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

178

ULTIMATE INTERSECTIONS.

Example 1. 250. To find the ultimate intersection of two consecutive
positions of a line which always includes equal areas between
it and the co-ordinate axes.
x y
Let - + 1 be the equation to the line, and 2c the
constant area; then af = c, and the equation to the line
becomes
07 ay
-+ - - 1 =0......(1)
a C
a being the variable parameter.
The partial derivative of this with respect to a is
w y
+- = 0......(2)
a2 c
from these equations we easily get
Ov X a
- + — 1 = 0, or <v= -;
a a 2
and then y=-.
2a
The value x = - shews that the line is bisected by the
2
point of ultimate intersection.
We may verify the correctness of this result thus.
Let a' be another value of a, then the equation to the
corresponding line is
iv ay
- + - 1 =0......(3),
a c
(3) a'- (1) a gives
~
(.'2 - a2) _ -a or y = -—;
and therefore the limiting value of y when a' approaches a
a
is -, and therefore by (1) that of x is -
2a c2