PROCEEDINGS OF THE A MERICAN ACADEMY OF ARTS AND SCIENCES. SELECTED FROM THE RECORDS. VOL. VII. Five hundred and Afty-fourth Meeting. August 9, 1865. — STATUTE MEETING. The following memoirs were presented by title : - the Products of the Destructive Distillation of Lime Soap. By C. M. WARREN and F. H. STORER. C. M. WARREN and F. H. STORER. Five hundred and fifty-fifth Meeting. September 12, 1865. — ADJOURNED STATUTE MEETING. The PRESIDENT in the chair. The President called the attention of the Academy to the recent decease of three of its members : Mr. George Livermore, the Treasurer of the Academy ; Bishop Alonzo Potter of Pennsylvania, of the Associate Fellows; and Sir William Jackson Hooker, of the Foreign Honorary Members. VOL. VII. and Joinen ногоо MR. FERREL made the following communication on certain Formulæ of Interpolation. The necessity of frequent interpolations in almost all kinds of computations renders it important that the most convenient formulæ possible should be devised for that purpose. The following formulæ are especially designed to facilitate interpolations where a number of them are to be made at equal intervals between values of a function given or computed for equal intervals of the variable: Let F, be any function of x, given or computed, for the equal intervals of x= -w,x=0, x=w, I=2w, &c., and let A!, A3, A', &c. express the different orders of finite differences. By writing A, for *(4, + 4,') = 4;' – 44, 4,9 for 4 (AS; + 4) = 4; - $40, &c., we have, (1.) F=F, + A, 2 + A,2° + Aş ..... in which 4= (1.' – $1,3 +3543 - do 4.....) 4, = 2:5-(' – 34+ rão a? .....) A= 2.3.4.5.6 (49 – ? .....) = 2.3.4.5.6.7.0 (' ....) 2.3.4 2.3.4.5 2.3.4.5 2. 3. 4. 5. 6. 7. The preceding formula may be used for interpolating F, for any value of x positive or negative within a certain range, but the greater the value of x the greater the effect of the neglected orders of differences upon the interpolated numbers, and if i orders of differences are used, it may become quite inaccurate if x is taken greater than jiw. If the value of x is confined within the limits of Ft w, instead of the preceding formula, we may use the following of only fout variable terms without sensible error : um and in which the maximum possible error E of any interpolated number arising from neglected terms is only The constants B, B2, &c. are so determined as to make the two preceding expressions of F, correspond for the four values of x = F1w, and x = F Lw; so that, corresponding at equal intervals of {w, they cannot differ much for any intermediate value of x, as is shown by (5). The advantage of this last formula over the preceding is, that with only four terms containing the variable you have nearly all the accuracy of seven or eight terms of the former. But it can only be used within the limit of fw before and after Fo; so that in interpolating it requires the constants B, B2, &c. to be computed for every interval or given value of Fg. As the unit of x is arbitrary, when the interpolations are made at equal intervals it can be taken equal to one of the equal parts of x corresponding to the interpolated values of F., and then w will represent the number of interpolated intervals contained in one of the original intervals ; that is, w- 1 will represent the number of interpolations in each original interval. In this case the value of x used in interpolating is always one of the numbers F 1, F2, F 3, &c.; and if the number of interpolations to each original interval is not too great, the different terms in the expression of F, are readily obtained after the constants B, B2, &c. have been computed. For all cases in which the value of x does not exceed 6, that is, in which w does not exceed 12, the preceding formula may be put into a form still more convenient for interpolating. The preceding expression of F, for all values of x from — 6 to + 6 gives. This again may be transformed into the following form : and in which the minus sign must be used for interpolated values of F. preceding F., and the plus sign for those following F,. In the latter transformations no small terms have been omitted ; so that this last form is of the same degree of accuracy as the preceding one, and |