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

159

133. Required the least triangle which can circumscribe
any given right segment of an ellipse.
134. Determine the dimensions of the greatest spheroid
and hemispheroid that can be inscribed in a given right cone.
135. Determine the least triangle that can circumscribe
any segment of an ellipse.
136. Find the content of the least cone that can be circumscribed about any right segment of a spheroid.
137. Find the least triangle that can be circumscribed
about an ellipse of given dimensions.
138. Required the dimensions of the least ellipse that can
be circumscribed about a given trapezoid.
139. Inscribe an ellipse in a given parallelogram so that
its area may be the greatest possible.
140. Inscribe the greatest ellipse in a given segment of
a circle, one axis of the ellipse being parallel to the base of
the segment.
141. In a given ellipse inscribe the greatest semi-ellipse,
the extremity of the major axis of the inscribed figure touching the extremity of the axis minor of the given ellipse.
142. Inscribe the greatest spheroid in a given paraboloid,
and compare their axes and volumes.
143. If a curve be defined by the equation amx'= by",
where x and y are the distances of any point of it from two
given points, it is required to draw a tangent and a normal
to it.
144. If m =n = 1, and the curve become a circle, it is
required to determine the position of the tangent, and to shew
that the normal always passes through the same point.
145. If the arc of a cycloid be considered a curvilineal
abscissa and the corresponding ordinate be always equal to
it, it is required to draw a tangent to the curve, and to find
where it meets the co-ordinate axes.