The vanishing of this invariant is the condition for equal roots.

It may denote a simultaneous orthogonal invariant of forms of orders n i, n2, n3,...; degree 0 of the covariant in the coefficients.

Possesses the invariant property.

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In addition, and transform each pair to a new pair by substitutions, having the same coefficients a ll, a12, a 21, a 22 and arrive at functions of the original coefficients and variables (of one or more quantics) which possess the abovedefinied invariant property.

Possess the invariant property, and we may write (AB) i (AC)'(BC) k ...A P E B C...

A binary form of order n contains n independent constants, three of which by linear transformation can be given determinate values; the remaining n-3 coefficients, together with the determinant of transformation, give us n -2 parameters, and in consequence one relation must exist between any n - I invariants of the form, and fixing upon n-2 invariants every other invariant is a rational function of its members.

If now the nti c denote a given pencil of lines, an invariant is the criterion of the pencil possessing some particular property which is independent alike of the axes and of the multiples, and a covariant expresses that the pencil of lines which it denotes is a fixed pencil whatever be the axes or the multiples.

This expression of R shows that, as will afterwards appear, the resultant is a simultaneous invariant of the two forms.

May be a simultaneous invariant of a number of different forms az', bx 2, cx 3, ..., where n1, n 2, n3, ...

A leading proposition states that, if an invariant of Xax and i ubi be considered as a form in the variables X and, u, and an invariant of the latter be taken, the result will be a combinant of cif and b1'.