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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21. In the spiral whose equation is
= /a2 b b2 /r2+ aV/2-b2
0 = b log - -,
r
b br
it is required to prove that sin P = - and p =
r 2 Va2~+2
-a ++
22. If a spiral be defined by /ac+b0 = a sec- a
a ar
then will sin P= and p = /*-: required a proof.
Vr2 2 — Vr2 — b2
23. If (r - a) 02= r be the equation of a spiral, it is
required to prove that
ar2 ar
sin P= and p=
V/a2r+4 (r-a)3 v/a2r + 4 (r-a)3
24. Find the values of tan P and p in an ellipse and
hyperbola referred to the centre, focus and vertex as a pole.
25. In any conic section, given the angle included between
a tangent and the diameter through the point of contact, to find
that point.
26. Required the same, when the angle included between
the tangent and the distance from the focus is given in each of
the conic sections.
27. The polar equation of a straight line being
r sin (a + 0) = a sin a;
it is required to prove that P = r - a - 0 and p = a sin a.
28. Determine the values of tan P and p in the spiral
defined by the equation r = a log tan (45 + 0).
29. In the spiral whereof the equation is (ac+rn) p2=r9+2,
it is required to prove that tan P= (-)
\al
30. Find the values of tan P and p in the spiral defined
by the equation r =a (1 + 2 sin 20).