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

Mu-tual Power of Two Circles. 117
circles is equal to the cosine of the arc joining their poles, a
relation identical with equation (408) for the six arcs joining
four points on a sphere 12, &c., denoting in this case the arcs
joining the points 1, 2, &c. If the three first points be the
vertices A, B, C of a spherical triangle, and the fourth any
arbitrary point D, whose distances from A, B, C are denoted
by a, /, y, respectively, we get1, os, cb cos, cos b cos a
cos c, 1, cos a, cos,
=0. (414)
cos b, cosa, 1, cos 1 c y
cos a, cos /, cosy, 1
112. If the first three circles of the second system in ~ 110
coincide with the first three circles of the first system, and
the poles of these circles be the angular points of a spherical
triangle ABC. Also, if s4/ be a great circle distinct from s4,
and the distances of the poles of these circles from the points
A, B, C be a,,, y; a', 3', y', respectively, and 8 the arc
joining their poles, we get1, cos c, cos b, cos a
cos c, 1, ca cos a, cos
= o. (415)
cos b, cos a, 1, cos y
cos a', cos /', cos y', cos
EXERCISES.-XXX.
1. The incircles of a triangle and its colunar triangles have a fourth
common tangential circle.-(HART.)
For if a, b, c be the sides of the original triangle, the direct common tangents of the incircles of the colunar triangles are (b + c), (c + a), (a + b),
respectively; and the transverse common tangents of the incircle of the