Suppose, for instance, that the values of tan x in § 6 extended only from x = 60° to x = 80°, we could then complete the table of differences by making the entries shown in italics below.
Example 7.
| x. | u = tan x. | δu. | δ2u. | δ3u. | δ4u. | δ5u. | δ6u. |
| + | + | + | + | + | + | ||
| 6775 | 34 | ||||||
| 60° | 1.73205 | 425 | 9 | ||||
| 7200 | 43 | ||||||
| 61° | 1.80405 | 468 | 9 | ||||
| 7668 | 52 | ||||||
| 62° | 1.88073 | 520 | 9 | ||||
| 8188 | 61 | ||||||
| 63° | 1.96261 | 581 | 10 | ||||
| 8769 | 71 | ||||||
| 64° | 2.05030 | · | 652 | · | 9 | ||
| · | · | · | · | · | · | · | · |
| · | · | · | · | · | · | · | · |
| · | · | · | · | · | · | · | · |
| 75° | 3.73205 | · | 3409 | · | 187 | · | 18 |
| 27873 | 788 | 73 | |||||
| 76° | 4.01078 | 4197 | 260 | 51 | |||
| 32070 | 1048 | 124 | |||||
| 77° | 4.33148 | 5245 | 384 | 64 | |||
| 37315 | 1432 | 188 | |||||
| 78° | 4.70463 | 6677 | 572 | 64 | |||
| 43992 | 2004 | 252 | |||||
| 79° | 5.14455 | 8681 | 824 | 64 | |||
| 52673 | 2828 | 316 | |||||
| 80° | 5.67128 | 11509 | 1140 | 64 | |||
| 64182 | 3968 | 380 |
For interpolating between x = 60° and x = 61° we should obtain the same result by applying Everett’s formula to this table as by using the advancing-difference formula; and similarly at the other end for the receding differences.
Interpolation by Substituted Tabulation.
10. The relation of u to x may be such that the successive differences of u increase rapidly, so that interpolation-formulae cannot be employed directly. Other methods have then to be used. The best method is to replace u by some expression v which is a function of u such that (i.) the value of v or of u can be determined for any given value of u or of v, and (ii.) when v is tabulated in terms of x the differences decrease rapidly. We can then calculate v, and thence u, for any intermediate value of x.
If, for instance, we require tan x for a value of x which is nearly 90°, it will be found that the table of tangents is not suitable for interpolation. We can, however, convert it into a table of cotangents to about the same number of significant figures; from this we can easily calculate cot x, and thence tan x.
11. This method is specially suitable for statistical data, where the successive values of u represent the area of a figure of frequency up to successive ordinates. We have first to determine, by inspection, a curve which bears a general similarity to the unknown curve of frequency, and whose area and abscissa are so related that either can be readily calculated when the other is known. This may be called the auxiliary curve. Denoting by ξ the abscissa of this curve which corresponds to area u, we find the value of ξ corresponding to each of the given values of u. Then, tabulating ξ in terms of x, we have a table in which, if the auxiliary curve has been well chosen, differences of ξ after the first or second are negligible. We can therefore find ξ, and thence u, for any intermediate value of x.
Extensions.
12. Construction of Formulae.—Any difference of u of the rth order involves r + 1 consecutive values of u, and it might be expressed by the suffixes which indicate these values. Thus we might write the table of differences