522 
[787 
787. 
FUNCTION. 
[From the Encyclopcedia Britannica, Ninth Edition, vol. ix. (1879), pp. 818—824.] 
Functionality, in Analysis, is dependence on a variable or variables; in the case 
of a single variable u, it is the same thing to say that v depends upon u, or to say 
that v is a function of u, only in the latter form of expression the mode of dependence 
is embodied in the term “function.” We have given or known functions such as u 2 
or sin u, and the general notation of the form <£«, where the letter <£ is used as a 
functional symbol to denote a function of u, known or unknown as the case may be: 
in each case u is the independent variable or argument of the function, but it is 
to be observed that, if v be a function of u, then v like u is a variable, the values 
of v regarded as known serve to determine those of u; that is, we may conversely 
regard u as a function of v. In the case of two or more independent variables, say 
when w depends on or is a function of u, v, &c., or w=(f)(u, v,...), then u, v,... are 
the independent variables or arguments of the function; frequently when one of these 
variables, say u, is principally or alone attended to, it is regarded as the independent 
variable or argument of the function, and the other variables v, &c., are regarded as 
parameters, the values of which serve to complete the definition of the function. We 
may have a set of quantities w, t, ... each of them a function of the same variables 
u, v, ... \ and this relation may be expressed by means of a single functional symbol </>, 
{w, t,...) = <f> (u, v,...); but, as to this, more hereafter. 
The notion of a function is applicable in geometry and mechanics as well as in 
analysis; for instance, a point Q, the position of which depends upon that of a 
variable point P, may be regarded as a function of the point P; but here, sub 
stituting for the points themselves the coordinates (of any kind whatever) which 
determine their positions, we may say that the coordinates of Q are each of them a 
function of the coordinates of P, and we thus return to the analytical notion of a 
function. And in what follows a function is regarded exclusively in this point of view,