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

Theory of Transversals. 69
DEFINITION XXI.-The anharmonie ratio of four great circles
(a, /3, y, 8) passing through the same point M is (a/ys3)
sin (ya) sin (Sa) sin CA. sin DA
sin(yyp) sin(8, )' siin CB sin:DB'
If (ap3y8) = - 1, tie pencil a, /3, y, 8 is said to be harmonic.
70. If a great circle intersects the sides of a triangle ABC in
the points A', B', C', then,
1~. (AB, C')(BC, A')(CA, B') = 1. (289)
2~. (ab, CC') (bc, AA')(ca, BB') = 1. (290)
B '.~ ~.
Fig. 24.
To prove 1~-If the sides AB, CA be eut internally, and
BC externally, and perpendiculars p', p", p"' be drawn to the
transversal, we have by the properties of right-angled triangles,
sin p' sin p" sin p
(AB,C)- si-', (B C, A')=, and ( CA, B') -sin p
sin " ps sin p'
Hence, by multiplication, we have
(AB, C') (BC. ') ( CA. B') = 1.
To prove 2~-We have, by equation (55), the equalities
sinA4 sin B
(AB C') = (ab, CC') sinB (BC, ')= (be, AA') sin C
sin C'
(CB') t (ca,BB') sin.
Hence, by multiplication, the proposition is proved.