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

Theory of Transversals. 73
DEF. XXII. —We shall call the normal co-ordinates of a point
M, with respect to a triangle ABC, quantities proportional to the
sines of arcs drawn from M perpendicular to the sides of the triangle, and denote them by &a, 8b, 8.
DEF. XXIII.-We shall call the triangular co-ordinates of M
half the products of the sines of the perpendiculars from M on the
sides, multiplied by the sines of the sides. The triangular coordinates of M are equal to the Staudtians of the triangles
A MB, B CU, CXA; we shall denote them by n,, no, n0.
8. If arcs AM, BM, CM meet the sides of ABC in A', B', C', respectively, prove that
(BC, A') = n,: nb; (CA, B') = na: n,; (AB, C') = nb: n,.
9. If two points be isotomic conjugates, they have reciprocal triangular
co-ordinates.
10. If three arcs drawn through A, B, C be the symmetriques of any three
arcs AM, BM, CM, with respect to the bisectors of the angles A, B, C,
they meet in a common point M', called the isogonal conjugate of M.
11. If two points be isogonal conjugates with respect to a triangle, their
normal co-ordinates are reciprocals.
12. If a transversal T cuts the sides of AEC in A', B', C', the symmetriques of A', B', C', with respect to the middle points of BC, CA, AB,are
upon the same arc of a great circle T, called the isotomic transversal of T.
13. In the same case, the symmetriques of the arcs AA', BB', CC', with
respect to the bisectors of the angles A, B, C, meet the sides of ABC in
points which lie on the same great circle T", called the isogonal transversal
of T.
14. Prove that the triangular co-ordinates of G, the point of intersection
of the medians of a triangle, are equal to one another.
15. If Al, Bi, Ci be the harmonic conjugates of the points A', B', C', in
which a transversal T cuts the sides of ABC with respect to the sides, then
the arcs AA,, BB1, CC, co-intersect in a point r, called the trilinear pole
of T.
T is called the trilinear polar of r.
16. If G be the intersection of the medians, M any point of the sphere,
n' the Staudtian of the triangle BGC; then
cos MA4 + cos MB + cos MC
A cos B c = constant = n - n'. (296)
cos MG