Notation of Differences and Sums.
8. It is convenient to denote the terms a, b, c, ... of the series by u0, u1, u2, u3, ... If we merely have the terms of the series, un may be regarded as meaning the (n + 1)th term. Usually, however, the terms are the values of a quantity u, which is a function of another quantity x, and the values of x, to which a, b, c, ... correspond, proceed by a constant difference h. If x0 and u0 are a pair of corresponding values of x and u, and if any other value x0 + mh of x and the corresponding value of u are denoted by xm and um, then the terms of the series will be ... un-2, un-1, un, un+1, un+2 ..., corresponding to values of x denoted by ... xn-2, xn-1, xn, xn+1, xn+2....
9. In the advancing-difference notation un+1 − un is denoted by Δun. The differences Δu0, Δu1, Δu2 ... may then be regarded as values of a function Δu corresponding to values of x proceeding by constant difference h; and therefore Δun+1 − Δun denoted by ΔΔun, or, more briefly, Δ²un; and so on. Hence the table of differences in § 2, with the corresponding values of x and of u placed opposite each other in the ordinary manner of mathematical tables, becomes
| x | u | 1st Diff. | 2nd Diff. | 3rd Diff. | 4th Diff. |
| · | · | · | · | · | · |
| · | · | · | · | · | · |
| · | · | · | · | · | · |
| xn-2 | un-2 | Δ²un-3 | Δ4un-4 ... | ||
| Δun-2 | Δ³un-3 | ||||
| xn-1 | un-1 | Δ²un-2 | Δ4un-3 ... | ||
| Δun-1 | Δ³un-2 | ||||
| xn | un | Δ²un-1 | Δ4un-2 ... | ||
| Δun | Δ³un-1 | ||||
| xn+1 | un+1 | Δ²un | Δ4un-1 ... | ||
| Δun+1 | Δ³un | ||||
| xn+2 | un+2 | Δ²un+1 | Δ4un ... | ||
| · | · | · | · | · | · |
| · | · | · | · | · | · |
| · | · | · | · | · | · |
The terms of the series of which ... un-1, un, un+1, ... are the first differences are denoted by Σu, with proper suffixes, so that this series is ... Σun-1, Σun, Σun+1.... The suffixes are chosen so that we may have ΔΣun = un, whatever n may be; and therefore (§ 4) Σun may be regarded as being the sum of the terms of the series up to and including un-1. Thus if we write Σun-1 = C + un-2, where C is any constant, we shall have
| Σun = Σun-1 + ΔΣun-1 = C + un-2 + un-1, |
| Σun+1 = C + un-2 + un-1 + un, |
and so on. This is true whatever C may be, so that the knowledge of ... un-1, un, ... gives us no knowledge of the exact value of Σun; in other words, C is an arbitrary constant, the value of which must be supposed to be the same throughout any operations in which we are concerned with values of Σu corresponding to different suffixes.
There is another symbol E, used in conjunction with u to denote the next term in the series. Thus Eun means un+1, so that Eun = un + Δun.
10. Corresponding to the advancing-difference notation there is a receding-difference notation, in which un+1 − un is regarded as a difference of un+1, and may be denoted by Δ′un+1, and similarly un+1 − 2un + un-1 may be denoted by Δ′²un+1. This notation is only required for certain special purposes, and the usage is not settled (§ 19 (ii.)).
11. The central-difference notation depends on treating un+1 − 2un − un-1 as the second difference of un, and therefore as corresponding to the value xn; but there is no settled system of notation. The following seems to be the most convenient. Since un is a function of xn, and the second difference un+2 − 2un+1 + un is a function of xn+1, the first difference un+1 − un must be regarded as a function of xn+1/2, i.e. of ½(xn + xn+1). We therefore write un+1 − un = δun+1/2, and each difference in the table in § 9 will have the same suffix as the value of x in the same horizontal line; or, if the difference is of an odd order, its suffix will be the means of those of the two nearest values of x. This is shown in the table below.