13. Ordinates not Equidistant.—When the successive ordinates in the graph of u are not equidistant, i.e. when the differences of successive values of x are not equal, the above principle still applies, provided the differences are adjusted in a particular way. Let the values of x for which u is tabulated be a = x0 + αh, b = x0 + βh, c = x0 + γh,... Then the table becomes
| x. | u. | Adjusted Differences | ||
| 1st Diff. | 2nd Diff. | &c. | ||
| · | · | · | · | |
| · | · | · | · | |
| · | · | · | · | |
| a = xα | uα | · | · | |
| (α, β) | ||||
| b = xβ | uβ | (α, β, γ) | ||
| (β, γ) | ||||
| c = xγ | uγ | · | · | |
| · | · | · | · | |
| · | · | · | · | |
| · | · | · | · | |
In this table, however, (α, β) does not mean uβ − uα, but uβ − uα ÷ (β − α); (α, β, γ) means {(β, γ) − (α, β)} ÷ ½(γ − α); and, generally any quantity (η, ... φ) in the column headed “rth diff.” is obtained by dividing the difference of the adjoining quantities in the preceding column by (φ − η)/r. If the table is formed in this way, we may apply the principle of § 12 so as to obtain formulae such as
| u = uα + | x − c | · (α, β) + | x − a | · | x − b | · (α, β, γ) + ... |
| h | h | 2h |
(21),
| u = uγ + | x − a | · (β, γ) + | x − c | · | x − b | · (α, β, γ) + ... |
| h | h | 2h |
(22).
The following example illustrates the method, h being taken to be 1°:—
Example 8.
| x. | u = sin x. | 1st Diff. (adjusted). | 2nd Diff. (adjusted). | 3rd Diff. (adjusted). |
| + | − | − | ||
| 20° | .3420201 | |||
| 162932 50 | ||||
| 22° | .3746066 | 1125 00 | ||
| 161245 00 | 48 75 | |||
| 23° | .3907311 | 1222 50 | ||
| 158800 00 | 48 30 | |||
| 26° | .4383711 | 1303 00 | ||
| 156194 00 | 47 49 | |||
| 27° | .4539905 | 1445 47 | ||
| 151857 60 | 46 00 | |||
| 32° | .5299193 | 1583 48 | ||
| 145523 67 | ||||
| 35° | .5735764 |