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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equally inclined to the normal will cut off from the axis of
x a portion represented by 2 (X + ) (p -Y) required a
J 2px —(1 - p2) y
proof.
110. Determine the limits of the length of the normal
in each of the conic sections, by means of the general equation.
111. Express the co-ordinates of a point in any curve
in terms of the angle which the normal at that point makes
with the axis of x.
112. If 0 be the angle made by the normal with the
major axis of an ellipse; then will the co-ordinates from the
centre be expressed by
a2 cos 0 b2 sin 0
and
/a2 cos20 + b2 sin 0 va2 cos20 + b2 sin20
required a proof.
113. On the same supposition, it is required to prove
that the subtangent and subnormal are represented by
b sin 0 tan 0 b" cos
-y, * -- and,
V/a2 cos2 0 + b2 sin2 0 V/a2 cos2 0 + b sin2 0
114. The same notation being retained, the tangent and
normal will be expressed by
b2 tan b2
and,/a2 cos 0 + b sin2 0 /a2 cos2 0 + b2 sin2 0
respectively: required a proof.
115. If the equation of a curve be a2y= b2 (2aax + 2i),
and the notation of Article (151) be retained; then will A W= a
and AZ= b: required a proof.
116. If y3 a2 -- 3 be the equation of a curve; then
will AW= -- a and AZ=-a: required a proof.
117. If (x - 1) y = ( + 1) x be the equation of a curve;
then will A W= 2 and AZ ==2: required a proof.
118. If (v - a) y2 = (i + a) x2 be the equation of a curve;
then will AW=a and AZ= + a: required a proof.