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

Poles and Polars. 111
staudtians of the triangles PAB, PBC, PCDZ, PDA being denoted by
na, nb, nc, nd, we have evidently
(P- BCD) = = sin a sin a. sin c sin y - sin b. sin j. sin d. sin
= sin ~ a sin ~ c - sin 1 b. sin d (equation (343)). (402)
3. If A, B, C, -D be four points on a small circle X, the spherical triangle
whose summits are the points of intersection of the arcs AB, CD; BC, DA,
and CA, 1DB, is such that each side is the harmonie polar of the opposite
vertex. This is called the harmonic triangle of the four points.
4. The harmonic polars of any point on the radical circle of two small
circles with respect to these circles intersect on the radical circle.
5. If X, Y are two small circles, Z a great circle perpendicular to the great
circle passing through the spherical centres of X, Y, the harmonic polars of
any point of Z intersect on a great circle.
6. If a spherical quadrilateral be inscribed in a small circle (X), and at its
angular points arcs of great circles be drawn touching X, their twelve points
of intersection lie four by four on the sides of the harmonic triangle.
7. PASCAL's THEOREM.-If a spherical hexagon be inscribed in a circle,
the opposite sides intersect in pairs on a great circle.
8. A, B, C; A', B', C', are two triads of points on two great circles;
prove that the intersections of the three pairs of arcs AB', A'B; BC', B'C;
CA', C'A lie on a great circle.
9. SALMON'S THEOREM.-Given any two points A and B and their harmonic polars, with respect to a small circle X, whose spherical centre is O.
Let fall a perpendicular AP from A on the polar of B, and a perpendicular
BQ from B on the polar of A; then, if A', B' be the harmonic conjugates
of A, B, with respect to X, prove that
cos OA: cos OB:: sin OA' sin AP: sin OB' sin BQ.
10. BRIANCHON'S THEOREM.-If a spherical hexagon be described about
a small circle X, the three arcs joining the opposite angular points are
concurrent.