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

47, 48]

THE LIMITING-CURVE

81

Let P and P' be any two different points upon the same ray of a
pencil of parallel lines; the Limiting-Curve through P is congruent
with the Limiting-Curve through P'.
P
S~~
FIG. 56.
We must first explain what we mean by two LimitingCurves being congruent.
We suppose a set of points obtained on the Limiting-Curve
which starts at P'; e.g. P', Q', R', S', etc., on any set of lines
1, 2, 3, 4,..., of the pencil.
We shall show that a set of points P, q, r, s, etc., exists on
the Limiting-Curve through P, such that the segments Pq,
P'Q' are equal, the segments qr, Q'R' are equal, etc., and
these related linear segments make equal angles with the lines
of the pencil which they respectively intersect.
To prove this, take the segment P'Q'.
At P make / 2Pp=L2QP'Q', and take Pq=P'Q'.
From q draw the ray parallel to Pf2.
Then, by ~ 26 (4), we know that L Pq2=L P'Q'2.
But P' and Q' are corresponding points.
Therefore P and q are corresponding points.
Proceeding now from Q' and q respectively, we find a point
r on the Limiting-Curve through P, such that the segments qr
and Q'R' are equal, while qr makes the same angles with the
N, -E.G. F