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

90

30. If the equation of a curve be ay= - /2ax- 2:
then corresponding to x=0 and y=o, as also to x la (3-1i)
and y =-a x/6 v/ — 9, there are points of contrary flexure:
required a proof.
31. In the companion of the cycloid whose equations are
x = aO and y = a (1 + cos 0), it is required to shew that there
is a point of inflexion when x = 7r a and y = a.
32. The quadratrix of Tschirnhausen being defined by
the equation - = sin 2, it is required to prove that there
a 2a'
are points of contrary flexure corresponding to x= =2ma and
y=-a sin mwr= O.
33. The syntractrix is defined by the equation
b + V^^ -/ -y
= a log + —y -/b2- y;
Y
and it has points of inflexion corresponding to
/alVa-b+/ab-b /a-b db a
a =alog //' a - b + /ab / and- b // 2aVn V/a 2a-b 2a-b
required a proof.
34. Of the Witch defined by the equation xy2 + a2x = a,
the points whose co-ordinates are x =- a and y=, are
points of inflexion: required a proof.
35. If a curve be defined by the equation y=aec~s,
then will its inflexions correspond to cos = — ( — 1 /5)
and y=ae(-'+-"): required a proof.
36. If a vers y =x be the equation of a curve, its inflexions correspond to x = a and y = (2m + 1) ~: required
a proof.