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

CHAP. VIII.

MISCELLANEOUS EXAMPLES.
IT is required by means of Articles (126)-(189) to effect
the solutions of the following Theorems and Problems.
1. If x and y be the co-ordinates of a parabola, whose
latus rectum is a, corresponding to the arc s, then will
ds N/a + 427 1 2+y
- I Ja+4x - ixa2+4y2;
d-~ =2 + -4y2
ds /a+4x 1 / —
and - == = a2 + 4y2
dy a a
2. In a common cycloid, the radius of whose generating
circle is a, it is required to prove that
ds /2a d2a
dax X dy 2a-x
3. If C2y =x-, be the equation to a curve; then will the
secant, passing through two points whose abscissa are a and b,
intersect the axis of x in an angle whose trigonometrical tangent
a2 + ab + b2
is -: required a proof.
c2
4. Find the angles in which a secant drawn through two
given points of the curve whose equation is y =ax2 - bx +fc,
intersects the co-ordinate axes.
5. Determine the angles in which a secant to any curve of
the second order intersects the co-ordinate axes.
6. Find the angle contained between the two straight lines
drawn from the vertex of an ellipse, to the extremities of the
latus rectum and axis minor.
7. Find the angle which a straight line, equally inclined
to each of the co-ordinate axes, drawn through the focus of an
ellipse or hyperbola makes with the distance from the centre.
G