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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78. Investigate the form of the curve expressed generally
by the equation xy2 + ay = b + c a: and distinguish particularly
the cases wherein c is positive, negative and evanescent.
Sir Isaac Newton in his Enumeratio Linearum tertii
Ordinis, designates the curve belonging to the equation last
given, by the name of the Hyperbolism of an Hyperbola, Ellipse
or Parabola according as c is positive, negative or evanescent;
and the reason he assigns for it is the following:
Solving the equation proposed with respect to y, we have
-a /a 2+4ba +4ecx2
y —
Y -
2z
and if the denominator of this expression were an invariable
quantity, it is obvious that the equation would belong to
an hyperbola, ellipse or parabola, according as c is positive,
negative or evanescent: whence, if such constant quantity be
replaced by the variable quantity 2x, the conic section becomes
hyperbolized by having in each case an infinite branch at the
origin of its co-ordinates.
79. Trace the curve defined by the polar equation
r cos2 0 = a sin 0. See the second Diagram of page (304).
80. Find the figure of the polar curve defined by the
equation r92 cos3 0= a" sin 0. See the Diagram of page (303).
81. Trace the rectangular hyperbola as defined by the
equation r2 sin 20 =2a". See the first Diagram of page (289).
82. Investigate the figure of the curve whose equation is
r" cos4 0 = d cos 20. See the Diagram of page (294).
83. Find the figure of the curve defined by the equation
r=- a sec 0- b. See the Diagram of page (312).
84. Describe the figure of the curve whose equation is
r = a2 sin 20. See the Diagram of (294), with the direction
of the fixed axis changed through half a right angle.
85. Trace the figure of the reciprocal spiral whose
equation is r-=a0-~.