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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37. Find the distances from the origin at which the rectilineal tangent of the curve defined by the equation ay"+"=
bx (a + x)', intersects the co-ordinate axes.
38. Determine the points in which the rectilineal tangent
to the curve defined by the equation x'y2- (a2- x2) (b -x)2,
intersects the axes.
39. In the logarithmic curve whose equation is y-=a",
the distances from the origin at which the rectilineal tangent
1 - x log a
meets the axes of x and y are and ( - log a) y,
log a
and the subtangents are and x y log a: required a proof.
log a
40. Find the points of intersection and the angles of inclination, of the rectilineal tangent to the co-ordinate axes, when
x2 + y2 = 2/2- y is the equation to the curve.
41. Determine where a rectilineal tangent to the quadratrix of Dinostratus defined by the equation y = (1 - x) tan I r x,
intersects the co-ordinate axes.
42. Find the subtangents in the quadratrix of Tschirnhausen whose equation is y = sin -7rx.
43. Determine where the rectilineal tangent to the cycloidal curve defined by the equations x= a (1 -cos 0) and
y = a (mO + n sin 0), intersects the co-ordinate axes.
44. Find expressions for the subtangents of the cardioide
defined by the equations x =_ a (2 cos 0 - cos 2 0) and
y = a (2 sin 0 - sin 2 0).
45. If the equation of a curve be xy (x + y) = a3; it is
required to shew that the rectilineal tangent meets the axes
at distances from the origin represented by
3a3 3a3
-^- - and
2wY + y2 2xy + x2
46. If the focus of a parabola be the origin of co-ordinates, it is required to prove that the perpendicular from it
upon the tangent = 2 /a2 (x"a + y2).