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Bs = 2 · 3 · 42 4,3 — († — rt of ‡) 4,5 + 1's 4, . . . . .),

B,

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B1 = 2 · 3 · 4 · w1 (41 — † 4o . . . . .),

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and in which the maximum possible error E of any interpolated number arising from neglected terms is only

(5.)

E = ± (3700 4,5 — 2800 46

τσεσσσ Δ,

.).

The constants B1, B2, &c. are so determined as to make the two preceding expressions of F, correspond for the four values of x = & w, and xw; so that, corresponding at equal intervals of w, they cannot differ much for any intermediate value of x, as is shown by (5). The advantage of this last formula over the preceding is, that with only four terms containing the variable you have nearly all the accuracy of seven or eight terms of the former. But it can only be used within the limit of before and after Fo; so that in interpolating it requires the constants B1, B2, &c. to be computed for every interval or given value of F

As the unit of x is arbitrary, when the interpolations are made at equal intervals it can be taken equal to one of the equal parts of x corresponding to the interpolated values of F, and then will represent the number of interpolated intervals contained in one of the original intervals; that is, w— 1 will represent the number of interpolations in each original interval. In this case the value of x used in interpolating is always one of the numbers 1, 2, 3, &c.; and if the number of interpolations to each original interval is not too great, the different terms in the expression of F, are readily obtained after the constants B1, B2, &c. have been computed.

For all cases in which the value of x does not exceed 6, that is, in which does not exceed 12, the preceding formula may be put into a form still more convenient for interpolating. The preceding expression of F, for all values of x from 6 to 6 gives

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This again may be transformed into the following form:
:-

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and in which the minus sign must be used for interpolated values of F preceding Fo, and the plus sign for those following F. In the latter transformations no small terms have been omitted; so that this last form is of the same degree of accuracy as the preceding one, and

it is so arranged that there are only four multiplications of the constants to be performed by numbers which are small and convenient for that purpose, in order to obtain any one of the interpolated values of F., whereas in the preceding form there are for the most part ten, the number being equal to the sum of the exponents of x in (3). This formula is accurate for all cases in which it is necessary to use eight orders of differences, and in all ordinary cases in which it is necessary to use only four or five orders of differences it is quite simple. In general, it is only necessary to compute the three constants A, B, and C, using B for B', for even then the maximum possible error is only of the order 40 4. This formula is applicable in all cases in which the number of interpolations does not exceed twelve. If we wish to interpolate to twelfths, o in the expression of the preceding constants A, B, C, &c. must be put equal to 12; if to tenths, equal to 10; and so for any other number. If we interpolate to twelfths, we must use (6) from F_ to F+e; if to tenths, from F-5 to F+5; and so on. In this way we get the middle interpolated number from two sets of constants; first, by going forward from Fo, and secondly, by going back from F, which is the F in the formula belonging to the next set of constants. This furnishes a very good check for the accuracy of the interpolations in addition to that of the regularity of the differences. In cases in which is less than twelve, the formula from F_ to F+ 6 would give several of the middle interpolations in duplicate, but it is unnecessary to take it so as to have more than one.

3

+39

In interpolating to sixths, it is evident that, instead of putting @ equal to six, and using the formula from F_ to F we can put it equal to twelve, and use only the functions of F with even subscript numbers from F-6 to F+6. By so doing, we have the advantage of using the functions F4 and F2, which are very simple, since three of numerical coefficients are ciphers in F4, and two of them in F2. By putting = 10, and using these same functions of x, we have a very convenient formula for interpolating to fifths; but it does not give any one of the interpolations in duplicate as a check, which, perhaps, is always unnecessary where the number of interpolations is so small, the regularity of the differences being a sufficient check. Also, in interpolating to fourths, instead of putting = 4 and using FF2 and F1, we can put = 12, and use FF6 and FF3; but it is much better to put = 8, and use FF4 and FF2, which, for reasons already stated, are much more simple. In interpolating to thirds we can

put = 12 and use F‡, which comprises only the two constants A and B, and hence is very simple.

As an example of the application of the preceding formula, let it be required to interpolate the moon's Right Ascension to twelfths, that is, to every second hour, having the Right Ascension and the differences given for each day at noon, as follows: :

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m. 8.

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Ap. 1 21 59 38.64 +50 46.58 +23.84 +18.97 +6.09 −2.22 −1.62

2 22 50 25.22 51 29.39 42.81 22.84 3.87 3.75 1.53 3 23 41 54.61 52 35.04 65.65 22.96 0.12 5.68 1.93

With these differences we get from (7), putting ✩ = 12,

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With these values (6) gives the following, in which the first column contains the multiples of a few of the last places of — A+B and AB, or of their complements, when the multiples are negative, the second, the multiples of BC and BC, or of their complements when the multiples are negative; and so on.

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D

In this example the part of the formula (6) depending upon is insensible. By combining the decimal part .640 of F in the first group with the first three columns, we get the decimal part of F, from F-6 to Fe. Thus .640.818+.863.931.252, only the first two decimals of which need be written down; also .640 +.181 +.359.978 = .158, for which .16 is written; and so on. From Foto F the resulting sum of the numbers is written in the line beneath the one in which the numbers are written. The two lines in brackets, the first combined with the decimal part of F, in the first - group, the second with that of F in the second group, both give Fo, which is also F-6 of the second group. A check for the accuracy of the interpolation, as has been stated, is that these should give the same result. The last two figures of the interpolated numbers being thus obtained, and also their differences, as represented in the second group, after the initial differences of the first group only are once obtained, the remaining part of the interpolated numbers is readily filled in, as represented in the first group of the preceding example.

As a second example, let it be required to interpolate the preceding Right Ascensions to fourths. Putting w = 8 in (7), we get from the given differences

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