The complete solution of numerical equations : in which, by one uniform process, the imaginery as well as the real roots are easily determined / by William Rutherford.

  • Rutherford, William, 1798?-1871.
Date:
1849
Catalogue details

Licence: Public Domain Mark

Credit: The complete solution of numerical equations : in which, by one uniform process, the imaginery as well as the real roots are easily determined / by William Rutherford. Source: Wellcome Collection.

    10/28 (page 10)
    10. If we compare the equation (1), divested of its second term, viz. -ir lx c — 0 (13), with the equation (9), it will he seen that the roots of (13) are just twice the roots of (9), and since the alternate terms of these two equations have contrary signs, it is evident that if r^, denote the roots of eq, (13) ; then will the roots of eq. (9) be ^ and Hence the values of a will be half those of x, with contrary signs. 11. We shall now apply these formulce to one or two examples, and when the proposed equation is to be divested of [ its second term, the approximation to the root of the transformed equation, will be continued in a connected form ,having the appearance of only one single operation. EXAMPLE I. Solve completely the cubic equation x^ — 6x — 6 = 0. Let r, a. and a — y' — /3 be the roots of the proposed equation, then since the last term is negative, the real root r will he positive, and the rational part a of the two other roots, whether they be real or imaginary, will be T negative and equal to — The value of /3 will indicate whether the two other roots are real or imaginary, according as its sign is — or When the second term is absent in any equation, the real root may first be found, as in the following operation.* 1 0 —6 — 6 ( 2-8473221019 2 4 — 4 2 —2 — 10 2 8 9152 4 6 — 848 2 544 714304 ^8 1144 — 133696 8 608 127795423 7 6 1752 —5900577 8 3376 5495637 8 44 178576 —404940 4 3392 366430 8 48 181968 —38510 4 596 8 9 36643 8 527 182564 8 9 —1867 7 597 3 8 1832 8 534 183162 2 7 —35 7 25 6 2 18 185141 183187 8 9 — 17 25 6 2 16 183213 5 — 1 1 7 183215 2 1 7 183217 Hence the real root of the equation is 2-8473221019, and therefore the rational part of the two other roots is r . _ . . ^ 3c a = — - = — 1-4236610509 2 By equation (12), /3 = -4- — 5 hence the following operation for finding /3. * The student is supposed to be acquainted with the usual methods of determining the initial figure of the root of an equation, as well as familiar with Homer’s method of continuous approximation. Those who may find themselves deficient in these operations, may profitably consult a valuable treatise on the “Theory and Solution of Equations”, by Professor Yodno,of Belfast.