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

iv

PREFACE.

which Lagrange's functional notation is employed, as
being the simplest to begin with. In Chap. VI the
Diferential notation of Leibnitz is explained, d/ is
dx
defined as the quote of the differentials dy and dx,
which however are not supposed to be infinitesimals,
but simply two arbitrary quantities in a certain ratio.
In the case of partial differential coefficients, some
du
modification of the common differential notation -
du
d, is clearly necessary: I have employed the suffix
notation d.u, dyu, as being frequently employed, though
not exactly in this manner. I should have much preferred the notation deu, dyu to denote partial differend,u du
tials, and d- - to denote partial diferential codx dy
efficients. Chap. VII relates to successive differentiation, and the change of the independant variable.
Chap. VIII contains certain very important Lemmas
upon which the use and application of the Differential Calculus in a great measure depends. Chap. IX
contains the theory of Series, based upon one of the
preceding Lemmas, without assuming that f(x + h)
can be developed in the form
A + Bh + C + &c...:
and here I have endeavoured to shew what the real
nature of a series is, and to prove rigorously the prin