PROCEEDINGS OF THE AMERICAN ACADEMY OP ARTS AND SCIENCES. SELECTED FROM THE RECORDS. VOL. VII. Five hundred and Afty-fourth Meeting. August 9, 1865. — STATUTE MEETING. The RECORDING SECRETARY in the chair. The following memoirs were presented by title:I. Examination of a Hydrocarbon-Naphtha, obtained from the Products of the Destructive Distillation of Lime Soap. By C. M. WARREN and F. H. STORER. II. Examination of Naphtha from Rangoon Petroleum. By C. M. WARREN and F. H. STORER. Five hundred and Afty-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. PROCEEDINGS OF THE AMERICAN ACADEMY 12. FERREL made the following communication on certain Formule of Interpolation. The secessity of frequens interpolations in almost all kinds of compois os redders it important that the most convenient formulæ posstjesbied be devised for that purpose. The following formulæ are espadly designed to facitate interpolations where a number of them are to be rede at equal intervals between values of a function given or coccied for equal intervals of the variable: Les F. be any faction of Z, giren or compated, for the equal interrals of 3-a I=0,=. i=2., &e- and let Al, A', A', &c. express the cerent orders of finite differences. By writing 4,' for ( +4)=4 -*4. ' for it ( + 4) = 4;- 44,4, & we bare, (1.) F.=F, + 40+ 4+ 4,7"..... in which The preceding formula may be used for interpolating P, for any value ofr 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 tiu. If the value of x is confined within the limits of Fje, instead of the preceding formula, we may use the following of only fout variable terms without sensible error: 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= F + w, and x = F fw; so that, corresponding at equal intervals of tw, 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 £w before and after Fo; so that in interpolating it requires the constants Bi, B2, &c. to be computed for every interval or given value of Fr. 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 Fr, 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, F 2, 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 , B , &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 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 Fr be any function of x, given or computed, for the equal intervals of x= - w, x=0, x=w, x=2w, &c., and let A, A2, A', &c. express the different orders of finite differences. By writing a," for (41, +4,+) = A' – 44, 4,for 4 (4 + 4) = 4,- 440*, &c., we have, (1.) F. = F. + 47x + 1x + A3 28..... in which 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 iw. If the value of x is confined within the limits of F1w, instead of the preceding formula, we may use the following of only four variable terms without sensible error: |