hd 
Vol. IV, No.8.] Geometrical Theory of a Plane Non-Oyclic Arc. 395 
(N:8.] 
eee below or above the given arc, as it cannot intersect the 
arc at a fourth point. 
Suppose the arc POQ is positive. Then the circle POO will 
fall entirely below the given arc. 
t let it lie entirely above, as represented by the dotted 
line (Fig. 3). 
Take any point Bon the given arc between Pand O. Join 
he and produce QR to meet the circle POO at §. Join PS, PR, 
ud QO. Then evidently angle SPO is less than angle 
0 
eo eww ee wee ww we we 
~ce5e., 
- 
-- 
= 


Fig.3. q 
SQO, as Q falls inside the circle. “eau Me PSQ is greater 
than angle PO Much more is the a PRQ greater than 
angle POQ, which is panne! to hypot Peis ~ 
Similarly, if the are POQ ben d teersane then the circle POO 
will lie entirely outside A given arc 
The converse theorem is also evidently trne, namely, the are 
POQ will be positive or negative according as the circle POO falls 
continually esa inside or outside ithe given arc, as O moves 
rom 
C. —If POQ be a non-cyclic arc, then it will fall between 
the circles POO and QOO. 
Cor. B.—If POQ be anon- cyclic arc, then the circle of cur- 
vature at P falls entirely within the circle of curvature at Q. 
Thus the radius of curvature at P is less than the radius of 
ne ee at 
orem IV.—If POQ be a positive wee ag arc and S be 
rg ni in it, then the minor arcs and SQ will be also posi- 
-tive, t.e., the angle POS will continuonsly increase as O moves 
from P to S, and the angle QOS will continuously increase as O 
moves from S to 
oin PS, Then since angle POQ continuously increases 
as O moves ahh P to Q, t circle POO continuously falls 
below the given Hence as O moves from P to 8S, the circle 
POO falls hiss "the arc PS, ad hence the angle vos continu- 
ously increases as 0 moves from P to S. 
sees if O be taken in are SQ, it can be proved that the 
angle SOQ continuously decreases as O moves from Q to 8, 1.¢., 
the “Cc — is negative. Therefore arc SQ is positive. 
r. A—If PQbe any positive non-cyclic arc, then any minor 
