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

CHAPTER VI.
SMALL CIRCLES ON THE SPHERE.
SECTION I. —COAXAL CIRCLES.
92. If an arc of a great circle passing through a fixed point O
cut a small circle X in the points A, B, tan - AO. tan - OB is
constant.
-\~ c \
Fig. 35.
DEM.-Let P be the pole of the small circle. Join OP. Let
fall the perpendicular PC; then, from the triangles A CP, O CP,
we have
cos A C cos AP
cos CO - cos OP'
Hence tan 4 (A C + CO) tan ~ (A C - CO)
= tan (AP + PO) tan i(AP - PO);
or, denoting the radius of X by p and OP by 8,
tan i OA. tan % OB = tan 2 (p + 8) tan (p - 8). 395)
112