When a homogeneous **polynomial** is transformed by general linear substitutions as hereafter explained, and is then expressed in the original form with new coefficients affecting the new variables, certain functions of the new coefficients and variables are numerical multiples of the same functions of the original coefficients and variables.

Discriminants.-The discriminant of a homogeneous **polynomial** in k variables is the resultant of the k **polynomials** formed by differentiations in regard to each of the variables.

THE Theory Of Binary Forms A binary form of order n is a homogeneous **polynomial** of the nth degree in two variables.

An expression denoting that two or more monomials are to be added or subtracted is a multinomial or **polynomial**, each of the monomials being a term of it.