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

138. The hyperbolic asymptote of the curve whose equaa2
tiondis x4 - y4 + a2y = 0, is defined by y' = x' +: required
a proof.
139. The hyperbolic asymptotes of the curve whose
equation is y3-2Xay2'+xy-a'=o, are defined by the equations
y-xs - \/a / O: required a proof.
140. In the curve whose equation is
x y2 _ a2 y = 3 +- 2 a + x + b,
the hyperbolic asymptotes are defined by the equations
1 1
y = '+1+ — and y'=- - - -:
2 x 2x'
required a proof.
141. In the curve whose equation is y3- Samy + = 0,
it is required to prove that the rectilineal and hyperbolic
asymptotes will be defined by the equations x'+ y'+ a =o
and x'y' + a'2 + ax' + a2 0, respectively.
142. The curve whose equation is
y2 - 2ay + 3x2- 2y - 42y? + 5 =0,
is concave or convex towards the axis of x, according as the
greater or less value of y is considered: required a proof.
143. In the curve whose equation is (x - 1) y = (x + 1) x,
it is required to prove that the branches have their concavities
upwards or downwards according as the abscissa is greater or
less than 1.
144. The curve whose tangent is of an invariable magnitude is always convex towards both the co-ordinate axes:
required a proof.
145. The branches of the curve whose equation is
(a + w) y= a2 + ax + x2, are convex towards both the coordinate axes: required a proof.