A)
are identical with the relations between the similarly placed terms in
(B); e.g. [beta] + [gamma] is the difference of [alpha] + 2[beta] +
[gamma] and [alpha] + [beta], just as c - b is the difference of c and
b: and d - c is the sum of c - b and d - 2c + b, just as [beta] +
2[gamma] + [delta] is the sum of [beta] + [gamma] and [gamma] +
[delta]. Hence if we take [beta], [gamma], [delta], ... of (B) as
being the same as b - a, c - 2b + a, d -3c + 3b - a, ... of (A), all
corresponding terms in the two diagrams will be the same.

Thus we obtain the two principal formulae connecting terms and
differences. If we provisionally describe b - a, c - 2b + a, ... as
the first, second, ... differences of the particular term a (S 7),
then (i.) the nth difference of a is

n.n - 1
l - nk + ... + (-1)^(n-2) ------- c + (-1)^(n-1) nb + (-1)^n a,
1.2

where l, k ... are the (n + 1)th, nth, ... terms of the series a, b,
c, ...; the coefficients being those of the terms in the expansion of
(y -x)^n: and (ii.) the (n + 1)th term of the series, i.e. the nth
term after a, is

n.n - 1
a + n[beta] + ------- [gamma] + ...
1.2

where [beta], [gamma], ... are the first, second, ... differences of
a; the coefficients being those of the terms in the expansion of (x +
y)^n.

4. Now suppose we treat the terms a, b, c, ... as being themselves the
first differences of another series. Then, if the first term of this
series is N, the subsequent terms are N + a, N + a + b, N + a + b + c,
...; i.e. the difference between the (n + 1)th term and the first term
is the sum of the first n terms of the original series. The term N, in
the diagram (A), will come above and to the left of a; and we see, by
(ii.) of S 3, that the sum of the first n terms of the original series
is

/ n.n - 1 \ n.n - 1 n.n - 1.n - 2
( N + na + ------- [beta] + ...) - N = na + ------- [beta] + ------------- [gamma] + ...
\ 1.2 / 1.2 1 . 2 . 3

5. As an example, take the arithmetical series

a, a + p, a + 2p, ...

The first differences are p, p, p, ... and the differences of any
higher order are zero. Hence, by (ii.) of S 3, the (n + 1)th term is a
+ np, and, by S 4, the sum of the first n terms is