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

66] ELEMENT OF ARC 113
Also, in the quadrilateral MNHP, the angles at M, N, and H
are right angles, and the sides beginning at M are
vI z, q, y.
These correspond to a, in', c, 1
in a right-angled triangle. [Cf. ~ 35.]
Yt ~~Q.
F P I
qH
0 X M N x
FiG. 77.
sinh-t6x k
Thus we have sinh k'. ~ [f, I.]
cosh -
k
Therefore q cosh Y x, to the lowest order.
Also, we have
cosh tanhY coth [Cf. ~59, V., and 58.]
k k k
Therefore y and z differ by a small quantity when 3x is small.
Put Z=y~+).
Then we have tanh c + ' cosh & tanhT.
k
NC EG. H