DISCRIMINANT.—If (in a system of equations the functions equated to zero are the derived functions of a single rational and integral homo geneous function with respect to each of the variables thereof, the resultant of the system is said to be the discriminant of the single function. The definition is easily made applicable to the case of a non-homogeneous function, the functions equated to zero are here the function itself and its derived functions with respect to each of the several variables. For a single function, it may be said that the dis criminant is the function which equate:11e zero gives the condition for a pair of equal roots of the equation obtained by putting the function equal to zero.
To fix the precise value of the discriminant of a given function, it is assumed that the coefficient of some ono selected term is + 1. Thus, the discriminant of a x'+2 b x y+c yl is a that of ax'+3 bzly+3exe+dylis —6abcd+4aca+4Pd-3 In quadratic forms (in the theory of numbers) the expression which is the determinant I b, c b with the sign reversed, is called the determinant of the form a e+ b xy + c And in like manner for ternary quadratic forms, there is the same reversal of sign. It may be
said as a convenient definition, that the determinant is the discriminant taken negatively.
Ptzxus.—It frequently happens, in problems of elimination and in other problems, that a given number of relations existing between a system of quantities can only be completely expressed by means of a greater number of equations. Thus, to take a very simple instance, if the unknown quantities x, y, are to be eliminated between the three equations a x+by=o, a'x+ =o, a"x+ b"y=o: this im plies two relations between the coefficients a, b, a', b', 6" ; but these relations cannot be completely expressed otherwise than by means of the three equations a f/—db=o, db"—a"6'= o, a"b — a b" = o ; for taking any two of these equations, e.g. the first and second, these would be satisfied by a'=o, b' =o, however do not satisfy the third equation and are not a solution. Such a system of equations, or generally the system of equations required for the complete expression of the relations existing between a set of quantities (and which are in general more numerous than the relations themselVes) is said to be a Plexus.