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

55, 56] ARCS ON CONCENTRIC LIMITING-CURVES 95

Therefore, with this unit of length we have the following
theorem:
If ABDC (Fig. 67) is a figure bounded by two Concentric
Limiting-Curves AC and BD, and two straight lines AB and CD,
the straight lines being axes of the curves, the lengths s and sx of
the arcs AC and BD are connected by the equation
sx = se -,
when the segments AB and CD are x units of length, and AC is
the external curve, BD the internal.
A,s S,
FIG. 67.
If another unit of length had been chosen, so that the
ratio of the arc AB (Fig. 65) to the arc A^Bl had been
a(a> 1), when AA= BB ==the unit of length, the equation
connecting s and sx would have been
x= sa -x.
1
Putting a =k,
we have s= se k.
The number k is the parameter of the Hyperbolic Geometry
depending upon the unit of length chosen.
~ 56. Since we can find p to satisfy the equation

there is a point Q on the Limiting-Curve through P, such that