396 ©. - Journal of the Asiatié Society of Bengal. [August, 1908. 
are P’Q’ is also positive, For, PQ’ is positive, therefore, P’Q’ is 
also positive 
Uor. B.—If in an are POQ there be a cyclic point, then angle 
Ans cannot continuously increase or decrease as O moves from 
if there be a cyclic point S, on arc PQ, then in ca 
Seiten Beook of 8, four distinct points, PRS eld, me 
exist lying on a circle. Hence in the are P’Q’, the angle P’ OQ! 
cannot continuously i increase or decrease as O moves from P’ to Q’. 
Hence in the arc POQ the angle POQ cannot continuously 
ously increase or decrease as 0 moved from P’ to . 
Cor. in an arc POQ there be acyclic point 8, thena 
— are P’SQ’ can always be found such that the tangents P’7’, 
8 ne Q’ are equal, 
in the neighbourhood of S, four distinct points P’, R’, 
S’ are obtainable lying on a circle, The point S will te between 
P’ and Q’. Keep R’s’ fixed and vary the circle till FR or 
coincide. Then keep these latter coincident points fixed, wit vary 
the circle till the other two points coincide. 
or. D.—If POQ be a positive non-cyclic arc, then the radius 
of curvature at O eeceueotnty i increases as 0 moves from P to Q. 
Cor. B.—lf in an arc POQ there be a cyclic point S, then 
the radius of curvature has a maximum or minimum value at S. 
For, the circle of curvature at § as it passes through four con- 
Thus if are PS be positive, arc SQ will be negati nd vice 
versa. The circles of curvature at P and Q will, Echunek pe. both 
be less or both be greater than the circle of curvature at S. 
Theovem V.—If POQ be a non-cyclic positive are, and § any 
fixed point on it, then vie POS will continuous y decrease as O 
If O be taken between § and Q, then the circle PSO will 
evidently fall below the given arc from P to S, and above the 
given i: from § to O, and again below the given arc from O to Q 
g. 4 
(Fi 
S 


a 
P Fig-t. 
if O' be another position of O nearer @: then evidently 
aoe PO’ S is less than angle POS. Hence angle POS continu- 
ously diminishes as O moves from § to 
