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

83, 84]

THE FUNCTION p (y)

143

This function is associated with the segment M'O, denoted
by y', and will be written as ((y').
Now we have seen that
~ (y/) > M'Q: z.
But in the quadrilateral OM'PM, the side PM plays the same
part as OM' in the quadrilateral OM'QM.

Thus

0(y)> M'P: x.
0'(?) < X < (y).

Thus we have

Since x' < x, the function / (y') is less than unity, except for
y'=0, when it becomes equal to unity.
~84. We shall now show that the function defined in the
previous section is continuous.
Let OS and Os be two lines meeting at 0, such that
OS= Os= and LSOs is acute.
S

0

FC B10
Fio. 103.

Then the angles at S and s are both right angles.
Let SB=x-y, SC=x, and SD=x+y.
Let the perpendiculars at B, C, and D to OS, meet Os at
b, c, and d.
Through b and d draw bm and dn perpendicular to Cc.