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

96, 97] SYSTEM OF ORTHOGONAL CIRCLES 161

We define the nominal line through any two nominal points
as the circle which passes through these two points and cuts the
fundamental circle orthogonally.
Two different nominal points A, B always determine a nominal
line AB, just as two different ordinary points A, B always
determine a straight line AB. The nominal points and lines
also obey the " axioms of order."
We define the angle between two intersecting nominal lines
as the angle between the tangents at the common point, within
the fundamental circle, of the circles with which the nominal
lines coincide.
We have now to consider in what way it will be proper to
define parallel nominal lines.
05 U
FIG. 108.
Let AM (Fig. 108) be the nominal line through A perpendicular
to the nominal line BC; in other words, the circle of the system
which passes through A and cuts the circle of the system through
BC orthogonally. Imagine AM to rotate about A so that these
nominal lines through A cut the nominal line through BC at a
gradually smaller angle. The circles through A which touch
the circle through BC at the points U and V, where it meets
the fundamental circle, are nominal lines. They separate the
lines of the pencil of nominal lines through A, which cut BC
from those which do not cut it. All the lines in the angle +
shaded in the figure do not cut the line BC; all those in the
angle f, unshaded, do cut this nominal line.
N.-E.G. L