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

21.8 APPENDIX.
ds = Va' sin 2t+ b' cos't. dt,
b
tan 4' = - - cot t,
a
cos xe
A/1 +- cot2 t
Equation to tangent is
x cos t y sin t
a b
(4) To find -the locus of the intersection of the tangent
and a perpendicular upon it from the focus of an ellipse.
The equation to the tangent is
b b
y - -cot t. x +
a sin t
The equation to the perpendicular upon this from the
focus is
y = tan t. (v - acu),
which equations may be put in the form
ay sin t + bx cos t ab,
by cos t - av sin t - a e sin t;
therefore, squaring and adding these equations,
(IV + y') (a' sin2 t + b' cos2 t) a2 (b2 + a'e2 sin' t)
- a (b' cos' t + a" sin' t).
Since a2e'- a - b..'- ~y'=a';
hence the locus required is the circle described on the major
axis as diameter.