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.

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    NOTE B. Let 1“ + ^ = 0 be an equation of the seventh degree, and if two of its roots be of the form ±: \/ — /3, then the value of a, the rational part of the roots, will be zero, and the values of A, B, C, D, E, F, G, given in equations (3) p. 7, will reduce to A = a, B =z &, C = c, D = d, E = e, F =y, G = y. Now if m = 7 in equations (6) art. 7; thenf^a = 0, and f-ja = 0, and we get (1), a^^-cl3^ + el3-ff = 0 (2). Eliminating /3 from these two equations, we get the relation {{ab—c)[cf—bg)—{ad^e){af—g]} {{ad—e){ef—dg)—[af—g){cf—hg)} = {{ah—c) [ef—dg)—{af—gjf.. (3). Hence if the coefficients of the unknown quantity in the several terms of an equation of the seventh degree be such as to satisfy the relation (3), then the equation will have two roots of the form ± ^ — /3, and the value of /3 deduced from the simultaneous equations (1) and (2) will be a — (a& — g)(g/— dg)—{af—gY _ {ad — e) {ef— dg) —{af —g) {cf— bg) {ab—c){of—bg)—{ad — e){af—g) {ab ~c){ef—dg)—{af—gY If the equation wants the second term, then a = 0, and (3) and (4) reduce to {cf— bg + de)[c{cf— bg) + eg} + {be — e)={cd —g) [2cef—g{cd—g) j (3'), A a — g(g/—/ _ ey—g{cf—bg + de) ^ c{cf—bg)-\- eg cef—g{cd—g) In the relation (3) let g =0, then we shall have {abc — a^d — c* + ae) {ade — acf— e^)={ahe — df— ce)^ which by partial multiplication, and cancelling equal terms from both members, gives {be — ad e){a{cf — de)-\- c\cf— de)={be — af)[a{af— be)-\-2ce} (5). If (2 = 0, then (5) reduces to (5'). Also the value of /3, when y = 0 in equation (4), becomes e{ab—c)—ay e{ad — e)—acf /3 = c{ab — c)—a{ad—e) e[ab — c) — af .(6), and when a = 0, /3 = — c .{&). Hence if the equation + ax^ + bx*‘ + cx^ + dx' -^ex -\-f=.0 be such that the coefficients a, b, c, etc., satisfy the relation (5), then two of its roots will be of the form ± ^ —/3, and their value will be found from (6), If the equation wants the second term, and if the relation (S') is satisfied, the equation will have two roots equal to ± V — e c' Again, in the relation (5) and the formula for the value of/3 (6) lety= 0, then we get and (ab — c) [cd — be) = {ad — ey..,. e{nb — c) ad — e c{ab — c)—a[ad—e) ab — c When a—0, then the formulas (7) and (8) reduce to c{be —cd) — ^ (7'), and /3 = — c .(7), .(8). (8').