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

ASYMPTOTES TO POLAR CURVES.

147

~215. We may also obtain this result as follows. Sanmeresult
obtained inLet a be the angle which the asymptote makes with the dependantprime radius; then it is evident that 0 = a ought to makely.
r infinite, and therefore f(a) must be zero. Let the rectangular equation to the asymptote be
y = tan a ( + c)............ (2),
which, putting r' cos 0 r' sin 0 for x and y, becomes
c sin a
sin (0 - a)'
r f e na
ene ci. hence - = c sin a.
r sin (0- a) (
Now the limiting value of - when 0 approaches a ought
to be unity, from the nature of an asymptote; but by the
usual method of vanishing fractions this limiting value is
c sin af'(a).
Hence we have c sin af'(a) = i;
and.-. c sina =f (a)
The equation to the asymptote is therefore {substituting
for c in (2)},
1
y = x tan a +
cos af (a)
We have here obtained the rectangular equation to the
asymptote, because it is easier to make use of it than the polar.
a
Let r -= be the equation to the curve; here Example l.
0
0 1 1
f(0) =; and.-. a = 0; also f'(0)- -;.. f'(a);
a a a
therefore the equation to the asymptote is y = a, which represents a line parallel to the axis of x at a distance a above it.
See fig. 20.
a (e' - 1)
Let r = (a hyperbola referred to focus), Example 2.
1 - e cos 0