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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90. The equations for determining the tangents to an
hyperbola, drawn from a given point a, t3 without it, are
a^y2 2- 2bX= ' and a2ty a bC a a2b2;
and that of the straight line joining the points of contact is
a23 y- bax = - a2 b: required a proof.
91. From a given point without a circle, two equal
straight lines may be drawn to touch it: prove this, and find
their inclination to each other.
92. If from any point in the directrix of a parabola,
two tangents be drawn to the curve, they shall be at right
angles to each other: required a proof.
93. From a given point in the axis of the curve whose
2 2 2
equation is (a)3 + (by)3= (a2 - b2)3, it is required to draw
a tangent to it; and to find the limitations under which this
is possible.
94. From a point whose co-ordinates are a and 2a, it
is required to draw a tangent to the curve whose equation is
27ay2=4 (x-2 a)3; and to determine its length.
95. Find the angle included between two tangents to
an ellipse drawn from a given point without it; and determine
the distance from its centre at which this is a right angle.
96. Find the portions of the axes, through the vertex
of an ellipse or hyperbola, cut off by the normal at any point
of the curve.
97. Find the points where the normal to a cycloid meets
the co-ordinate axes, and its lengths intercepted between them
and the point of contact.
98. Determine the portions of the axes, through the
centre of an ellipse or hyperbola, cut off by the normal at any
point; and prove that the rectangle of the parts of the normal
intercepted by them is invariable.
99. Find the equation to the normal of the curve
defined by x" +y"= a (x - y); and determine the position of a
fixed point through which it always passes.