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For interpolating to sixths, putting w = 6, we get

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

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

+ 110 For interpolating to twelfths, putting w=12, we get

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

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,

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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 82, or at least the last places of sa, are obtained ; and by means of it the interior places of gl are readily filled in, when the initial number of gl 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.


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. Svo 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. Svo . vol. Boston. 1856.

Annual Report of the Adjutant-General of the Commonwealth of Massachusetts, with Reports from the Quartermaster-General, etc.,

for 1861 – 1863. 3 vols. 8vo. Boston. 1862 - 64. Massachusetts Historical Society.

Collections. Vol. VII. Fourth Series. 8vo. Boston. 1865. Observatory of Harvard College.

On the Right Ascension of the Pole Star, as determined from Observation. By Truman Henry Safford, Assistant at the Observatory of Harvard College. (From the Proceedings of the American Academy of Arts and Sciences.. Vol. VL] 8vo pamph. Cam

bridge. 1864 Museum of Comparative Zoology.

Annual Report of the Trustees of the Museum of Comparative Zoology at Harvard College, in Cambridge, together with the Re

port of the Director, 1864. 8vo pamph. Boston. 1865. Boston Athenæum.

List of Books added to the Library of the Boston Athenæum from December 1, 1863, to December 1, 1864. 8vo pamph. Boston. Essex Institute.

Proceedings. Vol. III., IV. No. 2-4. 8vo. Salem. 1864. American Antiquarian Society.

Proceedings of the American Antiquarian Society, at the Annual Meeting held in Worcester, October 21, 1864. 8vo pamph. Bos

ton. 1864. Portland Society of Natural History.

Proceedings. Vol. I. pp. 97 – 128. 8vo. Portland. 1864. Dartmouth College

Catalogue of the Officers and Students .... for the Academical Year 1864 - 65. 8vo. Hanover, N. H. 1864.

Catalogus Senatus Academici et eorum qui Munera et Officia Academica gesserunt. .... 8ro. Hanoveræ, Neo Hant. 1864. Editors of American Journal of Science and Arts.

Journal. N. S. Vol. XXXVIII., XXXIX. 8vo. New Haven. 1864 - 65. Lyceum of Natural History of New York.

Annals. Vol. VII. No. 13-16. Vol. VIII. No 1. 8vo. New York. 1862 - 63. Albany Institute.

Transactions. Vol. IV. 8vo. Albany. 1858 - 64. Society for the Promotion of Useful Arts in the State of New York.

Transactions. Vol. IV. Part 2. Sro Albanr. 1819.

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