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

50

dy
8. If p=; then may the inclinations of the rectilineal
dx
tangent to the axes of co-ordinates be determined from the
p - 1
equations X= sin - P -= os-
//1 pc2 lCO + / 'p2
9. What are the inclinations to the axes, of the rectilineal
tangents at the extremities of the latus rectum of each of the
conic sections referred to rectangular co-ordinates?
10. At what angle does the rectilineal tangent of the
curve whose equation is ay =-a2x + x, meet the axis of x,
at a point where x a?
11. Find the angles which a tangent to the cycloid makes
with the axes; and prove it to be parallel to the corresponding
chord of the generating circle.
12. Determine the inclinations to the axes, of a tangent
to the trochoid whose equation is y= ma vers-' - + -/2a -,
a
(m +-1) a
at a point whose abscissa is
13. If any ordinate of the curve whose equation is
y' —3axy- =- 0 be drawn; prove that the sum of the trigonometrical tangents of the angles which the lines touching the
curve at the points of intersection make with the axis of x =0.
14. Find the angles at which the cissoid of Diodes is
inclined to the axes, where it intersects the generating circle.
15. Determine the points in a circle at which it is inclined to the axis of x at the angles 30~ and 60~; and find the
distance between them.
16. Required the inclination to the axis of x, of the curve
whose equation is log (a 4 2- y2) = + log y,
a
a aa
at a point where y= -7
V/2