the calculus of functions . 391 
If the second function enter, the equation rises to the second 
order: thus, 
4 r x = x 
(x -f 4> x) + (t)/ x — x) % = 0 
( 4 ^ x 4 “ 4 * ~) — ax 
A function of two variables admits of two second functions : 
thus ( x >y) becomes 4> (4< ( x >y)> and 4 / [ x > 4> ( x > y )) or 
they might be thus expressed 4' 2 ’ 1 (.r,y) and 4' 1,z (x,y). 
These express the second functions ; the first taken relative 
to x } the other relative to y. But besides these two there is 
another, which arises from taking the second function simul- 
taneously relative to x, and y; it is ^ j (t|/ xy ) , ip ( x,y ) j . This 
ought not to be written \p 2,z (x,y) for it is not the second 
function first taken relative to x and then to y, nor is it the 
converse of this. In fact, the notation is defective ; some 
method is wanting of indicating the order in which the suc- 
cessive substitutions are made. I shall for the present lay 
aside the consideration of functional equations, involving more 
than one variable. 
Those of the first order have long been known, but the 
method in which I have treated them is, I believe, entirely 
new. Equations of the second and higher orders have never 
been even mentioned ; it is these which present the most 
interesting speculations, and which are involved in the greatest 
difficulties. I shall first give some account of the enquiries 
which led me to this subject, and shall then treat of the vari- 
ous orders of functional equations. 
Some few years since, while considering a problem men- 
tioned by Pappus, relating to the inscription of a number 
