The transformed equation is z^3+ 6z^2-4.88z+.008 . . . . (3).
Equation (2) has a root between .2 and .3; the roots of equation (3) are less by .2 than those of equation (2); hence equation (3) has a root between 0 and .1. Neglecting in equation (3) the terms involving the higher powers, as was done in the case of the first transformed equation, we have
-4.88z + .008 = 0, or z= .001.
Diminishing the roots of (3), the second transformed equation, by .001, we have
1 +6 - 4.88 + .008 |.001 001 .000601 -.004879399 601 -4.879399 .003120601 .001 .000602 .002 - 4.878797 .001 .603
The transformed equation is v^3 + .603 v^2 - 4.878797 v + .003120601 \cdots (4).
Equation (3) has a root between .001 and .002; the roots of equation (4) are less than those of equation (3) by .001; therefore equation (4) has a root between 0 and .001. Neglecting the terms involving v^3 and v^4, and solving
- 4.878797 v + .003120601 = 0,
we have v= .0006. Diminishing the roots of (4), the third transformed equation, by .0006, we find this to be the correct figure for the fourth decimal place; hence 1.2016 is the value, to the fourth decimal place, of the root which lies between 1 and 2.
Denoting the coefficients of the successive transformed equations by (A), (B), (C), etc., the work is more compactly arranged thus:
