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

114 NON-EUCLIDEAN GEOMETRY [ca. v.
This gives, to the lowest order,
(tanh Y 1+ '= tanh Y I + k tanh
i.e. =-1 sinh I cosh -Y X2.
2k lE k
Therefore y and z differ by a quantity of the second order
when Sx is of the first order.
Now = (y + y) -
Therefore p = 8y, to the first order.
It follows from 8s2 =2 + q2, that
s2 = cosh2 Y Sx2 + dy2, to the lowest order.
Thus we have shown that the element of arc in Cartesian
Coordinates is given by
ds2 cosh2 dX2 + dy2.
k
~67. Element of Arc in Polar Coordinates.
In the Euclidean Plane we have for the element of arc in
Polar Coordinates, the equation
ds2 = dr2 r2d02.
We proceed to find the corresponding formula in the
Hyperbolic Plane.
It may be obtained in two ways. It could be deduced from
ds2 = cosh2 Y dx2 + dy2,
by using the relations connecting x, y and r, 0; viz.
r x y
cosh = cosh cosh,
tanh [Cf. ~ 63.]
tan 0 =
sinh x