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

it intersects the line drawn from the origin to the point of
contact is invariable.
73. Draw a tangent to the curve whose equation is
2amym l= y2m - a2m; and prove that the distance of the point
of contact, from a given point in the axis of x, varies as the
segment of the said axis between that point and the tangent.
74. If any tangent be drawn to an ellipse, and four perpendiculars be drawn to the axis major, from the centre of
the ellipse, the two extremities of the axis major and the point
of contact, these four perpendiculars are proportionals.
75. Find the general equation to the tangent at any point
of a curve when the co-ordinates are oblique; and apply it to
an ellipse where the co-ordinate axes are parallel to any pair
of conjugate diameters.
76. If a and b be the semi-axes of an ellipse or hyperbola; the product of the tangents of the angles at which any
system of conjugate diameters are inclined to the axis major is
b2 b2
-a2 or -: required a proof.
77. If a system of conjugate diameters of a conic section
be produced to meet a tangent whose position is given; the
rectangle of the parts of the tangent intercepted between those
diameters and the point of contact, is independent of their
position: required a proof.
78. The equation to a straight line drawn from the point
a, j3 of a curve whose equation is y =f(x), perpendicular
dx ie
to its tangent is y'-,f3=- (ad-a): required a proof.
dy
79. In the ellipse and hyperbola referred to the centres
and principal axes; prove that the equations to the perpendiculars upon the tangents from the points a, /3 of the curves are
(' - a) aly + (y' — ) b2 = 0.
80. The point of intersection of the rectilineal tangent
with the perpendicular upon it from a point a, 3 of any curve
H