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

116 NON-EUCLIDEAN GEOMETRY

[CH. V.

Then cosh + sinh =cosh I1+ 2)
1% k kE k\ $-2k2/
to the lowest order.
Thus = 2 coth I, to the lowest order;
i.e. r and z differ by a small quantity of the second order,
when 80 is of the first order.
But p = + r - z.
Therefore p = 5r, to the first order.
It follows that
8s2 = 8r2 + k2 sinh2 802, to the lowest order.
Therefore ds2 = dr2 + k2 sinh2 r d02
k
~68. The Element of Arc in Limiting-Curve Coordinates.
We shall now describe a system of coordinates peculiar to
the Hyperbolic Plane. The position of the point P is given
by the Limiting-Curve and axis on which it lies, the LimitingA
'0P'P
O P M
0
FIa. 79.
Curves being all concentric, their common centre being at
infinity 6n the axis of x.
Let the Limiting-Curve through P cut off a segment of
length (OPo) on the axis of x, and let the axis through P