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

CHAP. XIV.

MISCELLANEOUS EXAMPLES.
BY means of the Principles laid down and exemplified in
Articles (290)-(343), the solutions of the following Theorems
and Problems may be obtained.
1. If the equation of a sphere be x2 + y2 + z2 = a2, then
will the equation to its tangent plane be xx' - yy' +zz' =a2
required a proof.
2. Determine the inclinations of the tangent plane last
found to each of the co-ordinate planes.
3. Let it be required to construct the plane touching
a sphere by means of the equation above found.
4. To a given sphere it is required to draw a tangent
plane which shall pass through a given point, and to shew
how far the position of such plane is indeterminate.
5. Required the equation to a plane passing through two
given points and touching a given sphere.
6. Draw a tangent plane to a spheroid defined by the
V2 y2 2
equation + -Y + - =1: shew how it may be constructed
a2 a2 c
and determine its inclinations to the co-ordinate planes.
7. To the surface last mentioned it is required to draw
tangent planes which shall pass through one and two given
points.
8. A conical surface whose vertex is the origin of coordinates being defined by the equation = e/22+ y2,it is
required to prove that its tangent plane will be defined by
the equation zz' =e2 ( x' +yy').