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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    equating the first and second values of and also the first and third, gives yz — a:{y ^ z) (4), yAy +Z) +c =a'{y'‘+yz + z^) (5), Multiply (4) hy y z, and from the product subtract (5) ; then we have a!yz-^c=ib’(^ + «) (6). From (4) and (6) we get , ci'h' — c 6'^ — a'c + 2'*= P)- The equations (7) will furnish the values of y and z, and from (3) we get <£* — (]^ = r> or —ci!)-\-z — a' = 0 (8), Also from (1) we have X.®(« + yf + (a; + zf = 0, or \\x + = _ (a? + z)^hence X{x -\-y) = — (ir+ «) (9). Eliminate z from the equations (8) and (9), and divide both members of the resulting equation by X -}- 1 ; then we have ar = X(X — 1)(?/ — a)— a' (iQ). If the form of the equation x^ hx c = 0, the modified equations are 3^7 1) f y4-«=-^, y« = ^ = anda; = X(X—l)y (11). EXAMPLE. Find the value of x in the cubic equation ar® -j- 12a; — 30 = 0. 15 Here 0 = 0, 6=12, c = — 30, consequently we have y z =■ — and yz=. — 4 ; hence 1 [z \1 y — ^^ ^ = — 8, x = -ja =24/2, anda; = X(X — 1)2/= ,5'2{2,J'2—1)= 2^/4 — .3/ 2. EREATA. Page 7, line 4 from bottom,/br r[a + readr'(a* + /3). — 8, — 3 from top, for equation (3) read equation (2). — —, — 13 from bottom,ybr p. 4 read p. 6. — 12, — 4 from top, foi 434-67.... — 4*34(67.... — 20, — 16 from top, for fa = 0 readfa = 0. E. Jones, Printer, “Journal” Office, Thomas Street, Woolwich.