392 Journal of the Asiatic Society of Bengal. { August, 1908. 
Section ].—Finite Arc. 
e c 
then in their final position. thev are called so many consecutive 
points at O, lying on the line X of given species, and the line 
is said to pass chicuph so many consecuti\e points on PQ. 
Thus if X pass throngh * r’ consecutive points at - ~ in 
the neighbourhood ea O, ‘r’ distinet points must s be 
obtainable on PQ, at finite Pa from each nee age 
which the line X of aaron species 
The straight line passing Zacouch, two consecutive pointe on 
PQ at O, is the tanvent a 
If a straight line ie throngh three consecutive points on 
PQ at O; then O is called a point of inflexion. Thus in the 
ee Pow te F of a point of inflexion, three distinct points 
must mare be obtainable, throngh which a straight line 
asses. 
. cle passing throngh three Sooners points on PQ 
at O is ited the circle of curvature at 0. 
If acircle pass through four consecutive points on PQ at 0, 
then “sf ne be called a cyclic point. 
he radins of the circle of curvature at a cyclic point be 
infinitely large, it is enlled a point of undulution 
It is s hardly plea to define a point of undulation as one 
where the tangent meets the curve at four consecutive points. In 
the neighbourhood of a point of undulation asiraight line can meet 
the curve at only two real points, and therefore by the variation 
of such a straight line we can ultim: itely get only two consecutive 
points. Therefore, of the four points«t which the tangent is said 
to meet the curve, at a point of undulation, only two can be strictly 
called rie tora the other two being only conjugate points. 
10 
ae get four coneyclic pointe on i She curve, and by varying 
in ulti 
gen 
moves from P to Q, along line PQ, tien the line ren called a 
continuous curve. On a continuous curve, ther ems we cannot 
us we cannot have a noe or a cnyp anywher. 
A continuous curve FQ is called a convex ae or simply an 
if no straight line cut it at more than two points, ‘The 
sraight line PQ is walled its chord, Ane. other chord P’Q’ is 


