An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

17o

248. One circle rolls upon another equal to it, it is required to determine the locus of any point of its circumference.
249. Of two ellipses having at first their vertices coincident and their axes in the same directions, one rolls upon
the other, it is required to find the loci of the vertex and
focus of the moveable ellipse.
250. From the vertex of a parabola a straight line is
drawn inclined at an angle of 450 to the tangent at every
point: find the equation to the curve which is the locus of
their intersections.
251. To determine the greatest ellipse inscribable in a
given triangle touching one of the sides at a given point.
252. Prove that the line, joining the points of contact
of rectangular tangents to the cardioide, passes through a fixed
point.
253. Determine the locus of the intersections of rectangular tangents drawn to the cardioide.
254. If a circle be inscribed in an ellipse and a common
ordinate to the minor axis be drawn, it is required to prove
that the locus of the intersections of the normals at the corresponding points of the ellipse and circle is a concentric circle
whose radius is equal to the sum of the semiaxes.
255. If about a given point D the angle BDC revolve,
cutting the straight lines AB and AC given in position in the
points B and C: it is required to prove that the curve which is
always touched by the straight line BC is a conic section.
256. If tangents be drawn making given angles with the
axes of all conic sections having the same given foci, it is required to find the curve which is the locus of the points of
contact.
257. If a and /3 be two sides of a triangle subject to
the condition expressed by the equation a + a,3= c2, it is required to find the nature of the curve to which the third side