12 PROCEEDINGS OF THE AMERICAN ACADEMY 



In the first column the complements .8876, .8131, &c. of the deci- 

 mal part of B, since it is negative, axe 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 G 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 -\- G, gives the decimal part of the seconds in 

 5S from which 5^ or at least the last places of 5^, are obtained ; and 

 by means of it the interior places of 8^ are readily filled in, when the 

 initial number of 8^ is known for the first group only. The first dif- 

 ferences 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 re- 

 maining 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" resul^, 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, Nijmphcea 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. Bouvd was elected a member of the Finance Com: 

 mittee, 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. 



