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

148

ASYMPTOTIC CIRCLE.

I -- e cos 0
here f()= -
a (e2 - i)
and.'. cosa =-, sin a /1 -- = ---
e e2 e
tan a = ve - 1.
e sin 8 1
Also f'(O)= (; - f(a)= ~
a -l) ' av /- 1
hence the equation to the asymptote is
y = A / /e_- 1. hx. ae %6/e - 1,
b
or y= A - (x + ae),
a,
which shews that there are two asymptotes making angles
b b
tan1 -, and -tan' - with the axis of x, and meeting it at
a a
a distance ae behind the origin.
Asymptotic 216. It sometimes happens that r assumes a finite value,
circle,
what. c suppose, when we put 0 = co in the polar equation to a
curve: it is easy to see that in such a case if we describe
a circle round the pole with radius c, we may by continually
increasing 0 make the curve approach as near as we please
to this circle without ever actually meeting it. Such a circle
is called an asymptotic circle.
Example. Let r; then 0 = co makes r = a; therefore by
r + 0
continually increasing 0 we may diminish r~ a ad libitum,
and therefore make the curve approach as near to the circle
as we please without ever actually meeting it, since it requires
an infinite number of revolutions of r to make r = a.
The circle in this case is an exterior asymptote, since r
is evidently always less than a, (see fig. 21).
aO
If - - be the equation, r is always greater than
a (at least when 0 is taken large enough), and therefore
~the circle is an exterior asymptote.