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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165. If 0 be the angle which the normal at any point
of a curve makes with the axis of x, then will the radius
dy ddx
of curvature at that point be expressed by d or ---
d sin 0 d cos 0
required a proof.
166. Apply the expressions above written to prove that
in an ellipse the radius of curvature is equal to
a2b2
(a2 cos2 0 + b2 sin2 0)1
167. Determine the magnitude of the radius and the
co-ordinates of the centre, of the circle of curvature in each
of the curves defined by the following equations:
(1) y3=a2X: (2) (1+ -2)y=x:
(3) ay = (b3- 3): (4) a3 =, (v + b)2:
(5) a(y-b)2 = x(< -a)2: (6) ay2= ' (x + b)2:
(7) (a-?) y = (a + ) 2: (8) 2y2= 2 (a2- i2):
(9) 3 + y=a3: (10) y=sinx: (ll) y=tan x.
168. Determine the radius of curvature of the curve
whose equation is y3 = ax2 - x3, at the points where x =o and
X _ a.
169. Find that point in a parabola at which the curvature
is one fourth part of the greatest curvature.
170. If a circle of curvature to the vertex of a parabola
be described, and another circle touch that and the. arcs of
the parabola, and so on continually: it is required to prove
that the radii of these circles are as the odd numbers 1, 3, 5, 7,
&c.
171. Find the points in which the circles of curvature
at the extremities of the principal axes of an ellipse intersect
each other, and also the angles at which the circles are there
inclined to each other. Is the intersection always possible?