Page 20 - Foundations Of Differential Calculus

1. On Finite Differences 3
6. Once we have found the series of differences, if we again take the dif-
ference of each term and its successor, we obtain a series of differences of
differences, which are called second differences. We can most conveniently
represent these by the following notation:
I
∆∆y =∆y − ∆y,
I II I
∆∆y =∆y − ∆y ,
II III II
∆∆y =∆y − ∆y ,
III IV III
∆∆y =∆y − ∆y ,
....
I
We call ∆∆y the second difference of y,∆∆y the second difference of
I
y , and so forth. In a similar way, from the second differences, if we once
more take their differences, we obtain the third differences, which we write
3 I
4
3
as ∆ y,∆ y ,... . Furthermore, we can take the fourth differences ∆ y,
4 I
∆ y , ... , and even higher, as far as we wish.
7. Let us represent each of these series of differences by the following
scheme, in order that we can more easily see their respective relationships:
Arithmetic Progression:

x, x + ω, x +2ω, x +3ω, x +4ω, x +5ω, ...

Values of the Function:

III
IV
II
V
I
y, y , y , y , y , y , ...
First Differences:
I II III IV
∆y, ∆y , ∆y , ∆y , ∆y , ...
Second Differences:
I II III
∆∆y, ∆∆y , ∆∆y , ∆∆y , ...

Third Differences:
3 II
3
3 I
∆ y, ∆ y , ∆ y , ...
Fourth Differences:
4 I
4
∆ y, ∆ y , ...