
Vol. IV, No. 8.] Geometrical Theory of a Plane Non-Oyclic Arc. 397 
[N.S.] 
Similarly the angle QOS continuously diminishes as 0 moves 
from P to 8. 
Cor. A. —If PQ be any positive non-cyclic arc, and P’Q’ any 
minor arc, then angle P Og’ will continnoasly diminish as Q 
moves P to P’ or Q to Q’, and re eon of angle P’OQ’ will 
rete: diminish as O moves from P’ to Q'. This follows 
above theorem and Vor. A of Theorem TV. 
re B.—lf POQ be any positive non cyclic arc, in which 
anvle POQ is always obtuse and P’Q any minor arc, then the 
radius of the circle P’O ‘ continuously increases as O moves from 
PtoQ. For, the diameter of the circle P’OQ' is P’Q’/sin P OQ’. 
Cor. O.—I£ any three points O,, O,. Os be taken on a posi- 
tive non-cyclic arc POQ, in whith the angle POQ is always 
obtuse, then the radius of the circle O,0,0, is always increased 
if any of the three gers be moved hacinssea . 
Theoren VI.—\f P’Q’ be a minor chord "of the aie Fo 
parallel to the base PQ. & and R’ the midpoints of habe ted 
and 6 the complement of the angle between BRR’ i Po, then 
the distance between the centres of the circles PP! and P’Q’Q 
is equal to nd. 
Join PP,’ QQ’ and RR’ and produce t them to eee inT, Let L 
4 M be the 
(ae) 
aud let perpen- 
diculars to 

perpendicular 
to B'UV au 
P’Q’  respec- 
tively. 

Vv 
Then it follows easily from elementary geometry (Fig. 5) 
‘that WX=R'Y=NR, since WY and TN cut off equal intercepts 
MY’ and LN’ from LM. 
Therefore, tan one Lees UV op PQ tan = UV =distance 
N 
between the centres of the circles pra and FAG: aioe cess, 

. 
