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

TRACING OF CURVES. 153
points of contrary flexure; which indeed the figure shews, for
there must evidently be a change of flexure somewhere between
C and A, since the concavity of the curve is turned upward
at C and downward at A; also there is evidently a change of
flexure at B. There is no change of flexure elsewhere, and
therefore the figure represents the precise form of the curve.
221. In order to determine still more minutely the form We may
find the
of the curve, we may find the points where the ordinate is maximum
a maximum or a minimum; to do this in the present case and minimum ordiwe have nates, in
dy m3 - a - _a termine the
- ~ a - form of the
d3 a (,aO - a)2 curve more
minutely.
and therefore, when y= 0, we have
dx
- a -- a2C2 = 0,
which gives,: = 0, and x = - / + a2, or (1 /5);
2 4 2
which three values correspond to the points A, Q, and P.
222. When the curve has asymptotes, it is often useful Asymptotes
are useful in
to find them; for they enable us to see more clearly the nature givingusan
of the infinite branches of the curve; for example, take the idea o the
manner in
equation which the
branches
(aM' + a) of a curve
y= - (? go off to
(I - a)' infinity.
Forming a table as in the former case, we find xample
dy
tPI Y
-a 0 0
+
a To:~X t $c