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

Centres of Similitude. 105
Cor. 1-If a variable circle touch two fixed circles, the great
circle passing through the points of contact passes through a
fixed point, namely, a centre of similitude of the two circles;
for the points of contact are centres of similitude.
Cor. 2.-If a variable circle touch two fixed circles, the
tangent drawn to it from the centre of similitude, through
which the chord of contact passes, is constant.
99. DEF. XXXIII.-Being given afixed point S, and any line
whatever, y, on the sphere, if upon the arc of a great circlejoining
S to any point MW of y a point M1' be taken, such that tan SM:
tan SM' in a given ratio, the locus of M' is said to be homothetic to y.
DEF. XXXIV.-If JM' be taken, such that tan ~ SM1. tan SM'
is constant, the locus of M' is called the inverse of y.
This method of inversion was first employed in the Author's Memoir on
Cyclides and Sphero-quartics. (Read before the Royal Society in 1871.)
EXERCISES.-XXVIII.
1. If two small circles touch two others, the radical axis of either pair
passes through a centre of similitude of the other.
2. The figure homothetic to a circle is a circle.
3. The inverse of a circle is a circle.
4-5. S being the centre of similitude of two circles; M, N two inverse
points on these circles-1l, the tangents at M and N intersect on the radical
axis; 2~, these points are points of contact of two circles touching the two
given circles.
6. The angle of intersection of two circles on the sphere is equal to the
angle of intersection of the circles inverse to them.
7. Any two circles can be inverted into two equal circles.
8. Any three circles can be inverted into three equal circles.
9. If two circles be the inverses of two others, then any circle touching
three of them will also touch the fourth.