Page 33 - Foundations Of Differential Calculus

16 1. On Finite Differences
In a similar way the third difference will be

3 3 4 5
∆ y = Pω + Qω + Rω + ··· ,
and so forth.
We should note that these letters P, Q, R,... do not stand for deter-
mined values, nor does the same letter in different differences denote the
same function of x. Indeed, we use the same letters lest we run out of
symbols.
Furthermore, these forms of differences should be carefully noted, since
they are very useful in the analysis of the infinite.

23. According to the method we are using, the first difference of any
function is found, and from it we find the differences of the successive
I
II
orders. Indeed, from the values of successive functions of y, namely, y ,y ,
III
IV
y ,y ,..., we find in turn differences of y of any order. We recall that
I
y = y +∆y,
II 2
y = y +2∆y +∆ y,
2
3
y III = y +3∆y +3∆ y +∆ y,
IV 2 3 4
y = y +4∆y +6∆ y +4∆ y +∆ y,
and so forth, where the coefficients arise from the binomial expansion. Since
III
II
I
y , y ,y ,... are values of y that arise when we substitute for x the
successive values x + ω, x +2ω, x +3ω,... , we can immediately assign the
value of y (n) , which is produced if in place of x we write x + nω. The value
obtained is
n n (n − 1) 2 n (n − 1) (n − 2) 3
y + ∆y + ∆ y + ∆ y + ··· .
1 1 · 2 1 · 2 · 3
Furthermore, values of y can be obtained even if n is a negative integer.
Thus, if instead of x we put x − ω, the function y is in the form
2 3 4
y − ∆y +∆ y − ∆ y +∆ y − ··· .
If instead of x we put x − 2ω, the function y becomes

2 3 4
y − 2∆y +3∆ y − 4∆ y +5∆ y − ··· .

24. We will add a few things about the inverse problem. That is, if we
are given the difference of some function, we would like to investigate the
function itself. Since this is generally very difficult and frequently requires
analysis of the infinite, we will discuss only some of the easier cases. First of