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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89. In the same case determine the length of the tangent
intercepted between the point of contact and the plane of zy,
and also its inclination to that plane.
90. The equations of the helix being
- cos - and -= sin -
a b a b
it is required to determine the equations of its tangent line,
and thence to construct it and find all its circumstances.
91. Draw a tangent line to the curve of double curvature formed by the intersection of a sphere and cylinder
whose equations are 2 + + y 2 = a2 and (x- c)2 + y2 = b2, and
determine all its circumstances.
92. If the axes of two cylinders whose equations are
z2 + z2 = a2 and y2 + z2 = b2, intersect each other at right angles,
it is required to draw a tangent to the curve formed by their
surfaces, and to determine all its circumstances.
93. If the axes of two right cones whose equations are
a+ =m m/y2 + and b + y = n\/ 2, intersect each other
at right angles, it is required to draw a tangent to the curve of
double curvature thus formed, and to determine all its circumstances.
94. The equations of a curve of double curvature being
2 + y2 -+ z2 = a nd tan-' - = m cos- -, it is required to draw
O Ca
a tangent line to it, and to ascertain all its circumstances.
95. Draw a tangent to the spiral of Pappus as defined
by the equations
z Y x
- Ccos, - - = sin ( and - = cos dP,
a 4 r v/a2- V 2a
and determine all the circumstances attending it.
96. The equations to the screw of Archimedes being
y 2 ax - 2 and z = b vers-l -, it is required to draw a tana
gent line to it, and to ascertain all its circumstances.