  Covariant or invariant functions under a compact linear group can be
expressed in terms of functions defined in the orbit space of the group. The
semialgebraic relations defining the orbit spaces of all finite coregular real
linear groups with at most 4 basic invariants are determined. For each group
$G$ acting in $\real^n$, the results are obtained through the computation of a
metric matrix $\widehat P(p)$, which is defined only in terms of the scalar
products between the gradients of a set of basic polynomial invariants
$p_1(x),\dots p_q(x),\x\in\real^n$ of $G$; the semi-positivity conditions
$\widehat P(p)\ge 0$ are known to determine all the equalities and inequalities
defining the orbit space $\real^n/G$ of $G$ as a semi-algebraic variety in the
space $\real^q$ spanned by the variables $p_1,\dots ,p_q$. In a recent paper,
the $\widehat P$-matrices, for $q\le 4$, have been determined in an alternative
way, as solutions of a universal differential equation;the present paper yields
a partial, but significant, check on the correctness and completeness of these
solutions. Our results can be widely exploited,e.g. in the determination of
patterns of spontaneous symmetry breaking, in the analysis of structural phase
transitions (Landau's theory),in covariant bifurcation theory,in crystal field
theory and in solid state theory where symmetry adapted functions are used.
