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

84 Various Applications.
Cor. 6.-If A, B, C, &c., be any system of fixed points, and
1, m, n, &c., a system of multiples, and if Pa, Pb, Pc, &c., be the
perpendiculars from A, B, C, &c., on any great circle T; then,
if ( (1 sin pa) = constant, the envelope of T is a small circle.
ExERCIsEs. —XXIV.
1. If from the points Ia, lb, I, perpendiculars be drawn to the sides BC,
CA, AB, respectively, these perpendiculars are concurrent.
2. If H be the pole of Hart's Circle, prove that
cos AH - cos = cos b cos c - cos a. (350)
3. The arcs of connection of the vertices of a triangle to the points of
contact of the opposite sides, with the circles inscribed in the corresponding
colunar triangle, meet in the same point, called the NAGEL point of the
triangle.
4. If tl, t2, t3 be the t gents from A, B, C to Hart's Circle, prove
a b c
cos tl cos t2 cos t3 = cos - cos cos -.
2 2 2
5. The normal co-ordinates of the pole of Hart's Circle are
cos(B- C), cos(C-A), cos(A-B).
6. The normal co-ordinates of Nagel's Point are
sin2 A, sin2 B, sin2 C.
7-10 Prove the following relations:1~. sin (B + C)/sin A = (cos b + cos c)/(l + cos a).
2~. sin (B - C)/sin A = (cos c - cos b)/(l - cos a).
sin (B + C) cot ~ a (cos B + cos C ) cot A
30 =
cos b + cos c sin (b + c)
sin (B + C) tan ~ a (cos C- cos B) tan.A
cos b - cos e sin (b + c)