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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chord which connects the two points where the two tangents cut
the circle is a third tangent to the parabola.
293. If A and B be two fixed points and P a point in
any curve such that AP + BP is a minimum: it is required
to shew that the straight lines AP and BP make equal angles
with the normal at P.
294. If from two given points, two straight lines be
drawn to a point in a given curve of any order so that their
rectangle may be the greatest or least possible: it is required
to prove that these lines will have the same ratio as the cosines
of the angles which they make with the curve at the point
of intersection.
295. If (- = secmO be the equation to a spiral, it is
\a/
required to prove that the equation to the spiral which is
always touched by a perpendicular through the extremity
m
of the radius vector of the former is (-) = sec ( +
a \m+l 1
296. There are three geometric series, whose first terms
are 1, 2, 3, and if the fifth, eleventh and thirteenth terms
of the first, second and third series respectively be added
together, the sum amounts to 101: it is required to find the
three series such that the sum of the fourth, sixth and seventh
terms respectively may be the greatest possible.