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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33. The content of a hollow sphere is equal to the frustrum of a cone whose bases are equal to the exterior and
interior surfaces of the solid, and height is equal to the difference of their radii.
34. A sphere has to its inscribed equilateral cylinder,
its inscribed equilateral cone and its circumscribed equilateral
cone, the ratios 4//2: 3, 32: 9 and 4: 9 respectively.
35. The right cylinder and equilateral cone circumscribing
a sphere are to one another as 2: 3, whether we consider their
convex surfaces, whole surfaces, contents, bases or heights.
36. It is required to divide the area of a given circle
into n equal parts by means of concentric circles.
37. Divide the area of a circle into n equal parts whose
perimeters shall be equal.
38. Divide a given sphere into n equal parts by means of
concentric spherical surfaces.
39. The whole surface of a cone is three times as great
as the area of the base; find the vertical angle.
40. Find the radius of the sphere which can be inscribed
in a given cone.
41. Compare the sum of the contents of all the spheres
that can be inscribed in a cone, having their centres in its
axis, with that of the cone itself.
42. Divide a right cone into two parts, whose contents
shall be in the ratio of m: 1, by a plane parallel to its base.
Also, that their convex surfaces may be in the same ratio. -