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

102 Small Circles on the Sphere.
DEM.-Let O, O' be the spherical centres of Y, Z. Join OP,
O'P by arcs of great circles; then, if the radius of the sphere
be unity, the perpendicular from P on the plane of Y= cos OT
- cos OP = cos OT - cos OT. cos PT = cos OT. 2 sin2 PT.
Similarly, the perpendicular from P on the plane of Z= cos
O'T. 2 sinU PT'. But since the planes of X, Y, Z are collinear,
the perpendiculars have a given ratio. HIence the ratio of
cos OT. sinm PT: cos O'T'. sin2 PT' is given, and OT, O'T'
are given, being the spherical radii of Y and Z. Hence the
ratio of sin PT: sin J-PT' is given.
Cor.-If PT = PT', the locus of P is the radical circle of the
system.
EXERCISES.-XXVII.
1. The radical circles of three small circles taken in pairs are concurrent.
2. If there be a coaxal system of circles S, and a circle X distinct from
it, then the radical circles of X, combined with each circle of S, are concurrent.
3. If through a point on the radical circle of two small circles we draw a
spherical secant to each, the four points of intersection are concyclic.
4. Through two points of the sphere describe a small circle touching
a given great circle.
5. If through a fixed point A we draw a great circle, cutting a given
small circle in the points B, C, and if a point D be taken on it, such that
tan2 ~ AD = tan2 DB. tan2 1DC, prove that the locus of D is a great
circle.
6. If X, Y be two small circles; PT, PT' two tangents to them from a
point P, prove that the locus of P is a circle, if m cos PT + n cos PT' be
constant, m and n being given numbers.
7. The locus of the poles of small circles, intersecting two small circles
X, Y at the extremities of two spherical diameters, is the radical circle
of X, Y.
8. Describe a circle cutting three small circles at the extremities of three
spherical diameters.