The second differential coefficient will be 2^ositwe for
every value of x either greater or less than b. The
curvtf will therefore be convex toward the axis of x.
If we make Xz=:b-\-h, or a;=6— A, we shall have, in
either case,

y=a-f?;
and hence y will be greater for x=:b than for any other
value of a; either greater or less than b. Hence the value
x=b' is the abscissa of the greatest ordinate. The
branches PP' and PP" are not considered as parts of a
continuous curve, for the general relations between x
and y which determine each of those parts are entirely
broken at the point P, where x=zb.
Ex.206. Let {y—aY={x—bY; (1)

then



and



y=:a±{x—b)'^,
c?V_ 3



4(a!-5)2
If we put the first differential coefficient equal to zero,




182 DIFFERENTIAL CALCULUS.
we shall have a; =5, and substituting this value of a; in
the equation of the curve, we find y=:a. Hence the
point whose co-ordinates are x=h,y=ia, is a point of the
curve at which a tangent line is parallel to the axis of x.
Now for a;=:5— Athe ordinates are imaginary, and for
a;=5+A the values of
d'^y 3

Hence there is a cusp at P,
where the upper branch, PP',
which is convex towarqjthe axis

3
of X, is determined by + — 7=,

and the lower branch, PP", is concave, determined by
3

If for X in (Eq. 1) we substitute 5+ A, we shall have

3
y=a±Ji^;

and giving to A a succession of values, which shall have
small differences, we may trace the curve.
Ex. 207. Let {y—x') —x^ be the equation of the curve ;

6

then 2/=a!2±a;^, (2)

and ^=2±ii.x^.
