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

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16. In the curve whose equation is y4 -.2axy2 + x4 = 0o
it is required to prove that two values of y are equal when
_x =O and x = a; and that the corresponding ordinates are
tangents to the curve.
17. In the curve whose equation is y4-96a2y2 + 100a2x2
- = o, the values of y become equal when x = +6a and
x = ~ 8a, and the corresponding ordinates are tangents to the
curve: prove this, and shew that the values of x can never be
equal to one another.
18. If y2 = a2 - 2 be the equation of a curve; then, when
= 0, y= -~ a, a positive and negative maximum; and when
y= o, = + a, a positive and negative maximum: required
a proof.
19. If (y-a)2 - (x - b) = c2, then will the values of y
be a c, a maximum and a minimum when x=b: and the
values of x will be b + c, a maximum and a minimum, when
y =a: required a proof.
20. If the equation to a conic section be y2-2exy +x2
-ay=0, it is required to prove that when x=0, y=O, a
ae a
minimum; and when x= 1-e2 Y 1 — a maximum
l-e~ 1 -- e~
a a
also, that when y = )' - = ), a maximum;
2(1 - e) 2 ( - e)
a a
and when y= x= - a minimum or ne2 (1 - e) 2 (1 +- e)
gative maximum.
21. In the curve belonging to the equation (x2 +y2)y
= a (x + y)2, it is required to prove that when x = 2 a, y = a,
a maximum ordinate; and when x = oo 9 y = a, which is neither
a maximum nor a minimum.
22. In the curve whose equation is x4'- a2 S~ a2y2-=0
a
it is required to shew, that when x = -~~, y= + a,
which are maxima; and when y = 0, x = a, which are also
maxima.