To find u for x = 31°, we use the values for 26°, 27°, 32° and 35°, and obtain
| u = .4383711 00 + | 5 | (156194 00) + | 5 | · | 4 | (−1445 47) + | 5 | · | 4 | · | −1 | (−46 00) = .5150380, |
| 1 | 1 | 2 | 1 | 2 | 3 |
which is only wrong in the last figure.
If the values of u occurring in (21) or (22) are ualpha, ubeta, ugamma, ... ulambda, corresponding to values a, b, c, ... l of x, the formula may be more symmetrically written
| u = | (x − b) (x − c) ... (x − l) | uα + | (x − a) (x − c) ... (x − l) | uβ + ... |
| (a − b) (a − c) ... (a − l) | (b − a) (b − c) ... (b − l) |
| ... + | (x − a) (x − b) (x − c) ... | uλ |
| (l − a) (l − b) (l − c) ... |
(23).
This is known as Lagrange’s formula, but it is said to be due to Euler. It is not convenient for practical use, since it does not show how many terms have to be taken in any particular case.
14. Interpolation from Tables of Double Entry.—When u is a function of x and y, and is tabulated in terms of x and of y jointly, its calculation for a pair of values not given in the table may be effected either directly or by first forming a table of values of u in terms of y for the particular value of x and then determining u from this table for the particular value of y. For direct interpolation, consider that Δ represents differencing by changing x into x + 1, and Δ′ differencing by changing y into y + 1. Then the formula is
ux, y = (l + Delta)x (1 + Δ′)y u0,0;