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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181. In the curve whose equation is m (e -L- 1) x=-y9
it is required to prove that the portion of the ordinate included
between the curve and a straight line through the origin
making equal angles with the axes varies as the cotangent
of the curve's inclination to the axis of x.
182. Describe a circle with a given centre so as to touch
a given parabola; both when the point is within and without
the parabola.
183. A straight line drawn through the focus of any
conic section to the point in which a diameter meets the directrix will be perpendicular to a tangent at either extremity
of that diameter: required a proof.
184. If from any point in a line parallel to the axis
of a common parabola, two tangents be drawn to the curve;
it is required to prove that the sum of the cotangents of their
inclinations to the axis of x is invariable.
185. If y=mx+csin- be the equation to a curve;
a
it is required to shew that the sum of the tangents of the
angles in which two ordinates at the distance a from each
other cut the curve, is constant.
186. If with a radius equal to the line joining the extremities of the axes of an ellipse, a concentric circle be described,
two tangents drawn to the ellipse from any point in the circumference of this circle will be at right angles to each other:
required a proof.
187. If with the co-ordinates of any point in an elliptic
quadrant as semi-axes a concentric one be constructed, this
will be touched by the chord of the first: required a proof.
188. Shew that the ellipse whose equation is
Y2- 4 a + 22 = 0,
always cuts at right angles the parabola whose equation is
y2=m (a —x), whatever be its latus rectum.
189. Prove that the part of the tangent of an hyperbola
intercepted between the asymptotes is equal to the diameter to
which it is parallel, and is bisected by the point of contact.