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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40. Draw a tangent plane to the surface whose equation
is z =f (2 +y2), and find the portions it cuts off from the
co-ordinate axes.
41. Apply the equation deduced in the last example
to construct the plane touching at any point the surface whose
equation is (V/ + 2 — c)2 = a2 - 2.
42. The equation to a curve surface being
z2_ C2 = 2 (2y)
it is required to prove that the portion of the axis of z cut off
by the tangent plane at any point varies inversely as z.
43. Adapt the result of the last example to the cases
of the ellipsoid and hyperboloids of one and two sheets.
44. If bz- cy =f(bo- ay) be the equation of a curve
surface, then will its tangent plane at every point be perpendicular to the same fixed plane: required a proof.
45. The equation of a curve surface being
x - a \ o - a)
it is required to draw a tangent plane to it, and to prove that
it will always pass through the same fixed point.
46. If log T ) = d (), be the equation to a
curve surface, it is required to prove that the plane touching
it at any point cuts off from the axis of z a part propor1
tional to -
47. The square of the perpendicular let fall from the
centre of an ellipsoid upon the tangent plane at any point,
is equal to the sum of the squares of the projections of the
three semi-axes upon the normal at the same point.