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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106. The equation to an epicycloid being
e r2 -a2 e (r2 a2)
0 = tan- tan -tan- 1 —,
a e2 r a (e- r)'
e2 (r. - a2)
it is required to prove that p2 = e (- a, where e= a+ 2b,
-ae"
if a= the radius of the base and b = the radius of the generating circle.
107. It is required to find the evolutes of the epicycloid
and hypocycloid.
108. Find the evolute of the spiral defined by the equation r= aC0.
109. If the equation of a spiral be r- a=aO+a sec' -,
a
it is required to find that of its evolute.
110. If 0oc -, it is required to prove that the polar
subtangent at any point: the corresponding circular arc whose
radius is r:: m 1.
111. If Oocr, the number of revolutions made by the
radius vector varies as the square root of the polar subtangent:
required a proof.
112. If a spiral be defined by the equation r2 cos 2 0= a2,
it is required to prove that the sine of the angle at which the
radius vector is inclined to the tangent oc.
113. Given the radius vector at any point of a parabola
and the angle it makes with the curve, to find the position
of the vertex and the magnitude of the latus rectum.
114. If - lo =log gsin (a- 0 be the equation
7f a 7r
of a polar curve, it is required to shew that two tangents
drawn to the curve at the extremities of a line passing through
its pole will always include the same given angle.