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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57. Determine those conjugate diameters in an ellipse
or hyperbola whose rectangle is the greatest possible.
58. In a given ellipse, it is required to assign the positions
of those conjugate diameters which include the greatest and
least angles.
59. It is required to determine the magnitude of the
greatest parabola that can be made by cutting a given cone.
60. Inscribe in a portion of a parabola cut off by a
double ordinate the greatest possible rectangle.
61. In a given parabola, inscribe another parabola whose
area shall be the greatest possible, the vertex of the inscribed
parabola being any given point in the axis of the other.
62. Of all the straight lines which can be drawn from the
vertex of a given ellipse to the circumference of its circumscribed circle, find that of which the portion intercepted
between the circumferences of the ellipse and circle shall be
the greatest possible.
63. Two circles of given radii intersect each other, find
the longest straight line which can be drawn through either
point of intersection, and terminated by the circumferences.
64. In an ellipse, it is required to find the point at which
the normal makes the greatest angle with the line drawn from
it to the centre.
65. If M be any point in the diameter of a circle whose
centre is C, PMQ a chord and CP be joined; prove that the
chord PQ is a minimum and the angle CPQ a maximum,
when PQ is perpendicular to the diameter.
66. Perpendiculars A a, Bb are drawn from the extremities A and B of the axis major of a given ellipse, and
through a given point P in its circumference is drawn a
straight line aPb intersecting them in a and b respectively:
prove that the rectangle A a. Bb is a maximum when aPb
is a tangent to the ellipse at P.