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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80. In the same surface, it is required to prove that
so long as the distance of the tangent plane from the centre
is invariable, the rectangle of the greatest and least radii of
curvature is constant.
81. Determine the points in the surface of an ellipsoid
at which the curvature of every normal section is of the same
invariable magnitude.
82. Find the radii of greatest and least curvature at any
point of the curve surface defined by the equation xyz = a3.
83. Required the same for the elliptic paraboloid whose
X2 y2
equation is z=- +.
a b
84. In any curve surface, the radii of greatest and least
curvature will become equal in sign and magnitude when the
1 p-' 1 -_ +q'
conditions expressed by the equations = - = +r S t
are fulfilled: required a proof.
85. If the radii of greatest and least curvature, having
different signs, become equal in magnitude, it is required
to prove that (1 + p2) t + (1 + q') r = 2pqs.
86. It is required to prove that the. surface whose
equation is xS2 - y2 - ( - C)2 = 0, is at the same time a conical
surface and a surface of revolution, by satisfying the general
partial differential equation of each species of surface.
87. The equations of the curve of double curvature
formed by the intersection of the surface of a sphere with
that of the cylinder whose diameter is its radius, being
y2 2=2ax- -x2 and z2=4a2-2ax,
it is required to prove that the equations to the tangent line are
yy + x = a ( + x') and zz'= 4a2 - a (x a').
88. Shew how the line determined in the last example
may be constructed, and determine where it intersects each of
the co-ordinate planes, and also the loci of these intersections,