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

56

that the part of it between the point of contact and the axis of
x is of invariable length.
y2 - a2 a a
65. If the equation of a curve be = --- + -log-;
4a ~2 y
find the equation of its tangent, and thence prove that the
difference between the tangent and subtangent belonging to
the axis of I is invariable.
y
66. Of the curve defined by ea =x- 2 a2, find the equation
to the tangent; and thence prove that the sum of the tangent
and subtangent varies as the rectangle of the corresponding
co-ordinates.
67. Find the equation of the tangent to the curve defined
by (m, —)m-lym = mnam-l'x: and thence shew that if AD be
drawn from the origin perpendicular to the axis of x and meet
the tangent PT in D, ADm = a"-'AT.
68. Required the equation to the tangent of the curve
whose equation is v V/2 (IV2 + y2) = a (x2 - y2); and find where
it intersects the axes of co-ordinates.
69. In the curve whose equation is (1 + log x) y = x, find
the equation to the tangent; and thence prove that at any
point, the subtangent: the distance of the intersection of the
tangent with the axis of x from the origin:: the abscissa: the
ordinate.
70. From a point in a curve whose co-ordinates are x
and y, a straight line is drawn making with the curve an angle
whose tangent is a; prove that the equation to the line will be
a+p, ( dy
y' — Y - ('V - x), where p.
1 - ap d
71. If straight lines be drawn perpendicular to the major
axis of an ellipse at its extremities; find the points in which
a tangent at any point of the curve meets them, and prove
that the rectangle of the parts cut off is invariable.
72. In the curve defined by log v2 + y2 = a tan-1 Y find
the equation to the tangent; and prove that the'angle in which