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

87

7. The parabola defined by the equation y2 -4xyy+4 x2
-3 +\/41
-8y+3x -2=0, meets the axis of x when = —
8
-41 + 13\/41
and tan X= -- 6-: and the axis of y when y=4+ 3\/2
64
96 + 78 \/2
and tan y= -96 + 78 -: required a proof.
41
8. Determine where the curve whose equation is y3-X3
+ a3 = 0, meets the co-ordinate axes, and the angles it makes
with them.
9. Find where the curve whose equation is (a-x)y2
= (a + ) X2, intersects the axes of co-ordinates, and the angles
in which it cuts them.
10. The curve whose equation is ay2 (x - a) =-e, has
an infinite ordinate when = a, which is an asymptote to the
curve: required a proof.
11. The curve defined by the equation a2 y = a4-b4,
has an infinite ordinate through the origin, which is an
asymptote to it: required a proof.
12. If y (a + c + x) = b (c + x) be the equation of a curve,
it is required to prove that when = - (a + c), the ordinate
parallel to the axis of y is infinite, and becomes an asymptote;
and when y = b, the ordinate parallel to the axis of x becomes
infinite, and is then an asymptote.
13. The equation of a curve being (x +a) ys= b2 (x+2a),
it is required to show that when = - a, the ordinate parallel
to the axis of y is infinite; and when y= + b, the ordinates
parallel to the axis of x are infinite.
14. Find the positions of the infinite ordinates in the
curves whose equations are y = tan x and y = sec x.
15. The equation of an ellipse being 2y2 -4y+5a22-3S=o,
it is required to prove that the two values of y are equal to
one another when x =O and when v = 1: that the two values
of B are equal to each other when y = 1 (2 + V/l-); and that
the corresponding ordinates are tangents to the curve.