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

Ill

68. Investigate the form of the curve whose equation
is y = 2 sin 2 - sin x: and find after what values of x the form
of the figure recurs.
69. Let it be required to trace the curve whose equation
is 4 r- a x2y + by3 =o, distinguishing particularly its singular
points.
70. Trace the curve belonging to the equation
a3 (y - )= ( - b)4 (C-c),
explaining fully the natures of its singular points.
71. If the equation of a curve be
a' (y - a)2 = ( - b)' ( - c),
it is required to trace it, and to distinguish fully the natures of
of its singular points.
72. Trace the curve belonging to the equation
a3 y2 -2 a b '2 y = - 5;
and particularize the nature of it at the origin of co-ordinates.
73. Investigate the figure of the curve belonging to the
equation a4 + x4 + y = 2 a' (X2 + y2): and find the natures of
its multiple points.
74. Trace the curve whose equation is
(y2 + x2)2 - 6acy2 = ax2 (2x - a):.and find the natures of its singular points.
75. Trace the curve whose equation is
y2 = a ( - b) ( - c);
and distinguish the different cases depending upon the relative
values of b and c, regarded as possible and imaginary.
76. If the equation of a curve be y'=y2 (a-x) (x-b) (x-c),
it is required to trace it, and distinguish the various forms of
which it admits.
77. Investigate the figure of the curve called the Trident,
as defined by the equation cxy = ac3+ bc2 2+ c + e: and distinguish the different cases originating from the forms of the
roots of the equation a '+b2 +- C + e = O.