HELENE AIRAULT 0. Introduction. The quantum-mechanical problems of n mass points on the line interacting pairwise under the influence of a potential proportional to the inverse square of the distance or to the square of the distance were solved explicitly by F. Calogero [1]. This led him to conjecture that the classical problems would be integrable. This was established in [2] for the three-body problem. Then J. Moser [3] introduced matrices L and B, and writing the equations in P. Lax's form

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... ] , he solved the classical n-particle system on the line with the inverse square potential. He successfully applied the method to the potential sin -2 * and to the Toda lattice. This method was further extended by M. Adler [5] to potentials of the form x~2 + ax 2 . The question arose, to which potentials could this method be applied. In the case of the classical nbody problem characterized by the Hamiltonian F. Calogero [6] considered potentials of the form V(x) = a(x)a( -x) -f const. Writing P. Lax's condition with hk = 8 ikPj + (1 -ô jk) «(** -**) and N B jk = Sjh JJ ß(x, -x t ) m -(1 -8 jk) «'(*; -x k) he was led to solve the equation (related equations appear in [7, 8] ). (1) a'(y)a(z) -a(y)a'(z) = a(y + z)[ß(y) -ß(z)]. Functions such that a x (x) = bdn(ax)/sn(ax) and a 2 (x) = bcn(ax)/sn(ax) are solutions of (1) and they yield the same potential V(x) -XP(x) + JU, where X and JU are two constants and P is the Weierstrass P-function. In