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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97. It is required to draw a tangent line to the curve
of double curvature defined by the equations
yXv/as sin-1 -sn
y- _ 1 a a a
y= a X_ and —je a+e f f
and to determine all its circumstances.
98. Draw a tangent line to the curve of double curvature formed by the intersection of a right cone with the
cylinder erected on the radius of the base as a diameter, and
determine all its circumstances.
99. Determine the equation of the normal plane to each
of the curves of double curvature mentioned in the last twelve
examples and construct it:
100. Find the equation of the osculating plane in each of
the curves of double curvature just referred to, and to its
intersection with the normal plane.
101. In any curve of double curvature, if a, 3, 7 be the
co-ordinates of the centre of spherical curvature, it is required
to prove that they may be determined from the three equations
dy da- ( — y + (/3 -) d = 0,+
d,2 d2 1 (dy)2 (d) 2
( ~-y) + ( 3 I ) d = 2 + -d z
w y) +(7-^:7-1 iXj + 2IWJ
d' dz f. 3^ d dy dx y d dx dC2
and the radius of the sphere from the equation
2 =(a )2 + ( - y)2 + (y - Z)2.
102. If the proposed point of the curve be the origin
of co-ordinates and the tangent line be considered the axis
of x, it is required to prove that a =0,
d3z d3y
dX3 dx
d2zy d3 d d2zdy' 7 d2z d3y d2y d3z'
d 2 d w2 d2 da Yx d w d x2 d x3 dx d