Example 6.
| x. | u = tan x. | δu. | δ2u. | δ3u. | δ4u. |
| + | + | + | + | ||
| 11147 | 62 | ||||
| 74°.0 | 3.48741 | 700 | 8 | ||
| 11847 | 70 | ||||
| 74°.5 | 3.60588 | 770 | 9 | ||
| 12617 | 79 |
| x | u0 + θΔu0. | θV1. | 1−θV0. | θV1 + 1−θV0. | u. |
| 73°.6 | · | −22 35 | · | · | · |
| 73°.7 | · | −39 11 | · | · | · |
| 73°.8 | · | −44 71 | · | · | · |
| 73°.9 | · | −33 54 | · | · | · |
| 74°.0 | 3.48741 00 | 3.48741 | |||
| 74°.1 | 3.51110 40 | −24 58 | −33 54 | −58 12 | 3.51052 |
| 74°.2 | 3.53479 80 | −43 02 | −44 71 | −87 73 | 3.53392 |
| 74°.3 | 3.55849 20 | −49 18 | −39 11 | −88 29 | 3.55761 |
| 74°.4 | 3.58218 60 | −36 89 | −22 35 | −59 24 | 3.58159 |
| 74°.5 | 3.60588 00 | 3.60588 |
The following are the values of the coefficients of u1, δ2u1, δ4u1, and δ6u1 in (9) for certain values of n. For calculating the four terms due to δ2u1 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. δ2u. | co. δ4u. | co. δ6u. |
| + | − | + | − |
| .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. δ2u. | co. δ4u. | co. δ6u. |
| + | − | + | − |
| .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. δ2u. | co. δ4u. | co. δ6u. |
| + | − | + | − |
| 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 |