For interpolating to sixths, putting & 6, we get

For interpolating to eighths, putting = 8, we get

δι

82

For interpolating to tenths, putting = 10, we get

For interpolating to twelfths, putting = 12, we get

3

1

+2

+6

+9

+4

0

3

6

6 C.

6

810

811

812

0

9

+4

In this last form it was necessary to add the small constant С 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 ∞ = 12,

With these values of the constants (20) gives the following re

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 Cis .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 s', from which 8, or at least the last places of 82, are obtained; and by means of it the interior places of d1 are readily filled in, when the initial number of ¿1 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.

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