Also, since by the continual diminution of h the first
d^y
term -^-j . h^ will become greater than the sum of all the

succeeding terms of the series, it follows that the alge-
braic sign of this series will be the same as that of its
first term, and therefore, according as a curve is convex
or concave to the axis of abscissas, the second difieren-

/d^y\
tial coefficient ( ^2) will be positive or negative.
SINGULAR POINTS OF CUETES. 1^7

Hence, conversely, a curve will be convex or concave
to the axis of abscissas according as the second differen-
tial coefficient of the ordinate is positive or negative.
Or, more generally,

A curve will be convex or concave to the axis of ab-
scissas according as the ordinate and its second differen-
tial coefficient have the same or different algebraic signs.
121. A singular point of a curve is a point which has
some peculiarity which the other points do not generally
have ; as, for instance,

1. The point at which a tangent line is either parallel
or perpendicular to either axis.
2. The point at which the curve changes from being
convex to concave to the axis of x, or from being con-
cave to convex. The point at which this change of curv-
ature takes place is called a point of inflection.
3. A multiple point; that is, a point at which two or
more branches of a curve intersect each other.
