TRANSFORMATION
, in Geometry, is the changing or reducing of a figure, or of a body, into another of the same area, or the same solidity, but of a different form. As, to Transform or reduce a triangle to a square, or a pyramid to a parallelopipedon.
Transformation of Equations, in Algebra, is the changing equations into others of a different form, but of equal value. This operation is often necessary, to prepare equations for a more easy solution, some of the principal cases of which are as follow.—1. The signs of the roots of an equation are changed, viz, the positive roots into negative, and the negative roots into positive ones, by only changing the signs of the 2d, 4th, and all the other even terms of the equation. Thus, the roots of the equation ; whereas the roots of the same equation having only the signs of the 2d and 4th terms changed, viz, of .
2. To Transform an equation into another that shall have its roots greater or less than the roots of the proposed equation by some given difference, proceed as follows. Let the proposed equation be the cubic ; and let it be required to Transform it into another, whose roots shall be less than the roots of this equation by some given difference d; if the root y of the new equation must be the less, take it , and hence ; then instead of x and its powers substitute y + d and its powers, and there will arise this new equation
| (A) y3 + 3dy2 + 3d2y + d3 | } = 0, |
| - ay2 - 2ady - ad2 | |
| + by + bd | |
| - c |
3. To take away the 2d or any other particular term out of an equation; or to Transform an equation, so as the new equation may want its 2d, or 3d, or 4th, &c term of the given equation , which is transformed into the equation (A) in the last article Now to make any term of this equation (A) vanish, is only to make the coefficient of that term = 0, which will form an equation that will give the value of the assumed quantity d, so as to produce the desired effect, viz, to make that term vanish. So, to take away the 2d term, make , which makes the assumed quantity . To take away the 3d term, we must put the sum of the coefficients of that term = 0, that is , or ; then by resolving this quadratic equation, there is found the assumed quantity , by the substitution of which for d, the 3d term will be taken away out of the equation.
In like manner, to take away the 4th term, we must make the sum of its coefficients ; | and so on for any other term whatever. And in the same manner we must also proceed when the proposed equation is not a cubic, but of any height whatever, as : this is first, by substituting y + d for x, to be Transformed to this new equation
| yn + ndyn-1 + (1/2)n.(―(n - 1)).d2yn-2 &c | } = 0; |
| - ayn-1 - a.(―(n - 1)).dyn-2 &c | |
| + byn-2 &c |
From whence it appears that, to take away the 2d term of an equation, we must resolve a simple equation; for the 3d term, a quadratic equation; for the 4th term, a cubic equation, and so on.
4. To multiply or divide the roots of an equation by any quantity; or to Transform a given equation to another, that shall have its roots equal to any multiple or submultiple of those of the proposed equation. This is done by substituting, for x and its powers, y/m or py, and their powers, viz, y/m for x, to multiply the roots by m; and py for x, to divide the roots by p.
Thus, to multiply the roots by m, substituting y/m for x in the proposed equation ; or multiply all by mn, then is , an equation that hath its roots equal to m times the roots of the proposed equation.
In like manner, substituting py for x, in the proposed equation, &c, it becomes , an equation that hath its roots equal to those of the proposed equation divided by p.
From whence it appears, that to multiply the roots of an equation by any quantity m, we must multiply its terms, beginning at the 2d term, respectively by the terms of the geometrical series, m, m2, m3, m4, &c. And to divide the roots of an equation by any quantity p, that we must divide its terms, beginning at the 2d, by the corresponding terms of this series p, p2, p3, p4, &c.
5. And sometimes, by these Transformations, equations are cleared of fractions, or even of <*>urds. Thus the equation , by putting , or multiplying the terms, from the 2d, by the geometricals √p, p, p√p, is Transformed to .
6. An equation, as , may be Transformed into another, whose roots shall be the reciprocals of the roots of the given equation, by substituting 1/y for x; by which it becomes , or, multiplying all by y3, the same becomes .
On this subject, see Newton's Alg. on the Transmutation of Equations; Maclaurin's Algeb. pt. 2, chap. 3 and 4. Saunderson's Algebra, vol. 2, pa. 687, &c.
