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

Coaxal Circles. 101
and conversely, if the planes of any number of circles pass
through a common point, they have a common orthogonal
circle.
94. If the planes of a system of circles S have a common line
L of intersection, the circles S have an infinite number of common
orthogonal circles.
DEM.-Take any point P in the line L, and through it draw
tangent lines to the sphere; these will touch along a circle
which (~ 93) cuts each circle of the system S orthogonally;
and since the same thing holds for each point on L, we have an
infinite number of circles forming a system S', each of which
cuts each circle of 8 orthogonally.
Cor.-The planes of the circles of the system S' have a common line of intersection.
For, take any two of them, say P and Q. Now (~ 93) the
plane of each passes through the vertex of each of the cones,
touching the sphere along the circles of the system S. Hence
the vertices are collinear, and the plane of each circle of S'
passes through the line of collinearity.
DEF. XXVIII.-A system of circles S, whose planes pass
through a common line L, is called a COAXAL SYSTEM.
DEF. XXIX.-The circle of the system S, whose plane passes
through the centre of the sphere, is called the RADICAL CIRCLE of the
system.
DEF. XXX.-If through L two tangent planes be drawn to
the sphere, their points of contact, regarded as infinitely small
circles, are the limiting points of the system.
Cor.-Each circle of the system S' passes through the limiting
points of S.
95. If X, Y, Z be three circles of a coaxal system, and from
any point P in X tangents PT, PT' be drawn to Y and Z; then
sin PT: sin - PT' ir a given ratio.