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

101]

THE ANGLE OF PARALLELISM

169

becomes the circle au, touching the radius mu at u, and cutting
ma at an angle II(p). These radii mu, mb are also nominal
lines of the system.
Let the nominal length of AM be p.
AB/MB
Then we have p=log (AB / )
a/ab lmb\ /ab\
= log (- — )=log - *(
\acl mc/ \ac/
u
b _c
FIG. 115.
But from the geometry of Fig. 115, remembering that au
cuts be at the angle HI(p), we have
k {1 Lt^7r rI (P)
ab = k1 +tan - I )},
where k is the radius of the fundamental circle.
Therefore p = log cot (II2))
and e-P = tan (P))
Finally, in this geometry there will be three kinds of circles.
There will be (i) the circle with its centre at a finite distance;