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

*4 -

FUNCTIONS

to represent log sin a, \f' (x) to represent y in the curve, fig. 1,
&c.... &c.
Functions 7. The result of performing a certain operation or set
of several
variables, of operations upon several variables x, y, z, &c. is, in like
as well as
of one. manner, said to be a certain function of x, y, z, &c., and is
represented similarly by the notation f(x, y, a,...). Thus
X2+ y'2 4 z, x log (y + z), are the results of performing certain
operations upon x, y, z, and are accordingly said to be certain
functions of x, y, z, and we denote them by functional letters
written before x, y, z, thus, viz.
xi y+ y f 2 (x, y, z) S log (y + z) = i ((, y, z.)
Geometri- 8. We may evidently draw the curve BPQ (fig. 1) in
cal representation'of such a manner that y shall be any function we please of xi; and
a function, thus by means of a curve we may denote any function, and as
it were represent it to the eye, which is often a very good
method of illustrating general theorems respecting functions.
Functions 9. In a function of several variables f(x, y, z...) it may
of dependantand of happen that the variables x, y, z...are connected with each
andepend- other in some manner, so that we cannot change one without
ables. at the same time changing the others. Or it may happen that
x, y, z...are not at all connected with each other, so that we
may assign to each of them any value we please independently
of the rest. In the former casef (x, y, z...) is said to be a
function of several mutually dependant variables, and in the
latter case f (, y, z) is said to be a function of several independant variables.
Functions 10. A quantity y is said to be an explicit function of
explicit and
implicit. another, a, when we can state the precise operations by which
y may be deduced from x; if not, y is said to be an implicit
function of x. Thus if we are given the equation
y5 - 34y + X= 0,
we know that there must be a certain set of operations by
which y may be deduced from x, but what these operations
are we cannot precisely state; in such a case y is called an
implicit function of x.