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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86. Determine the figure of the logarithmic spiral whose
equation is y = a.
87. Describe the lituus as defined by the equation r20=a2.
88. Trace the circle as defined by the equation
r = a (cos 0- sin 0):
and find the values of its maximum and minimum radius vector.
89. Describe the figure of the rectangular hyperbola as
expressed by the equation r = a (cosec 0- sec 0).
90. Trace out the figure of the cissoid of Dioces by
means of the equation r cos 0= 2 a sin2 0.
91. Find the form of the cardioide by means of its
equation r = a (1 + cos 0).
92. Required the form of the figure called the trisectrix,
whose equation is r = a (2 cos 0 + 1).
93. It is required to trace out the figure of a rectangular
hyperbola by means of the equation r cos 20=2 a cos 0.
94. Required to trace out the figure belonging to the
equation r sin (a + 0) = a sin a.
95. It is required to give graphical representations of
the curves defined by the following equations:
(1) r=aO: (2) r=asin20:
(3) r2=-a2 tan 20' (4) r= a (sec - tan 0):
(5) r=acos 0: (6) r=a sec 0 tan2 0:
(7) r=atan0: (8) r= a sec 0
(9) r2=a2(tan2 0-1): (10) r=a(+2 sin 10):
(11) r= a log tan (45 + -0) (12) r(e + e- )2 a:
(13) r2(e —1)2=4a2eO: (14) r(0 2- )=a02:
(15) r sin 20=a sin 0 sin 30: (16) r (cos 0 + cot 0) =2 a.
96. Trace the curve whose equation is y3 -3 axy + 3 = 0,
3ay
by assuming y =, and therefore x = 32.
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