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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157. Determine the locus of the centre of a circle which
shall touch a given straight line and a given circle: also when
it touches two given circles.
158. Required the locus of the intersection of two tangents to a given parabola, when the angle included between
them is invariable.
159 Required the locus of the intersection of two tangents to a parabola inclined to its axis at angles, the product
of whose trigonometrical tangents is invariable.
160. Determine the locus of a point from which two tangents being drawn to a given parabola, their sum or difference
shall be invariable.
161. Find the locus of a point from which two tangents
being drawn to a given parabola, their lengths shall have
an invariable ratio.
162. Required the locus of the point from which the
rectangle of two tangents, drawn to a given parabola, shall
be of invariable magnitude.
163. Determine the locus of the intersections of the tangents to an ellipse or hyperbola, which shall be inclined to the
principal axes at angles, the products of whose trigonometrical
tangents are invariable.
164. Required the nature of the loci, when the tangents
to the ellipse or hyperbola intersect each other at right
angles.
165. Find the locus of the intersections of two tangents
to an ellipse or hyperbola, when the sum of the angles made
by them with the principal axes, is equal to a right angle.
166. Required the loci of the intersections of the tangents to an ellipse or hyperbola, which are parallel to conjugate
diameters.
167. Determine the locus of a point from which tangents
drawn to a given ellipse, shall have an assigned ratio.
X