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

101

110. If the equation of a curve be
4+ x3y xy2 + ay 2 y + ay02 + a'2y = O,
it is required to prove the existence of a point of complete
Embrassement at the origin. See the Diagram of page (326).
111. In the curve defined by the equation
a3y2 _ 2 ab2 y =,
the origin of co-ordinates is a point both of inflexion and
osculation, commonly called a point of OsculinJfexion: required a proof.
112. The curve whose equation is 4- b xa3y2, has
at the origin of the co-ordinates a point of Rebroussement
with a double inflexion; required a proof.
113. The curve defined by the equation
cy2 = (a + b) cy c33-aba + cx33y23
has a multiple point at the origin of co-ordinates: find its
nature and circumstances,
114. The curve whose equation is
a2y3 -a by2 + a22y + X5 = 0,
has a point of Osculinflexion at the origin: required a proof.
115. In the curve defined by the equation y5+a 4b2'xy"2
there is at the origin of co-ordinates a point of inflexion with
a cusp of the first species required a proof.
116. The curve defined by the equation y4 + x4+2 bx2y=0o
has at the origin of co-ordinates a triple point with a cusp of
the first species: required a proofd
117. The curve whose equation is
xa5 - ay4 + 2b xy + b"y2 = 0,
has at the origin of co-ordinates a triple point with a cusp of
the second species: required a proof,