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.

    19/28 page 19
    EXAMPLE III. Solve completely the equation a;'+ 312«' +23337a;-- -14874a;+2360 = 0. Increase the roots of the equation by 78, to divest it of the second term. 1 + 312 + 23337 — 14874 + 2360 (—78 — 70 — 16940 — 447790 32386480 242 6397 — 462664 32388840 — 70 — 12046 395010 — 289168 172 _ 5643 — 67654 32099672 — 70 — 7140 103800 102 — 12783 36146 The coefficients of the transformed equation — 70 — 192 104824 are 32 — 12975 140970 II 1 CO — 8 — 128 c= 140979, 24 — 13103 (2=32099672. — 8 — 64 16 — 13167 — 8 8 — 8 0 The equation in a* is consequently a® - 6583-5a‘ + 28l0700-0625a — 310508451-5625 = 0, which lias three real positive roots. We shall develope one of them. ) —6583-5 + 2810700-06-25 — 310508451 -5625 (225 = ; 200 — 1276700 306800012-5 —6383-5 1534000-06-25 — 3708439-0625 200 — 1236700 3560601-25 ..a=±15. —6183-5 297300-0625 — 147837-8125 200 — 119-270 147837-8125 —5983-5 178030-06-25 20 — 118870 —5963-5 59160-0625 20 — 29592-5 —5943-5 29567-5625 20 —5923-5 5 —5918-5 Hence the rational parts of the four roots of the proposed equation are — 78 ± 15 = — 63 and 93 ; and fore we have /3 = a* + -| +^ = 225 — 6583-5 + 2349-5 = — 4009; /3=:«* + - — = 225 — 6583-5 — 2349-5 = — 8708. 4a
    digitised image 1
    digitised image 2
    digitised image 3