The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

99] INVERSION AND REFLECTION

165

Therefore we have the following result:
Inversion with regard to any circle of the system changes any
point A into a point B, such that the nominal line AB is perpendicular to and " bisected " by the nominal line with which the
circle of inversion coincides..0
M
FiG. 111.
In other words,
Any nominal point takes up the position of its image in the
nominal line coinciding with the circle of inversion.
We shall now examine what effect such an inversion has upon
a nominal line.
Since a circle orthogonal to the fundamental circle inverts
into a circle also orthogonal to the fundamental circle, any
nominal line AB inverts into a nominal line ab, and the points
U and V for AB invert into the points u and v for ab (Fig. 112).
When the circle of inversion and the nominal line AB intersect, the lines AB and ab meet on the circle of inversion.
Denoting this point by M, it is easy to show that the nominal
lengths of AM and BM are respectively equal to the nominal
lengths of aM and bM. It follows that the nominal length of
the segment AB is unaltered by inversion with regard to any
circle of the system.
The same result can be obtained immediately from the
corresponding figure when the nominal line AB does not cut
the circle of inversion.