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

122 Inversions.
of the cone, and of the point m of the circle, on the plane of the
primitive. Let O be the pole of the primitive, and let Pm
produced meet a tangent plane to the sphere at O in T; then,
since the plane of the primitive and the tangent plane at O are
parallel, the plane OmP cuts them in parallel lines (Euc. XI.,
xvI.). Hence the angle P'm'O is equal to m' OT; but the angle
m'OT is equal to OmT, since the tangents mT, OT are equal.
Hence the angle P'm'O is equal to the supplement of Pm 0,
and the angle O is common to the two triangles PmO, P'm'O;
therefore OP: Pm:: OP': P'm'; and since the three first terms
of this proportion are given, the fourth, P'm', is given. Hlence
the locus of m' is a circle whose centre is collinear with the pole of
the primitive and the vertex of the cone.-(CHASLEs.)
Cor.-If the cones circumscribed to a sphere along a system
of circles have their vertices in a line passing through the pole
of the primitive, their stereographic projections is a system of
concentric circles.
2~. In the case of a great circle.
Let A be the pole of the primitive, and let the plane of the
circle AIB be perpendicular to the line of intersection of the
I CD
Fig. 47.
plane of the primitive with the plane of the circle to be projected, and let it intersect the plane of that circle in the line