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

176

ULTIMATE INTERSECTIONS.

then (2) gives a - A = +p P
and then (1) gives R = (1 + p),
q
which equations give us B, A, and R, and thus the circle is
completely determined.
Circle of This circle is commonly called the osculating circle, or
curvature
Its radius the circle of curvature. Since a may belong to any point
is theindex whatever of the curve, we may put x instead of a, and.
of curva-.,
ture p. See dy d2 y
(181.) y', d ' and d2 instead of b, p, q respectively, and then
we have
+1 +f
{ ( (dxt _: ds3
d'y dx dy'
dx'
hence it appears that R = p (see 184), and hence the point
P' (185) is the center of the osculating circle.
It follows therefore that the limiting position of the intersection of two consecutive normals to a curve when they
approach each other is the center of the osculating circle.
We may also arrive at this result by means of the following article.
Ofthe lti- 249. Let the equations to two curves be
mate intersection of
curves. f (xy a)= 0...... (1),
f (ya')= 0.....(2),
(2) differing from (1) only in having a' in place of a, a and
a' being two constants to which we may assign any values
we please. Then if we determine x and y from these two
equations taken together, the resulting values will be the
co-ordinates of a point of intersection of the two curves.
Now this point, which we shall call P, must be some definite
point as long as a' is different from a, but if a' becomes