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, & 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+6; 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_6 to F would give several of the middle interpolations in duplicate, but it is unnecessary to take it so as to have more than one.

-6

-3

-6

+6

In interpolating to sixths, it is evident that, instead of putting w equal to six, and using the formula from F to F+3, 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 F 干 2, which are very simple, since three of numerical coefficients are ciphers in F4, and two of them in FBy 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 F, and F1, we can put ∞ = 12, and use F+6 and F3; 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 F4, 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:

d. h.

m. S.

m.

8.

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,

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 B+ C, or of their complements when the multiples are negative; and so on.

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 Fx 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 F, 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

The following very convenient method of interpolation was published several years ago in an imperfect form in the Mathematical Monthly. The expressions for the constants have been here obtained in a quite different manner, and are given in a different function of the differences, which are more exact, and the whole matter is given in an improved form.

If we put in (1)

(8.)

FFB x + B2 x2 + B x3,

and determine B1, B, and B, so as to make the expression coincide with (1) when x = }},x= and x= 1, we obtain very nearly, reinstating AA2 for A,1, A3-A for A,3, &c.,

The greatest possible error, E, in using (8) with these constants instead of (1), is only

x2·

(11.) d2 = B1 + (2 x − 1) B2 + (3 x3 — 3 x + 1) B3,

which may be put into the following form :

w

(12.) d, = A + (w + 1 − 2 x) B + (} w2 + } ∞ +1+3x2 -3x-3x) C,

By this formula we obtain the first differences, d1, d2, d3, &c., of the interpolated numbers, and from them the numbers themselves. The formula can be used for all values of ; but when it is taken greater than 12, the numerical coefficients of B and C obtained from (12) for all values of x are not all small enough to be convenient in computation. For all values of a not greater than 12, by making some slight modifications, we can obtain numerical coefficients convenient in computation. The following are some of the forms most frequently used. For interpolating to thirds, putting = 3, (12) gives,

For interpolating to fourths, putting = 4, we get