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If none of the differences of this order expressed in units of the last place of decimals exceed the limit

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—that is, the value of the largest binomial coefficient of the nth orderthe given numbers may be assumed to follow a regular law, and therefore to be correct to a unit in the last figure.

If some differences exceed this limit, their quotient by the above binomial coefficient may be considered to show the maximum error with which the number opposite it is probably affected.

We can thus detect an isolated error in a series of numbers with great certainty. Suppose, for example, an error of 2 in some number of the series. Differencing the series 0, 0, 0, 2, 0, 0, 0, we shall find the four largest differences of the fifth order to be 10, 20, — 20, +10, which would enable us to hit at once upon the erroneous number and judge of the magnitude of its error.

An error near the beginning and end of the series of numbers of which the differences are taken cannot be detected by the differences unless it is considerable. If, for instance, the first or last number is in error by 1, the error of each order of differences will only be 1, as we may easily see by the following example:

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It is only in those differences which are on or near the same lineas the numbers which are magnified in the way we have shown. But at the beginning and end of the series we cannot determine these differences.

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Examining the various tables of differences, we see that n numbers. have n 1 first differences, n 2 second differences, and so on, the number diminishing by 1 with each succeeding order. Hence, unless the number of given functions exceeds the index expressing the order of differences which we have to form, no certain conclusion can be drawn.

What is here said of the correctness of the numbers when the differences run properly must be understood as applicable to isolated errors only. If all the numbers were subject to an error following a regular law, this error would not be detected by the differences because, from the nature of the case, the latter only show deviations. from some regular law.

27. Fundamental Formula of Interpolation.

We suppose a series of numbers to be differenced in the way already shown, and the various differences to be designated as in the following scheme, which is supposed to be a selection from a series preceding and following it.

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It will be seen that the lower indices are chosen so as to show on which line a difference of any order falls. Thus all quantities with index 2 are on one horizontal line, those with index § = 2 are half a line below, etc. This notation is a little different from that used in algebra, but the change need not cause any confusion. It is shown in algebra that if n be any index, we have

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the notation being changed as in the preceding scheme.

Now the fundamental hypothesis of interpolation is that this formula, which can be demonstrated only for integral values of n, is true also for fractional values; that is, for values of the function u between those given in the table or in the above scheme. We therefore suppose this formula to express the value of the function u for any value of n between 0 and 1.

For values between + 1 and + 2 we have only to increase the indices of u and its differences by unity, thus:

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and by supposing n to increase from 0 to 1 in this formula we shall have values of u from u, to u2.

Increasing the indices again—that is, applying our general formula to a row of quantities one line lower-we shall have

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The equation (a) is known as Newton's formula of interpolation.

28. Transformations of the Formula of Interpolation.

In the equation (a) and those following it, the formula of interpolation is not in its most convenient form. We shall therefore transform it so that the differences employed shall be symmetrical with respect to the functions between which the interpolation is to be made.

In working these transformations we shall suppose the sixth and following orders of differences to be so small as not to affect the result. These differences being supposed zero, any two consecutive fifth differences may be supposed equal.

First transformation. Let us first find what the original formula (a) will become when, instead of using the series of differences 4' 4" 4"", 4, etc.,

we use

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To effect the transformation we must find the values of the first series of differences in terms of the second, and substitute them in the formula (a).

We find, by the mode of forming the differences,

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for which, because we suppose the values of 4 to be equal, we may put

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Reducing by collecting the coefficients of equal differences, we find

1.2

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Second transformation. Next, instead of the series of this last

formula, (b),

let us use

4't, 4", 4"'"', 4, etc.,

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2)

Δ

1.2.3.4. 5

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To effect this transformation we substitute in (b) for 4', 4′′, etc.,

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The series (b) then changes into

(n+1)n (n − 1)

1.2

0

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1.2.3

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Third transformation. Stirling's formula. We effect a third transformation by taking the half sum of the equations (b) and (c), which gives us a formula perfectly symmetrical with respect to the lines of differences, namely,

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2
- 1) (n2 - 4) 4°—†+4°†
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which is known as Stirling's formula of interpolation.
It will be seen that we have put

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Fourth transformation. In the equation (b), instead of the series of differences

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Fifth transformation. Bessel's formula. Let us take half the sum of the equations (e) and (b). We then have

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which is commonly known as Bessel's formula of interpolation, and which is the one most convenient to use in practice.

In applying this formula to find a value of the function intermediate between two given values, we must always suppose the index O to apply to the given value next preceding that to be found, and the index 1 to apply to that next following. The quantity n will then be a positive proper fraction.

29. Example of interpolation to halves. If we increase the logarithms of 30, 31, etc., already given, by unity, we shall have the logarithms of 300, 310, 320, etc. It is required to find, by interpolation, the logarithms of the numbers half way between the given ones. (omitting the first and last), namely, the logarithms of 315, 325, 335, etc.

Here, the required quantities depending upon arguments half way between the given ones, we have n = 4, and the values of the Besselian coefficient, so far as wanted, are

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