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

110 Small Circles on the Sphere.
perpendiculars from A on the sides CO, OB, BC of the triangle
OB C be denoted by x, y, z; and the sines of the perpendiculars
from A' by x', y', ', respectively, then we have
x::: sin 0CA: sinA CB;
that is, x::: sin(B - E): sin C.
Similarly, y: s:: sin ( C- E): sin B;
xy sin (B - E) sin (C- E) =cos.** - == -—:-p. ^ - = COS2 ' a.
z2 sin B sin C
xiy'
In like manner, - = co2 a.
z,2
Hence xy: x'y': 2: 2;
but
xy: x'':: sin2 OA: sin2 OA', and z2: z'2:: sin2AN: sin2A'N;. sin OA: OA':: sin AX: sin NA'. (401)
EXERCISES.-XXIX.
1. If four points A, B, C, D lie on a great circle a, their anharmonic
ratio is equal to that of their harmonic polars, with respect to any small
circle X.
For if O be the spherical centre of X, P the harmonic pole of a, the
perpendiculars from P on the circles OA, OB, OC, OD will be the harmonic
polars of A, B, C, D, and will pass through the poles A', B', C', D' of the
great circles OA, OB, OC, OD. Now it is evident that
(P- A'B'C'D') = (A'B'C'D') = (O - ABCD) = (ABCD).
2. If A, B, C, D be four points on a small circle X, and if the arcs
AB, BC, C'D, DA be denoted by a, b, c, d, respectively; then if P be any
variable point on X, the anharmonic ratio
(P- BCD) = sin 2 a. sin c sin s b sin 2 d.
For if the perpendiculars from P on AB, BC, &c., be a, B, &c., and the