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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155. If x', y' be the co-ordinates of the circle of curvature at a point x, y of any curve: it is required to prove
that the equation to the circle is
(( - q +(l +P2) 2 + (y -) q -(1 +p2)}2(l p2)3
156. Required the equation of the circle of curvature of
the curvedefined by the equation - = cos (-)
a a
157. Determine the equation of the circle of curvature
of the curve defined by x = sec 2y; and thence find its magnitude and the position of its centre.
158. Find the equation to the circle of curvature at any
point of the tractrix.
1dy / s2
159. If ym- =V/a - ym be the differential equation
of a curve, it is required to shew that the radius of curvature
at any point is m times as great as the normal.
160. Determine the radius and chords of curvature of
the curve, whose ordinate is equal to the circular arc of which
its abscissa is the versed sine.
161. Required the radius and chords of curvature parallel
to the axes, of the curve traced out by taking a portion of the
radius of a circular arc drawn through its extremity, always
equal to the sine of the arc.
162. In any curve, find an expression for the distance of
the centre of curvature from any given point a, b; and also
from the vertex.
163. Determine expressions for the portions of the axes
of any curve included within the circle of curvature; and
apply them to the case of the common parabola.
164. Apply the expression for the chord of curvature
through a given point, to find the diameter and principal
chords of curvature in each of the: conic sections.