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.

    8/28 (page 8)
    In equation (3) substitute for a, b, c, etc,, their values as obtained in (5): then we shall have the following remarkable relations among A, B, C, etc., the coefficients of the several powers of the unknown quantity in the transformed equation (3), whose roots are less by a than the roots of the given equation (1), namely, where A =/ia B = fzu /3 C — f^a -1- A/3 D=:/,a + B/3 — E =f,a + C/3 — A^^ F=/ea +D^— G =/7a + E/3 — C/3^ + A/3' etc. etc. f^a = {m — 2)a + a . {m—2)(m — 3) , , n, , , i, f^a. — “ + (w — 3) a a -|- h (6). ^ _ (m—2)(ot — 3)(m — 4) /3a _ 1.2.3 {m — 3)(7« — 4) 1.2 aV+ {m — 4)b'a + c' /i“ = (m — 2)(ot — 3)(ot — 4){m — 5) . , (m — 3){m — 4)(m — 5) ,, (m — 4){m—5) , — a + ^ aa + 7-;: 0 a 1.2.3.4 1.2.3 ““ 1.2 + (m — 5)ca -I- d' . (m — 2)(m — 3)(ni — 4)(m — 5){m — 6) (m—3)(m — 4)(»?—5)(m—6) , y 5“ = r rTo'T'c “ 1 . <2 a' 1.2.3.4.5 1.2.3.4 , (m — 4)(m — 5)(m — 6) ,,3 , (m — 5Xm—6) -1 j 2 3 ^ “ t |“2 ^ “ I* ^» which may be continued at pleasure. 8. From these remarkable relations of the coefficients A, B, C, etc., which have resulted from the transformation of the proposed equation into another whose -roots are each less than the roots of that equation by a, the real part of the imaginary root, we shall obtain the necessary equations for the determination of the imaginary roots. We must not omit to remark that the two equations which we shall obtain from the results of the preceding transformation of the given equation are precisely those mentioned by La.gra.nge, and which we have given in art. 5, p. 4. These equations may readily be obtained by substituting a ^ — /3 or a— ^ — /3 for ar in the proposed equation, and separating the real terms from those which are multiplied by v/ — 1, and thus forming two equations involving a and /3. The preceding formulas, however, will furnish these equations, in all cases, without any substitution, and we shall now proceed to apply them to the complete solution of some numerical equations of the third, fourth, and higher degrees. I. CUBIC EQUATIONS. 9. Let the cubic equation be + aa?' -\-bx c =.0 (1), and let r,a —ft and a — ^ — /3 be its three roots, one of which (r) is necessarily real, and the two others will be real or imaginary, according as the sign of the value of ft is — or -I*. Then making w = 3 in equations (6), and recollecting that, in this case, all the coefficients of the several powers of a: in the expression (^) are zero, with the exception of a', we have./^a = 0 and/3a = 0 ; therefore we have B = ^ and C = A/3, (2).