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

Stereographic Projection. 121
3. DUPUIS' THEOREM.-If a variable sphere touch three fixed spheres,
the locus of its point of contact with each fixed sphere is a circle.
For if a variable sphere be inscribed in a trihedral angle, the locus of its
point of contact on each face of the trihedral is a right line; and when we
invert, the planes become spheres, and the right lines circles.
4. Prove that four unequal spheres can be inverted into four equal
spheres.
SECTION II.-STEREOGRAPHIC PROJECTION.
116. DEF. XXXVIII.-Stereographic projection is the drawing of the circles of the sphere upon the plane of one of its great
circles (called the plane of the primitive) by Uines drawn from the
pole of that great circle to all the points of the circles to be projected.
It is evident that the plane is the projection in space of the
sphere, the value of the constant k2 being 2R2.
117. The stereographic projection of any circle is a circle.
This follows at once from ~ 115, Cor. 1, but we here give
an independent proof.
DEM.-l1. In the case of a small circle.
Let Pm be a generator of the cone, touching the sphere along
p
P C
o
Fig. 46.
the given circle, and let P', m' be the projections of the summit