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

AN EQUATION WHICH BECOMES ILLUSORY.

107

When x = a, and therefore y = a, in virtue of (1), this
0
equation gives p =- therefore, differentiating again, considering p constant, we have
2ap2 - 12a. + 6a 0;
which, when x = a, gives
p2 - =;.. p= /3.
tt 157. There are two objections to this method of find- Objections
ing the values of p. 1st. It is generally very complicated and method of
findingp by
troublesome when we have to go beyond a second differen- successive
tiation. 2nd. We have no right to assume that the terms diferentiation.
d2y d'y
containing r-e'... vanish; for although the coefficients
cofntaese dx'ncg d,2y d'y
of these terms vanish, yet, since d ' dY... may and often
do at the same time become infinite, we cannot tell but that
0
these terms may assume the form 0. co or -; and therefore
0
we cannot assume them to be zero. For instance, in the first
example, (153), in the result of the second differentiation we
have the term (2a2x + a2y) q, which we assume to be zero
because 2 a + a2y = o when = 0; but we have no right to
do this, since we. cannot tell whether q is infinite or not.
tt 158. The following method will be found free from A method
f free from
this objection, and very simple in practice, especially when it these obis ourmp jections
is our object to find p for the values x = 0 and y =. explained.
(1) Suppose that we wish to find p for the value x = 0
from the equation
A4x + By + Cx4 + Dxy + Ey2 + Fax... &c. o..... (1).
Since y = 0 when x = 0, it is evident that the value
of p we seek is the limiting value of Y when x approaches
x
zero (see 148): if therefore we put - = u and find the limiting value of u, u, suppose, when tx approaches zero, u0 is the
quantity we wish to determine.