394 Journal of the Asiatic Society of Bengal. (August, 1908. 
By repeating the above viata a sufficient number of 
es it is evident we can make P, Q, R, S come as close together 
we like and ultimately coincide at some point OU, lying 
ieiuks the original positions “ en Q, R, 8, bat not coinciding 
with P or S at their original posi 
Thus there will be a cyclic ene on the given arc, which is 
contrary to hypothesis. 
or, A—If a circle meet a convex are at four distinct 
eo P, Q, R, 8, then there exists a cyclic point between 
and §, 
Cor. B.—Every closed convex curve, that is a curve of which 
every arc is convex, has at least two cyclic points on 
For, a circle, through any three points of thes figure, wi will 
meet the figure avain, ata fourth gg dividing the figure into 
at least four convex ares, ren there will be a cyclic point in 
every three consecutive arcs. Thus there will be at least two 
cyclic 2s on the figure 
Cor. U.—If a closed curve has a node or cusp, the remainder 
being convex, ee there will be at least one cyclic point in the 
remainder. 
Cor. D mae non-cyclic curve must be necessarily spiral in 
form. This sts an obvious general geometrical definition 
of spirals. it eviaently follows from Corollaries B and C. 
Theorem II.—If POQ be a non-cyclic arc, then angle POQ 
will rs increase or decrease as O moves along the arc 
from P to 
If not, ‘then two positions O, and O, can be found for QO, 
between P and age such that angle PO,Q is equal to angle PO,Q. 
Therefore, P, O,, O,, Q are concyclic and te is a cyclic point 
between rs and Q, whi ch is against hypothesi 
Cor. A.—If the tangents PT and Q’’ at P and Q are equal, 
then thas ‘mast exist a cyclic point on the arc POQ. For, the 
angles 7’PQ and 7QP, —— the limiting values of the supple- 
ment of the-angle POQ. when 0 coine i with Pand Q, respec- 
tively, ngueeed be equal in a non-cyclic ar 
Cor. B,—lf the angle POQ antoowenaly 3 increase as O moves 
from P to @, then the circle PO will fall below the arc from P 
to O and above the are from O to Q. 
Def—An arc POQ will be called positive, if the angle 
POQ continuously i increase, as O moves from P to Q along the 
seta and it wil called ‘negutive if the angle POQ continu- 
ously decrease, as 0 moves from P to Q. If the arc POQ be 
eas then evidently the arc QOP is negative and vice 
c= C.—If the tangents at P and Q toa positive non-cyclic 
are PQ, meet above nl arc, then QT is greater than PT. 
Theorem III—\f O be. any point on a non-cyclic arc POQ, 
then on circle Poo. passing through P and two consecutive 
points at O, will fall entirely below or oe the given are, 
according as the are POQ is positive or nega 
t place, it is evident that oe site POO will lie 
