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

182

ELIMINATION OF CONSTANTS

and let it be differentiated n times, and so give the following
equations; viz.
dU=.........(2)
d2 = 0......... (3),
dnU = o......... (n + );
then from these n + 1 equations we may in general eliminate
the n quantities ac, ac, c... a., and so obtain an equation
dy d2y d y
between B r, y -, -... - not containing any of these
dTx dx dan
constants.
Example. Let the given equation be
y - a#x + ab = O........ (1),
differentiating twice, we have
p - 2Cax + ab = 0......... (2), P = dyY
q 2 = O...d......... (3); d,2.
hence, 2a = q, and therefore, by (2),
ab = qxt -p;
and therefore, by (1),
- + qx Px = 0,
2
or y + — - O 0;

which is an equation between x, y, p, and q, not
constants a and b.
A number 255. It is evident that there be n + m
of different
equations a2... a, a,,... a,,n in the equation U=o, we
may be
formed in any n of them we please between the equations
this way.
U = 09
du = 0

containing the
constants a,
may eliminate