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

158 Miscellaneous Exercises.
*20. If A, B, C, D be four points on the same great circle, and if p be
the angle of intersection of the small circles, whose spherical diameters are
AC and BD, prove that the six anharmonic ratios of the points A, B, C, D
are
sin2 ~, cos2 ~, -tan2 ~, cosec2 (p, sec2-, -cot22.
t21. The mutual power of two circles on the sphere is unaltered by
inversion.
22. Prove the relation (412) by inversion.
23. If from a fixed point O on a great circle three pairs of arcs OA, OA';
OB, OB'; OC, OC' be measured, such that tan OA.tan OA'= tan OB. tan OB'
-tan OC. tan OC' = k2, where k is a constant; then the anharmonic ratio
of any four of the six points A, A', &c., which contains only one pair of
conjugates, such as (ABCC'), is equal to the anharmonic ratio of their four
conjugates (A'B'C'C).-(Compare Sequel to Euclid, p. 132.)
Draw a tangent to the great circle, and produce the radii through the
points A, A', &c., to meet the tangent.
DEF. I.-A system of pairs of points, such as AA', BB', CC', fufi/ling
the conditions that the anharmonic ratio of any four being equal to that of
their four conjugates, is called a system in involution.
DEF. II.-If two points D, D' be taken in opposite directions from O, such
that tan2 OD = tan2 OD' = k2, each point being evidently its own conjugate, is
called a double point.
DEF. III.-If a system of points in involution on a great circle X be joined
by arcs of great circles to any point P not on X, the six joining arcs having
evidently the anharmonic ratio of the pencil formed by any four equal to that
formed by their four conjugates, is called a pencil in involution.
24. The double points D, D' are anharmonic conjugates to any pair AA'
of conjugate points.
25. The six arcs joining any point on a sphere to the intersection of the
sides of a spherical quadrilateral form a pencil in involution.
*This theorem in plano was first published by the author in the Philosophical Transactions, 1871, p. 704.
t This theorem, in a different form, viz., "the ratio of the sine squared of
half the common tangent of two small circles to the product of the tangents
of their radii is unaltered by inversion," was first given by the author in a
Memoir " On the Equations of Circles," in the Proceedings of the Royal
Irish Academy, 1866.