New Series of the Mathematical Repository Volume 1 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1806 Excerpt: ...for the year 1798, Part I. Art. IX.) Or, the fluent of the above expression may be computed by a descending series which shall converge by the powers of-= Tv nearly. But, since the fluent is easily attainable by logram this and circular arches, which are series already computed, I shall now treat of that method. It is easy to perceive that the fluent of may be taken in this form, viz.-----r +:, 7 + / T, in which the, the coefficients A, B, and C, may be computed with great facility. But a better form for numerical calculation may be derived from Madam A%ncsi's Theorem in her Analytical Institutions, Book III. Sect. I. Art. 56; observing, first, that the indices of x, in the numerator of the fraction which she assumes, will always be an arithmetical progression decreasing by the constant difference m: And, feco-id'y, that when n (still using her notation isgreater than m, let m be taken from n as often as it may, which call p. times, and put the remainder, n--pm--r; then will y- XXn _-Bxr+um-zm+l__QxT+m-3m+i _Y)xr+um-4M+t + 8cC. (A"" + am)u xm + a"')u--1 y Ax#r.. '---:, from the fluxion of which equation the values x"-+ a- of A, B, C, D, &c. will easily be found, and then no more remains to be done than to compute the fluent of the term Axxr: by the well known method of logarithms and circular arches. Now, by means of this equation we shall find /--Tt t0 be = r--1 3.. + T-, 7-; and since the correct fluent of =-266866 in the present case where x =2, or 1 1--x6 = 26 = 1122462. Proceeding now to the second expression, put 3 = v, ----= 02987187, when = 2, 4 therefore 12998824, or 13 nearly the difference of the pre I/ X X ceding values of is the fluent of, generated 4 (--i) whilst x from--increases to 2. 10 Third Solution, by Mr...