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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190. If two equal parabolas have a common axis, prove
that a straight line touching the interior and bounded by the
exterior will be bisected in the point of contact.
191. Two parabolas having a common axis, it is required
to find a point in one of them, from which two tangents drawn
to the other shall include a given angle.
192. A curve is constructed by cutting off from the
ordinate of a circle a portion equal to the difference between
its abscissa and ordinate: find the angles in which it intersects
the diameter and the circumference of the circle.
193. If a straight line equal to the sum of the semi-axes
of an ellipse have its extremities in the axes and cut the curve,
and the parallelogram of which this is the diagonal be completed; it is required to prove that the line drawn from the
angle of the parallelogram to the point of intersection will
be a normal to the ellipse.
194. If between a rectangular hyperbola and its asymptotes any number of concentric elliptic quadrants be inscribed,
the rectangle of their axes will be invariable: required a proof.
195. The ordinate MP of an ellipse whose major axis
is AB is bisected in Q and AQ is joined and produced so
as to meet the tangent at B in T: prove that the straight
line TP will be a tangent to the curve.
196. An ellipse and hyperbola being constructed on the
same axes, if from any point in one of the curves two tangents
be drawn to the other, the straight line which passes through
the points of contact will be a tangent to the first curve:
required a proof.
197. AP is a portion of a common parabola, PT a tangent at P, PG a normal and TR a perpendicular to the axis at
T: prove that if GP be produced to meet TR in R, GR
will be- equal to the radius of curvature at P.