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

SINGULAR POINTS OF CURVES.

161.

has no real value, and f(x2) two real values; it will be merely
reversed in position: thus, instead of fig. 68, we shall have
fig. 70; and similarly the other figures.
These are all the cases that can occur when f(x) has two
real values, and only two, for some values of xa and no real
values for others.
230. It sometimes happens that f (m) and f (,) are both Conjugate
impossible, no matter how near xz and x2 may be taken to a, points.
and yet f(a) a real quantity; in such a case the point P is a
point belonging to the curve, since its co-ordinates satisfy the
equation to the curve, and we define the curve to be the
assemblage of all the points whose co-ordinates satisfy that
equation: but no points in the immediate vicinity of P belong
to the curve; P therefore is an isolated point of the curve,
completely detached from the other points. Such a point is
usually called a conjugate point.
231. If f(,v) and f/(x) have each several values while Multiple
f(a) has only one value, then several branches of the curvepoints.
must meet at the point P; in such a case P is called a mnul
tiple point.
We shall not extend this enumeration of cases any farther,
as it will be easy, after what has been explained, to make out
the form of the curve in any case that may present itself.
232. The following are examples of the cases that we Examples.
have just discussed.
a3
(I) y=a+ (-a
( - a)
Here f (a) = co, f () = positive, f(x) = positive; therefore fig. 50 represents the curve.
x - 2a
(2) =Y = - 2
Here f(a) = co, f(vx) = negative, f(te) = negative;
(fig. 51).
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