542 
FUNCTION. 
[787 
as a function of n, that is, of x. As regards the equation u n+1 = au n , considered in 
itself apart from what precedes, observe that this is satisfied by writing u n = a. 11 («), 
or the question of solving this equation of finite differences is, in fact, identical with 
that of finding the wth function a n (x), where a (x) is a given function of x. It of 
course depends on the form of a(x) whether this question admits of solution in any 
proper sense; thus, for a function such as log x, the rath logarithm is expressible in 
its original function log"x, (= log log...«), and not in any other form. But there 
are forms, for instance ax = 
a 4- bx 
c + dx 
, where the ??-th function a n x is a function of the 
like form a n x = 
A -p Bx 
C + Dx’ 
in which the actual value can be expressed 
as a function 
of n; if a be such a form, then <fia<£ -1 , whatever <¡6 may be, is a like form, for we 
obviously have ($a</r 1 ) n = <f>a. n ^r\ The determination of the nth function is, in fact, a 
leading question in the calculus of functions. 
It is to be observed that considering the case of two variables, if for instance 
a (x, y) denote a given function of x, y, the notation cr («, y) is altogether meaningless ; 
in order to generalize the question, we require an extended notation wherein a single 
functional symbol is used to denote two functions of the two variables. Thus 
(j> (x, y) — a (x, y), /3 («, y), a and /3 given functions ; writing for shortness x x = a («, y), 
y 1 = (3 (x, y), then (j) 2 (x, y) will denote cj) (x 1} yd, that is, two functions a (x u y^), ¡3 (x ly y^), 
say these are x 2 , y 2 ; (f> 3 (x, y) will denote </>(« 2 , y 2 ), and so on, so that <fi n (x, y) will 
have a determinate meaning. And the like is obviously the case in regard to any 
number of variables, the single functional symbol denoting in each case a set of 
functions equal in number to the variables.