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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57. The equation of the parabola being r = a cosec2 -0;
it is required to prove that SG= -r / and PG=r \/
a a
58. In any conic section, if normals be drawn at the
extremities of a chord passing through the focus so as to
intersect each other; the sum of the squares of the parts of
the normals between the curve and their intersection will have
a given ratio to the square of the chord: required a proof.
59. The notation of Article (200) being retained, if the
equation of a spiral be r = a + b sec 0, it is required to shew
that L LSX= 900 and ST= - b.
60. If the equation of a spiral be rs =a2 tan 20; it is
required to prove that z LSX-=45 and ST=o.
61. If a spiral be defined by the equation r2 = a2 sec 20;
it is required to prove that / LSX==450 and ST= o.
62. In a spiral defined by the equation r cos 0+a cos 20=0,
it is required to prove that z LSX= 900 and ST= a.
63. If the equation of a polar curve be r cos 2 0 = 2 a cos 0;
a
then will Z LSX= 45~ and ST= —: required a proof.
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64. The equation of a polar curve being r cos 0 = a cos 20;
it is required to prove that / LSX= 900 and ST= - a.
65. In the curve defined by the equation
r (sin3 0 + cos3 0) = a cos2 0,
it is required to prove that Z LSX= 1350 and ST= - a
66. If a curve be defined by the equation r2=a2(tan20-1);
then will z LSX= 90 and ST = + a: required a proof.
67. If the equation of a curve be
2r (sin3 0 + cos3 0) = a sin 20;
then will z LSX-=135~ and ST=-: required a proofe
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