In this notation, instead of using the symbol E, we use a symbol μ to denote the mean of two consecutive values of u, or of two consecutive differences of the same order, the suffixes being assigned on the same principle as in the case of the differences. Thus
μun+1/2 = ½(un + un+1, μδun = ½(δun-1/2 + δun+1/2, &c.
If we take the means of the differences of odd order immediately above and below the horizontal line through any value of x, these means, with the differences of even order in that line, constitute the central differences of the corresponding value of u. Thus the table of central differences is as follows, the values obtained as means being placed in brackets to distinguish them from the actual differences:—
| x | u | 1st Diff. | 2nd Diff. | 3rd Diff. | 4th Diff. |
| · | · | · | · | · | · |
| · | · | · | · | · | · |
| · | · | · | · | · | · |
| xn-2 | un-2 | (μδun-2) | δ²un-2 | (μδ³un-2) | δ4un-2 ... |
| δun-3/2 | δ³un-3/2 | ||||
| xn-1 | un-1 | (μδun-1) | δ²un-1 | (μδ³un-1) | δ4un-1 ... |
| δun-1/2 | δ³un-2 | ||||
| xn | un | (μδun) | δ²un | (μδ³un) | δ4un ... |
| δun+1/2 | δ³un+1/2 | ||||
| xn+1 | un+1 | (μδun+1) | δ²un+1 | (μδ³un+1) | δ4un+1 ... |
| δun+3/2 | δ³un+3/2 | ||||
| xn+2 | un+2 | (μδun+2) | δ²un+2 | (μδ³un+2) | δ4un+2 ... |
| · | · | · | · | · | · |
| · | · | · | · | · | · |
| · | · | · | · | · | · |
Similarly, by taking the means of consecutive values of u and also of consecutive differences of even order, we should get a series of terms and differences central to the intervals xn-2 to xn-1, xn-1 to xn, ....
The terms of the series of which the values of u are the first differences are denoted by σu, with suffixes on the same principle; the suffixes being chosen so that δσun shall be equal to un. Thus, if
σun-3/2 = C + un-2,
then
σun-1/2 = C + un-2 + un-1, σn+1/2 = C + un-2 + un-1 + un, &c.,
and also