Imágenes de páginas
PDF
EPUB

exceed half a unit of the last decimal in the given quantities, or five units in the additional decimal added on in dividing.

To correct these little imperfections after the interpolation of the second differences, but before that of the first differences, the sum of the last two figures in each triplet of second differences should be formed, and if it does not agree with the difference of the first differences, the last figures of the second difference should each be slightly altered, to make the sum exact.

The first differences can then be formed by addition.

In the same way, the sum of three consecutive first differences should be equal to the difference between the given quantities. If, as is generally the case, this condition is not exactly fulfilled, the differences should be altered accordingly. This alteration may, however, be made mentally while adding to form the required interpolated functions.

As an exercise for the student we give the continuance of the sun's declination for the remainder of the month, to be interpolated for the intermediate dates from July 15th onward:

[merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

As another exercise the logarithms of the intermediate numbers from 998 to 1014 may be interpolated by the following table:

[blocks in formation]

32. Interpolation to fifths. Let us next investigate the formulæ when every fifth quantity is given and the intermediate ones are to be found by interpolation. By putting n = in the Besselian formula, we shall have the value of the interpolation function second

following one of the given ones, and by putting n = that third following. The difference will be the middle interpolated first difference of the interpolated series. Putting n = in (ƒ), we have

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

The difference of these expressions, being reduced, gives

[merged small][merged small][merged small][ocr errors][subsumed][merged small][merged small][merged small][subsumed][merged small][ocr errors][merged small][ocr errors][subsumed]

The term in 4 will not produce any effect unless the fifth differences are considerable, and then we may nearly always, in practice, put instead of 14.

The interpolated second differences opposite the given functions are most readily obtained by Stirling's formula, (d). Putting n = }, we have the following value of the interpolated first differences immediately following a given value of the function:

[ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[ocr errors]
[ocr errors][merged small][merged small][merged small]

Again, putting n = 1, and changing the signs, we find for the first difference next preceding a given function

[ocr errors][merged small][subsumed][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

The difference of these quantities gives the required second dif

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]

As an example and exercise we show the interpolation of logarithms when every fifth logarithm is given:

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

FORMULÆ

FOR THE SOLUTION OF

PLAIN AND SPHERICAL TRIANGLES.

REMARKS.

1. It is better to determine an angle by its tangent than by its sine or cosine, because a small angle or an angle near 180° cannot be accurately determined by its cosine, nor one near either 90° or 270° by its sine.

Sometimes, however, the data of the problem are such that the angle can be determined only through its sine or cosine. Any uncertainty which may then arise from the source pointed out is then inherent in the problem; e.g., if the hypothenuse and one side of a right triangle are 0.39808 and 0.39806 respectively (sixth and following decimals being omitted), the value of the included angle may be anywhere between 0° 25′ and 0° 42', no matter what method of computation be adopted.

2. If the sine and cosine can be independently computed, their agreement as to the angle will generally serve as a check on the accuracy of the computation. If they agree, their quotient will give the tangent.

3. It is desirable, when possible, to have a check upon the accuracy of the computation; that is, to make a computation which must give a certain result if the work is right. But no check can give a positive assurance of accuracy: all it can do is to make it more or less improbable that a mistake exceeding a certain limit exists.

4. In the following list several formulæ are sometimes given as applicable to the same problem. In such cases, the most convenient for the special purpose must be chosen.

« AnteriorContinuar »