the calculus of functions. 
183 
and may for the sake of distinction be called the second simul- 
taneous function relative to x and y ; it differs from the two 
preceding ones, and in order to denote it with brevity, I shall 
put a line over the two indices thus, 
V i*,y) = 
This method of distinguishing it is equally applicable when 
there are more variables. 
i. 
There is only one other modification of the symbol denot- 
ing function to which I shall at present allude. Suppose (after 
any number of operations have been performed on a function 
of two variables for example) y becomes equal to x, and the 
result only is given: this will naturally be represented in a 
manner analogous to that in which Euler denoted the limits 
between which the integral of a quantity is to be taken. 
2, 2 
I 2 
Thus the equation ^ ’ (a;, y) = f x [y = x\ arises from 
the following question : What is the form of a function of x 
and y, such that taking the second function relative to x , and 
then the second relative toy, the result on making y equal to 
x shall be a given function of x ? 
It might be proposed, that after putting y equal to x, the 
whole should be considered merely as a function of a;, and 
that its n th function should be taken on this hypothesis, and 
the result only should be given. 
Such operations I would denote thus : 
2,3 
%f/’ 2 (x,y) =zf(x) [j=.r] or perhaps ■ 
V” } (*>?)=/(*)!>= 
and in a similar manner all other relations of the same kind 
may be expressed. 
