which may similarly be regarded as an extension of the theorem
that, if n is a positive integer,

u.=u.+nau;+%Qa2u, + a3u;+ . (16).

There are other central-difference formulae besides those mentioned above; the general symbolical expression is 1¢9 = (cosh 0hD-l-sinh 0hD)u, , (17), where

cosh éhl) =;4, sinh § hD =%6 (18).

8. Comparative Accuracy:-Central-difference formulae are usually more accurate than advancing-difference formulae, whether we consider the inaccuracy due to omission of the “ remainder ” mentioned in the last paragraph or the error due to the approximative character of the tabulated values. The latter is the more important. If each tabulated value of u is within ibn of the corresponding true value, and if the differences used in the formulae are the tabular differences, i.e. the actual successive differences of the tabulated values of u, then the ratio of the limit of error of ug, as calculated from the first 7 terms of the series in (I), to ép is the sum of the first r terms of the series I +0-l-9(I -0) -l-9(I -0)(2 -9) +r%9(I -0)(2 -0)(3-9) + i9(1 "0)(2 -0)(3-@)(4-0)+1z"a'n0(1 -0) - - -(5-9)-lf - - ~. while the corresponding ratio for the use of differences up to 5”'u0 inclusive in (4) or up to 52Pu1 and ofpuo in (9) (i.e. in effect, up to 52P+1u5) is the sum of the first p-i-1 terms of the series I +615;0)+<1+0>@<(;;f)<2-0>+

(2 +0)(I +@)0(I -0)(2-0)(3-0)

-'“T31V*'*+~~-1

it 'being supposed in each case that 0 lies between 0 and 1. The following table gives a comparison of the respective limits of error; the lines I. and Il. give the errors due to the advancing-difference and the central-difference formulae, and the coefficient p is omitted throughout.

only from x=6o° to x=8o°, we could then complete the table of differences by making the entries shown in italics elow. Example 7.

x. it = tan x. 5u. 5214. 53u. 6414. 5%. Bfu + -l- -l~ + + +

o 5775 34

60 I '73205 425 9

7200 4 3

61 ° 1 -80405 668 468 5 9

7 2

62° 1 -88073 520 9

8188 61

63° 1»96261 581 I0

8769 71

64° 2-05030 652 9

75° 3-75205 7é73 3469 7f88 Ié7;3

2

76° 4-01078 4197 260

32070 1048 124

77° 4-33148 1 5245 334

0 373 I 5 1432 188

73 470463 6677 572

43992 2004 252

79° 5-14-455 8681 824

52673 2828 316

80° 5-67128 11509 1140

64182 3968 380

TABLE 4.

Error due to use of Differences up to and including

1st. 2nd. 3rd. 4th. 5th. 6th. 7th.

I. -500 625 813 l 1-086 1-497 2-132 3-147 5 II. -500 625 625 -696 -696 -745 -745 2 I. -500 580 724 -960 1-343 1-976 3-042 II. -500 580 580 -624 -624 -653 -653 I. -500 620 812 1-104 1-553 2-265 3°422 4 II. -500 620 620 ~688 -688 '734 '734 6 I. -500 620 788 I'O24 1-366 1-886 2-700 II. -500 620 620 -688 -688 -734 -734 8 I. -500 580 676 -800 -969 1-213 1-582 Q II. -500 580 580 -624 -624 L ~653 -653 ln some cases the differences tabulated are not the t 1 abu ar differences,

but the corrected differences; i.e. each difference, like each value of u, is correct within isp. It does not follow that these differences should be used for interpolation. Whatever formula is employed, the first difference should always be the tabular first difference, not the corrected first difference; and, further, if a central difference formula is used, each difference of odd order should be the tabular difference of the corrected differences of the next lower order. (This last result is indirectly achieved if Everett's formula-is used.) With these precautions (i) the central-difference formula is slightly improved by using corrected instead of tabular differences, and (ii.) the advancing-difference formula is greatly improved, being better than the central-difference formula with tabular differences, but still not so good as the latter with corrected differences. For 8= -5, for instance, supposing we have to go to fifth differences, the limits 11-497 and ='='6Q6, asgiven above, become =*=-627 and =¢=-575 respectively.

9. Completion of Table of Diferenoes.-If no values of it outside the range within which we have to interpolate are given, the series of differences will be incomplete at both ends. It may be continued in each direction by treating as constant the extreme difference of the highest order involved; and central-difference formulae can then be employed uniformly throughout the whole ran e.

Sfippose, for instance, that the values of tan x in § 6 extended 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 it to x may be such that the successive differences of it increase rapidly, so that interpolation-formulae cannot be employed directly. Other methods have then to be used. The best method is 'to replace it by some expression 'v which is a function of u such that (i.) the value of 71 or of it can be determined for any given value of u or of 1), and (ii.) when v is tabulated in terms of x the differences decrease rapidly. We can then calculate v, and thence 14, 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 signihcant figures; from this we can easily calculate cot x, and thence tan x. II. This method is specially suitable for statistical data, where the successive values of it 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 with the other is known. This may be called the auxiliary curve. Denoting by E the abscissa of this curve which corresponds to area u, we find the value of E corresponding to each of the given values of it. Then, tabulating E in terms 0f' x, we have a table in which, if the auxiliary curve has been well chosen, differences of £ after the first or second are negligible. We can therefore find 5, and thence u, for any intermediate value of x.

Extensions.

12. Construction of Formulas.-Any difference of u of the rth order involves r+I consecutive values of u, and it might be expressed by the suffixes which indicate these values. Thus we might write the table of differences

x. u. 151 Diff. mi Diff. T 3-11 Diff. 4th Diff. "7 “ 'il

I 1 (-i, o) i (*2, -'I,0, 1)

xo Wo f'I»0» I) (-2.-I,0.l.2)

(01 I) (Iy Ov If 2)

xl ul (0, I,2) (-1,0,1,2,3)

(1, 2) <<>. 1, 2. 3)

x2 742 (Iv 21 (Or 11 2131

- - (2, 3) (I. 2, 3. 4) .