and diagonals
AB, BC, CD, DDA, AC, BD by a, b, c, d, e, f, respectively; then taking
the antipodes of A for the pole of the primitive, the arcs AB, AD, AC will
project into right lines ab, ac, ad; and the arcs BC, CD into arcs of circles