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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178. In two straight lines at right angles to each other
two points are given in position, and two other points at equal
distances from them are taken: it is required to find the nature
of the curve to which the straight line joining the latter points
is always a tangent.
179. If normals and tangents to a parabola be always
drawn at the extremities of any parameter, it is required to
find the loci of their intersections.
180. Determine the nature of the curve which shall include all isosceles triangles of a given area, having their bases
and perpendiculars coincident in directions.
181. Two straight lines ate drawn through two given
points whose equations are = m and x= n, perpendicular to
the axis of x, and their rectangle is of invariable magnitude:
it is required to find the curve to which the line joining their
extremities is always a tangent, both when they are drawn
in the same and in opposite directions.
182. From two given points two straight lines are drawn
parallel to each other so that the difference of their squares
is of invariable magnitude, it is required to find the nature
of the curve to which the straight line joining their extremities is always a tangent.
183. Required the nature of the curve analogous to that
of the last example, when the sum of the squares of the lines
cut off by the tangent is invariable.
184. Two points are assumed in two given lines and a
straight line intersects them so that the rectangle of the distances of its intersections from the said points is invariable:
required the curve to which the straight line is always a
tangent.
185. Three points A, B and C being given in position,
and AB, AC being drawn, a straight line TV meeting these
lines is so situated that the rectangle BT x C V has to the
rectangle AVx AT an invariable ratio: required the nature
of the curve to which this line is always a tangent.