An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

80

75. The equation of a circle being
r = b cos 0 + V/a2 -b2 sin 0;
it is required to find when the circumference is concave or
convex towards its radius vector.
76. Find whether the spiral defined by the equation
r-=aO, is convex or concave towards its pole.
77. Determine whether the curve whose polar equation is
r = a (cosec 0- sec 0), is convex or concave towards its pole.
78. If the centre of the generating circle be the pole, the
equation to a cycloid is r2= a2 (1 + 02+20 sin 0): find the
direction of its curvature.
79. Find the direction of the curvature of the curve
defined by the equation r = a (cos 0 - sin 0).
80. Determine whether the curve defined by the equation
r cos 2 = a cos 0, is convex or concave towards its pole.
81. Find the direction of the curvature of the curve
whose polar equation is r2 -2 a2 cosec 20.
82. Prove that the curve whose polar equation is
r cos 0 =a cos 20,
is always concave towards the pole.
83. Shew from the equation r cos 0 =2 a sin2 0, that the
arc of the cissoid of Diodes is always concave towards its.
radius vector.
84. If the notation of Article (205) be retained, prove
that in the spiral of Archimedes, the radius of curvature
(a2+r2) 2r (a2+r2)
CP= - and the chord of curvature PV-= (a+2)
2 a + r2 2a2+85. In the reciprocal spiral whose equation is r = a -, it
is required to prove that
CP- r (a2 + and PV 2a (a2+r)
a" a2