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

DEVELOPEMENT OF FUNCTIONS GIVEN BY EQUATIONS. 93
If f(a) = co, then if we assume appearance of ne-((S _~~~ a)S~~f( = +~ ~gative
(, - a)sf() = '(p (), powers in
the series.
and so determine s that p((a) shall not be infinite, we may
develope (p() as above in the form
1 2, (a) = A, + Al (a - a) + A (i - a)r...&c.
and therefore, restoring f(f), we have
f(I) = -A (I - a)-' + A1 (I - a)S+... &c.
Hence, f(x) becoming infinite when = a, indicates the
appearance of negative powers in the series. (For examples,
see Appendix H.)
ttJ 144. Suppose we have a relation between x and y, Developement of a
viz. f (xy) =0, to developer y in a perfect series of powers function
of x. given by an
equation.
To do this all that is necessary in general is, to write down
to the equation f (y) and its successive derivatives, and then
dy d2y
to put -O = 0, and so find the values of y, d- ' &c. when
doo doX2
V = 0, and thus obtain the perfect developement of y in powers
of x by Taylor's Theorem. An example will best explain
this method.
Let the relation between x and y be Example.
y3 -xy + 2 = 1. (1);
then differentiating successively, we have
(3y2 - a) p-y + 2 = o... (2),
(3y2 ) q + 6yp2 p = 0... (3),
when p, q, r... are, for brevity, put for -y dy d3y
dx' d)+' d)'
(3y2 _- ) r + (18yp - 3) q + 6p3 = o... (4)

&c. &c.