Page 26 - Foundations Of Differential Calculus

1. On Finite Differences 9
Then

4 4 n−4
∆ y = n (n − 1) (n − 2) (n − 3) ω x + ··· .

14. In order that we may more easily see the law by which these differences
of powers of x are formed, let us for the sake of brevity use the following:
n
A = ,
1
n (n − 1)
B = ,
1 · 2
n (n − 1) (n − 2)
C = ,
1 · 2 · 3
n (n − 1) (n − 2) (n − 3)
D = ,
1 · 2 · 3 · 4
n (n − 1) (n − 2) (n − 3) (n − 4)
E = ,
1 · 2 · 3 · 4 · 5
....

We will use the following table for each of the differences:

y 1000 0 0 0 0 0 ...
∆y 0111 1 1 1 1 1 ...

2
∆ y 002614 30 62 126 254 ...
3
∆ y 000636150 540 1,806 5,796 ...
4
∆ y 0000242401,560 8,400 40,824 ...
5
∆ y 0000 01201,800 16,800 126,000 ...
6
∆ y 0000 0 0 72015,120 191,520 ...
7
∆ y 0000 0 0 0 5,040 141,120 ...

Each number in a row of the table is found by taking the sum of the
preceding number in that row and the number directly above that preceding
number and multiplying that sum by the exponent on ∆. For example, in
5
the row for ∆ y the number 16,800 is found by taking the sum of the
preceding 1800 and the 1560 in the preceding row to obtain 3360, which is
multiplied by 5.