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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37. If the equation of a curve be y a vers 2a, then
will its points of inflexion belong to x = (2m+ 1) ~ and y=a:
required a proof.
38. It is required to ascertain the positions of the points
of inflexion in. the curves belonging to the following equations:
(1) a2y = br a: (2) a'y= 2X(a2-_2):
(3) 4I a2y2+x2y2: (4) (X2+y )2=ara2+b2y2:
(5) (a2+x')y, a': (6) ay2=a2(a -X):
(7) (a-'v)y2=a2x: (8) =ayx=av- b)Qx- c):
(9) (a-x) 2 y'=(2ax _V)2: (10) (a - X)y2-a2r(2a- x):
(x1 ( - a) yS=(a - x) x (1!2) (a )yy= c~-m)2
(is) logy=aX: (14) y=x?-tcosx:
(15) a2y2=,4_(b2+c2) aI2+b2c2: (16) a' y3 + b3&'= a' b3
(17) (2a - x) (XI +y2) b2,V~ (18) X4 _ d2X2 + a~y2 = 0
(19) (2 a -X) _V (X a) X_ 3 a)
39. In the curve whose equation is ay2 - X3+ ba X=0,
it is required to prove that the origin is a conjugate point.
40. If the equation of a curve be (a - e) y2 = X"-6 b,
prove that the origin of co-ordinates is a conjugate point.
41. If the equation of a curve be iy' = (x -9) (X - 2)2,
it is required to prove the existence of a conjugate point
corresponding to x =2 and y =0.
42. The equation of a curve being a (y - a)' = a' (w - a),
it is required to prove that there is a conjugate point when
x=0 and y=a.
43. If the equation of a curve be a (y - b) = (a- C)3,
it is required to prove the existence of a conjugate point
when x c and y = b.
44. In the curve whose equation is ay2 = (b + X)2 it
is required to shew that there is a conjugate point when
M= -b and y =o.