An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

FLEXURE OF CURVES.

127

1 downNow -- 2 is essentially a positive quantity; wards, to
jdy \ the right or
1 + I to the left.
day
hence d* has always the same sign as d -
dx
Now if dk be positive, xI increases when x increases, and
dx
therefore the concavity of the curve must be turned upwards,
as in (fig. 9), but if dx be negative, + diminishes when w
dx
increases, and therefore the concavity of the curve is turned
downwards, as in (fig. 10). Hence the concavity of the curve
d2 y
is turned upwards or downwards according as d- is positive
or negative.
And in the same way it may be shewn that the concavity
of the curve is turned to the right or to the left according as
de
de is positive or negative.
180. Hence if at any point -- changes its sign, the Proposidx" tion.
To deterconcavity of the curve will change its direction, if the change mine pointe
of sign is from + to - the concavity which was before turned of contrary
' -n i. i,, - flexure.
upwards will begin to be turned downwards, as at the point P
(fig. 11); and if the change be from - to + the reverse will
be the case, as at the point P (fig. 12).
A point where the concavity of the curve thus changes its
direction is called a point of contrary flexure.
dsy
Hence, by Lemma XVII, Y must be 0 or co at a point
of contrary flexure; and therefore, to determine where such
points are on the curve, we have only to find what values of
d2y
0x make -d 0 or co, and try whether x, in passing through