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

Centres of Similitude. 103
9. If two rectangular secants intersecting in M cut a small circle in the
pairs of points A, B; C, D, prove that
tan2 M + tan2 MB + tan2 MC + tan2XD) = M OSp + COS (397)
(cos p + cos 3)2
where 8 is the distance of M from the pole of the small circle.-(NE3uBERG.)
10. If from any point P of a great circle MP tangents PT, PT' be drawn
to a small circle, prove that tan MPPT. tan ~ MPT' is constant.
11. The difference of the cosines of the tangent arcs, drawn from any
point P on the surface of a sphere to two small circles X, Y, is proportional to the sine of the perpendicular drawn from P to the radical axis
of X and Y.
SECTION II. —CENTRES rF SIMILITUDE.
96. DEF. XXXI. —Two points, S, S', which divide the arc
PP', joining the poles of twuo small circles Y, Z externally and
internally in the spherical ratio of the sines of the radii, are called
the centres of similitude of the small circles.
MP/ \Fig. 37. \
Fig. 37.
Cor.-Common tangents to the small circles pass through the
centres of similitude, viz., the direct common tangents through
the external centre, and the inverse common tangent through
the internal centre.
DEF. XXXII.-If through a centre of similitude we draw a
secant cutting the circles, then the pairs of points M, M'; N, N'
are said to be homothetic, and M, N'; 1f', N are inverse.