Imágenes de páginas
PDF
EPUB
[ocr errors]

(13.) B= {-12—44°+(6+16 )AP+340–304....}

C=bl{4..... By this formula we obtain the first differences, 81, 82, 83, &c., of the interpolated numbers, and from them the numbers themselves. The formula can be used for all values of w; 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 w 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 w= 3, (12) gives,

+1) (14.) 82} = A

02B

- 2}c.

- 1 For interpolating to fourths, putting w= 4, we get

[ocr errors]
[ocr errors]

+

+

[ocr errors]
[ocr errors]

(15.)
= A

3
For interpolating to fifths, putting w=5, we get

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

VOL. VII.

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

ver correo er

[ocr errors]
[ocr errors]

Ee ver co

[ocr errors]
[ocr errors]
[ocr errors]

+ 5)

- 3 | For interpolating to sixths, putting w= 6, we get

For interpolating to eighths, putting ๒ = 8, we get

For interpolating to tenths, putting w= 10, we get

For interpolating to twelfths, putting w = 12, we get

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

* * * * * * * * * * *

(18.)

(19.)

(20.)

In this last form it was necessary to add the small constant C to A
in order to obtain small and convenient numerical coefficients of the
constant 6 O.

The following is an application of the last form to the example of

the moon's Right Ascension already given. From (13) we obtain,

putting w=12,

[ocr errors][ocr errors][subsumed][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][subsumed][ocr errors][ocr errors][ocr errors]

In the first column the complements .8876, .8131, &c. of the decimal part of B, since it is negative, are first written down in every twelfth place, and then immediately under, the decimal part itself, after which the multiples are readily filled in. In the second column C is written in one of the blank places of the formula, and its complement in the other, for the convenience of taking the multiples, and placed in parentheses to denote that they are not to be used in the summation of the terms. These two columns then being combined with the decimal part of the constant A + C, gives the decimal part of the seconds in 81, from which s?, or at least the last places of 89, are obtained; and by means of it the interior places of gl are readily filled in, when the initial number of 8l is known for the first group only. The first differences being thus obtained, the interpolated numbers are readily found. The last group of the preceding example shows the manner in which the computation is carried out at first ;. after which the remaining part is filled in, as shown in the first group. The computation might be readily carried out by first computing only the last two places of the differences. These have to be computed for one place more than the number of places required in the interpolated results, since small errors in the former might be accumulative in the latter.

In this method of interpolation the regularity of the differences is not a complete check, since a very small error in the value of B might not produce a sensible derangement of the differences. There must, therefore, be a separate check for B, and then the differences furnish a complete check against errors arising from any other source.

Professor Gray made a communication on a new Water Lily, Nymphæa tuberosa, of J. A. Paine, from the western part of the State of New York, remarkable for its tuberiferous rootstocks.

Mr. J. I. Bowditch was elected Treasurer, to fill the vacancy made by the decease of Mr. Livermore.

Mr. T. T. Bouvé was elected a member of the Finance Committee, to fill the vacancy left at the annual election.

Mr. Charles J. Sprague was appointed, on the nomination of the President, a member of the Auditing Committee.

DONATIONS TO THE LIBRARY,

FROM MAY 28, 1864, TO MAY 30, 1865. State of Massachusetts.

Nineteenth Report to the Legislature of Massachusetts relating to the Registry and Return of Births, Marriages, and Deaths in the Commonwealth for the Year ending December 31, 1860. By Oliver Warner, Secretary. 8vo vol. Boston. 1862.

Twentieth Report .... for 1861. .... 8vo vol. Boston. 1863.

Abstract of the Census of Massachusetts, 1860, from the Eighth United States Census, with Remarks on the Same. 8vo vol. Boston. 1863.

Returns of the Railroad Corporations in Massachusetts, together with Abstracts of the Same, 1860-1863. 4 vols. 8vo. Boston. 1861 – 64.

General Statutes of the Commonwealth of Massachusetts : revised by Commissioners. . ... 8vo vol. Boston. 1860.

Reports of Controverted Elections in the House of Representatives of the Commonwealth of Massachusetts from 1780 to 1852. 8vo vol. Boston. 1853.

Acts and Resolves passed by the General Court of Massachusetts in the Years 1860 – 1864. 5 vols. 8vo. Boston. 1860-64.

Journal and Documents of the Valuation Committee of the Year 1860 ; together with Acts of 1861 to secure a uniform Description and Appraisal of Estates in the Commonwealth. Published by Oliver Warner, Secretary. 8vo vol. Boston. 1861.

Report to the Governor and Council concerning the Indians of the Commonwealth, under the Act of April 6, 1859. By John Milton Earle, Commissioner. Svo vol. 1861.

Report of the Joint Committee of 1860 upon the proposed Canal to unite Barnstable and Buzzard's Bays. 8vo vol. Boston. 1864.

Statistics of the Condition and Products of Certain Branches of Industry in Massachusetts for the Year ending April 1, 1845. By John G. Palfrey. 8vo vol. Boston. 1846.

Statistical Information relating to Certain Branches of Industry .... for the Year ending June 1, 1855. By Francis De Witt. 8vo vol. Boston. 1856.

Annual Report of the Adjutant-General of the Commonwealth of

V

« AnteriorContinuar »