Page 45 - Foundations Of Differential Calculus

28 2. On the Use of Differences in the Theory of Series
or finally,

1 a II 2a I a
(x − 1) (x − 2) (x − 3) − + .
2 x − 3 x − 2 x − 1

It follows that the general term is defined by the first three terms of the
series.
II
I
III
IV
43. Let a, a , a , a , a ,... be the terms of a series of the third order,
II
I
III
III
I
II
let b, b , b , b ,... be its first differences, and let c, c , c , c ,... be
its second differences, while d, d, d,... are its third differences, which of
course are constants:
Indices: 1, 2, 3, 4, 5, 6, ...
III
I
II
IV
V
Terms: a, a , a , a , a , a , ...
I II III IV
First Differences: b, b , b , b , b , ...
I
III
II
Second Differences: c, c , c , c , ...
Third Differences: d, d, d, . . .
II
I
Since a = a + b, a = a +2b + c, a III = a +3b +3c + d, a IV = a +4b +
6c +4d,... , the general term, or the term whose index is x,is
(x − 1) (x − 1) (x − 2) (x − 1) (x − 2) (x − 3)
a + b + c + d,
1 1 · 2 1 · 2 · 3
so that the general term is formed from the differences. Since we have
I II I III II I
b = a − a, c = a − 2a + a, d = a − 3a +3a − a,
when these values are substituted, the general term will be
III (x − 1) (x − 2) (x − 3) II (x − 1) (x − 2) (x − 4)
a − 3a
1 · 2 · 3 1 · 2 · 3
I (x − 1) (x − 3) (x − 4) (x − 2) (x − 3) (x − 4)
+3a − a .
1 · 2 · 3 1 · 2 · 3
This can also be expressed as
(x − 1) (x − 2) (x − 3) (x − 4) a III 3a II 3a I a
− + − .
1 · 2 · 3 x − 4 x − 3 x − 2 x − 1

44. Now let a series of any order be given:
Indices: 1, 2, 3, 4, 5, 6, ...