(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 + + (15.) 3 VOL. VII. ver correo er Ee ver co + 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 * * * * * * * * * * * (18.) (19.) (20.) In this last form it was necessary to add the small constant C to A The following is an application of the last form to the example of the moon's Right Ascension already given. From (13) we obtain, 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. 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