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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146. Prove that the curve defined by the equation
(y - )3a =- a - 2 as + a,2,
is always concave towards the axis of x; and find how it is
situated with respect to the axis of y.
147. Required the direction of the curvature of the curve
whose equation is (a2 - v2) y = a3.
148. Find whether the curve whose equation is (y - b)
=- (x - a)2, is convex or concave towards the co-ordinate axes.
149. Find the direction of the curvature of each of the
curves defined by the following equations:
(1) a2y =b2( (-2a): (2) ( -) )y2 =(a+ ) 2:
(3) (+ a)y2=b2 (w+ 2a): (4) y3+e=3axy:
(5) y4+2 a (y2)3 (6) (2+ y2)3 P-4a22y:
(7) XyS = ab2 —b (2a + b) X + (a 2b)x2- -3:
(8) y4 + 2a (o - sa) y2 _ 2aa+t3 + a2r2 = O:
(9) x4- ax2y - 1a2y2- aY3 0.
150. Determine whether the curve whose equation is
(a- )2 ('2 - y2) = a2y2, is convex or concave towards the axes,
at the points where the values of x are a and 3a.
151. If yoccxm in one curve and yocxa in another, m
being greater than n; it is required to prove that the former
curve always falls between the latter and its rectilineal tangent.
152. Determine the conical parabola which shall have
contact of the second order with an ellipse at a given point,
and have its axis parallel to the axis major of the ellipse.
153. Define the circle of curvature of the curve whose
equation is 2y = ex + e-'; and prove that its radius is equal to
the normal.
154. Determine the position and magnitude of the circle
)f curvature, when the equation of the curve is x2- ax = ay _y2^
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