PREFACE
This introduction to the classical theory of invariants of
algebraic forms is divided into three parts of approximately
equal length.
Part I treats of linear transformations both from the stand
point of a change of the two points of reference or the triangle
of reference used in the definition of the homogeneous coor
dinates of points in a line or plane, and also from the stand
point of projective geometry. Examples are given of invariants
of forms / of low degrees in two or three variables, and the
vanishing of an invariant of / is shown to give a geometrical
property of the locus /=0, which, on the one hand, is inde
pendent of the points of reference or triangle of reference,
and, on the other hand, is unchanged by projection. Certain
covariants such as Jacobians and Hessians are discussed and
their algebraic and geometrical interpretations given; in
particular, the use of the Hessian in the solution of a cubic
equation and in the discussion of the points of inflexion of
a plane cubic curve. In brief, beginning with ample illustra
tions from plane analytics, the reader is led by easy stages
to the standpoint of linear transformations, their invariants
and interpretations, employed in analytic projective geometry
and modern algebra.
Part II treats of the algebraic properties of invariants
and covariants, chiefly of binary forms; homogeneity, weight,
annihilators, seminvariant leaders of covariants, law of reciproc
ity, fundamental systems, properties as functions of the roots,
and production by means of differential operators. Any
quartic equation is solved by reducing it to a canonical form
by means of the Hessian (§33). Irrational invariants are
illustrated by a carefully selected set of exercises (§35).
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