Let A be a root system in the sense of Bourbaki [1] and W be its Wey] group. A spans an / dimensional real vector space V on which W acts as a finitelinear group. By extension of the transpose action, W acts on the symmetric algebra S(V) of the dual space V*. There is a well known theorem of Chevalley [2], that is, the ring of W-invariant elements of S(V) is generated by / algebraicallyindependent homogeneous polynomials. How will the situation change when A is an infiniteroot system and W is an infinite group acting in V defined by a generalized Cartan matrix of nonfinitetype? With regard to this, there is a work of Moody [4], that the indefinitequadratic form by itself generates the entire ring of invariants, when A is defined by a generalized Cartan matrix of hyperbolic type. In this paper, we study the ring of polynomial invariants of the Weyl group of a Euclidean Lie algebra. In thiscase, V is not isomorphic to F* as TV-module. So we have to consider both the ring of invariants of S(V) and of S(V*), the symmetric algebra of V*. It becomes clear that the unique invariant vector, called null root, by itself generates the entire ring of invariants of S(V*) (Theorem 1) while the ring of invariants of S(V) is isomorphic to that of corresponding finite type, which is generated by algebraicallyindependent homogeneous elements (Theorem 2). This indicates that the polynomial invariants of Euclidean Lie algebras are critically situated between those of finitetypes and those of hyperbolic types. When A is a root system in the sense of Bourbaki [1], thissubject has some relationwith classicalHarish-Chandra's theorem, which states that the center of the universal enveloping algebra of corresponding finite dimensional complex simple Lie algebra isomorphic to the ring of ^-invariants of S(F*). This theorem cannot be extended when A is an infiniteroot system. For example, we cannot use a number of propositions with respect to even the Casimir operator in this case. We hope to discuss this case in near feature.