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

75
31. Required the values of tanP and p in the curve
whose polar equation is r = a sec 0 tan2 0.
32. One of the polar equations to a rectangular hyperbola being r = (cosec 0 —sec 0); it is required to find the
corresponding values of tan P and p.
33. In the rectangular hyperbola as defined by the equation r cos 20= 2a cos 0, it is required to find the values of
tan P and p.
34. If the notation of Article (196) be retained, it is required to prove that in the spiral of Archimedes whose equation
r2
is r = a, the polar subtangent ST= - and the polar tangent
a
35. In the circle referred to a point in its circumference
2
when r = a cos 0, it is required to prove that ST = -
a
and PT=
36. If a spiral be defined by the equation r =asec0;
it is required to prove that
r2, r"2a2
ST= 2 and PT=r V r
/r2 r2 2
37. If r= a0m be the equation of a spiral; it is required
rO r
to prove that ST= - and PT= + 02.
m m
38. If a spiral be defined by the equation
a + V\a- T2
0 = log ---
it is required to shew that
ST= -/ ---ar 'and PT a
T/a2 -, a2 2a