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

ULTIMATE INTERSECTIONS. 179
251. Hence we may find the ultimate intersection of Example2
two consecutive normals as follows.
The equation to the normal to a curve y =(x) at the
point whose abscissa is a is
f'(a) y -f(a ) + v - a = 0............ (1),
taking its partial derivative with respect to a we have
f'(a)yf() \y - { f'(a)}- 1 = 0.....(2),
and the values of x and y got from these equations are the
co-ordinates of the point required. These values are
Y f()+ {f( a)} I + 1. f2(a)
=, _ f' (. {W- f (a) 2 +
Comparing these values with those of A and B in (249) we
see that the point of ultimate intersection of two consecutive
normals is the center of the circle of curvature.
252. We may find the locus of all the points of ultimate How to
find the
intersection of the set of curves represented by the equation locus of the
points of
f (xy a) = 0, ultimate
intersection
by eliminating a between this equation and its partial deri- crves~f
vative
f'(xy&) o.
Let us take the first example, last article, in which Example.
f(vya) = - + -y 1 0,
f' (xya) - - +- = 0
a C
a c
Eliminating a, we find
a = 2x, and.~. xy=c,
which gives a rectangular hyperbola referred to its asymptotes.

12-2