COEFFICIENTS
, in Algebra, are numbers, or given quantities, usually prefixed to letters, or unknown quantities, by which it is supposed they are multiplied; and so, with such letters, or quantities, making a product, or coefficient production; whence the name.
Thus, in 3a the coefficient is 3, in bx it is b, and in cx2 it is c. If a quantity have no number prefixed, unity or 1 is understood; as a is the same as 1a, and be the same as 1bc. The name coefficient was first given by Vieta.
In any equation so reduced as that its highest power or term has 1 for its coefficient; then the coefficient of the 2d term is equal to the sum of all the roots, both | positive and negative; so that if the 2d term is wanting in an equation, the sum of the positive roots of that equation is equal to the sum of the negative roots; as they mutually balance and cancel each other. Also the coefficient of the 3d term of an equation is equal to the sum of all the rectangles arising by the multiplication of every two of the roots, how many ways soever they can be combined by twos; as once in the quadratic, 3 times in the cubic, 6 times in the biquadratic equation, &c. And the coefficient of the 4th term of an equation, is the sum of all the solids made by the continual multiplication of every three of the roots, how often soever such a ternary can be had; as once in a cubic, 4 times in a biquadratic, 10 times in an equation of 5 dimensions, &c. And thus it will go on infinitely.
Coefficients of the same Order, is a term sometimes used for the coefficients prefixed to the same unknown quantities, in different equations. the coefficients a, d, g, are of the same order, being the coefficients of the same letter x; also b, e, h are of the same order, being the coefficients of y; and so on.
Opposite Coefficients, such as are taken each from a different equation, and from a different order of coefficients. Thus, in the foregoing equations, a, e, k, or a, h, f, or d, b, k, &c, are opposite coefficients.
COELESTIAL. See Celestial.
