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

184

ELIMINATION OF FUNCTIONS

and proceeding in this manner we may eliminate al, a,, a3.
successively.
Examtiple. Thus in the example just considered, if we solve for a we
find
Y,' - b a
therefore differentiating we have
p (&v - bxt) -y (2w - b)
= 0,
(('-< bv)2
or (IV - b) p - y (2w - b) =0,
which equation does not contain a.
Again, solving for b, we get
-X2_ - -2y
- = b;
wp - Y
and therefore differentiating we find
(X2q - 2y) (xp - y) - (*2p - 2xy) x q = 0,
or xqy - 2y (p - y) 0,
or t2y = 2 (xp - y) = o,
which is the same equation as before.
This method is generally longer-than the former.
Elimina- 257. In the same way that we eliminate constants from
tions of
functions equations between two variables I and y, we may eliminate
from equa- functions from equations between three variables x, y, n.
For let u = 0... (1)
be an equation between I, y, and z, containing the n functions
Ib (v1) 02(V2), (PA3()... (q(vn), v1 v2... v being any
functions of x y z. Then in virtue of (1) we may consider
z as a function of the two independant variables x and y, and
we have by successively differentiating with respect to I and
y the following set of equations:
u = 0,