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

56, 57] EQUATION OF THE LIMITING-CURVE 97

It follows from ~ 55 that
arc A1C: arc AC = e+tt.
Therefore S - s = Se-(+t).........................(1)
Next, produce the arc AB through B to the point P, such that
the are BP=S (Fig. 70).
\ fl
t.
Du
FIG. 70.
Let the tangent at B as before cut the axis through A at
D, and let AD=u and BD=t.
On A2, on the opposite side of A from D, take the point Q,
such that DQ=t.
Then the perpendicular through Q to the axis is parallel to
DB, and, therefore, to P2'.
Let the Limiting-Curve through Q cut the axis P2 in R.
Since the tangent at Q is parallel to the axis through R,
ar QR = S.
But AQ=t-u.
Therefore S + s= Set- t............................(2)
From (1) and (2), we have
ell = cosh t,...........................(3)
and s = S tanh t.........................(4)
~ 57. The Equation of the Limiting-Curve.
Let Ox and Oy be two lines at right angles, and let P be the
point (x, y) on the Limiting-Curve through 0, with Ox for
axis (Fig. 71).
Draw PM perpendicular to the axis Ox, and let the Concentric Limiting-Curve through M cut the axis through P in N.
Then OM = PN=x, MP = y.
Let arc OP =s, and arc MN =s'.
From the construction it follows that s'< 8.
N.-E.G. G