([theta] + 1)[theta]([theta] - 1)
---------------------------------[delta]^2u1
3!
([theta] + 2)([theta] + 1)[theta]([theta] - 1)([theta] - 2)
+ -----------------------------------------------------------[delta]^4u1 + ... (10)
5!
by _([theta])V1, we have, for interpolation between u0 and u1,
u_([theta]) = u0 + [theta][Delta]u0 + _([theta])V1 + _(1 - [theta])V0 (11),
the successive values of [theta] being 1/n, 2/n, ... (n-1)/n. For
interpolation between u1 and u2 we have, with the same succession of
values of [theta],
u_(1+[theta]) = u1 + _([theta])V1, V2 + _(1-[theta])V1 (12).
The values of _(1-[theta])V1 in (12) are exactly the same as those of
([theta])V1 in (11), but in the reverse order. The process is
therefore that (i.) we find the successive values of u0 +
[theta][Delta]u0, &c., i.e. we construct a table, with the required
intervals of x, as if we had only to take first differences into
account; (ii.) we construct, in a parallel column, a table giving the
values of _([theta])V1, &c.; (iii.) we repeat these latter values,
placing the set belonging to each interval h in the interval next
following it, and writing the values in the reverse order; and (iv.)
by adding horizontally we get the final values for the new table.
As an example, take the values of tan x by intervals of 1/2 deg. in x,
as found above (Ex. 5). The first diagram below is a portion of this
table, with the differences, and the second shows the calculation of
the terms of (11) so as to get a table in which the intervals are 0.1
of 1 deg. The last column but one in the second diagram is introduced
for convenience of calculation.
_Example 6._
+---------+-----------+---------+-----------+-----------+-----------+
| x. | u = tan x.|[delta]u.|[delta]^2u.|[delta]^3u.|[delta]^4u.|
+---------+-----------+---------+-----------+-----------+-----------+
| | | + | + | + | + |
| | | 11147 | | 62 | |
| 74 deg.0| 3.48741 | | 700 | | 8 |
| | | 11847 | | 70 | |
| 74 deg.5| 3.60588 | | 770 | | 9 |
| | | 12617 | | 79 | |
+---------+-----------+---------+-----------+-----------+-----------+
+----------+------------------+--------------+----------------+----------------+---------+
| | u0 + | | | _([theta])V1 + | |
| x. | [theta][Delta]u0.| _([theta])V1.| _(1-[theta])V0.| _(1-[theta])V0.| u. |
+----------+------------------+--------------+----------------+----------------+---------+
| 73 deg.6 | . | -22 35 | . | . | . |
| 73 deg.7 | . | -39 11 | . | . | . |
| 73 deg.8 | . | -44 71 | . | . | . |
| 73 deg.9 | . | -33 54 | . | . | . |
| 74 deg.0 | 3.48741 00 | | | | 3.48741 |
| 74 deg.1 | 3.51110 40 | -24 58 | -33 54 | -58 12 | 3.51052 |
| 74 deg.2 | 3.53479 80 | -43 02 | -44 71 | -87 73 | 3.53392 |
| 74 deg.3 | 3.55849 20 | -49 18 | -39 11 | -88 29 | 3.55761 |
| 74 deg.4 | 3.58218 60 | -36 89 | -22 35 | -59 24 | 3.58159 |
| 74 deg.5 | 3.60588 00 | | | | 3.60588 |
+----------+------------------+--------------+----------------+----------------+---------+
The following are the values of the coefficients of u1, [delta]^2u1,
[delta]^4u1, and [delta]^6u1 in (9) for certain values of n. For
calculating the four terms due to [delta]^2u1 in the case of n = 5 it
should be noticed that the third term is twice the first, the fourth
is the mean of the first and the third, and the second is the mean of
the third and the fourth. In table 3, and in the last column of table
2, the coefficients are corrected in the last figure.
TABLE 1.--n = 5.
+------+---------------+---------------+--------------------------+
|co. u.|co. [delta]^2u.|co. [delta]^4u.| co. [delta]^6u. |
+------+---------------+---------------+--------------------------+
| + | - | + | - |
| .2 | .032 | .006336 | .00135168 = 1/740 approx.|
| .4 | .056 | .010752 | .00226304 = 1/442 " |
| .6 | .064 | .011648 | .00239616 = 1/417 " |
| .8 | .048 | .008064 | .00160512 = 1/623 " |
+------+---------------+---------------+--------------------------+
TABLE 2.--n = 10.
+------+---------------+---------------+---------------+
|co. u.|co. [delta]^2u.|co. [delta]^4u.|co. [delta]^6u.|
+------+---------------+---------------+---------------+
| + | - | + | - |
| .1 | .0165 | .00329175 | .000704591 |
| .2 | .0320 | .00633600 | .001351680 |
| .3 | .0455 | .00889525 | .001887064 |
| .4 | .0560 | .01075200 | .002263040 |
| .5 | .0625 | .01171875 | .002441406 |
| .6 | .0640 | .01164800 | .002396160 |
| .7 | .0595 | .01044225 | .002115799 |
| .8 | .0480 | .00806400 | .001605120 |
| .9 | .0285 | .00454575 | .000886421 |
+------+---------------+---------------+---------------+
TABLE 3.--n = 12.
+------+---------------+---------------+---------------+
|co. u.|co. [delta]^2u.|co. [delta]^4u.|co. [delta]^6u.|
+------+---------------+---------------+---------------+
| + | - | + | - |
| 1/12 | .013792438 | .002753699 | .000589623 |
| 2/12 | .027006173 | .005363726 | .001145822 |
| 3/12 | .039062500 | .007690430 | .001636505 |
| 4/12 | .049382716 | .009602195 | .002032211 |
| 5/12 | .057388117 | .010979463 | .002307357 |
| 6/12 | .062500000 | .011718750 | .002441406 |
| 7/12 | .064139660 | .011736667 | .002419911 |
| 8/12 | .061728395 | .010973937 | .002235432 |
| 9/12 | .054687500 | .009399414 | .001888275 |
|10/12 | .042438272 | .007014103 | .001387048 |
|11/12 | .024402006 | .003855178 | .000748981 |
+------+---------------+---------------+---------------+