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The theorem of Harnack referred to in the title is the following: If u is a positive harmonic function in a domain D, then in any compact set D' contained in D the inequality rnax u 5 c min u, D D holds where the constant c > 1 depends on D and D' only. Equivalently, if 21 is normalized to take on the value 1 at some point of D' one has c-l 5 u 5 c in D' with the same constant c.This important theorem can be considered as a sharpened and quantitative version of the maximum principle. It is used in several existence theorems of potential theory since it establishes the compactness of a family of bounded harmonic functions,In this paper we want to derive such a theorem for the solutions of uniformly elliptic differential equations of second order in selfadjoint form where z = (xl, x2, -* , xn) and the matrix u = (avh(x)) is symmetric and positive definite. Moreover, the positive eigenvalues of u are assumed to lie between two positive constants, say between 1-1 and I , where the constant il > 1 is fixed for the following considerations. Otherwise the elements uvp(x) are only required to be Lebesque integrable functions.The generality of the assumptions on the coefficients is called for if one studies nonlinear differential equations, where the coefficients uvp in (1.2) may depend on the solution u and its first derivatives for which no smoothness properties are available. In fact, one is lead to differential equations of this form if one studies the extremals of a variational problem *This paper represents results obtained under Contract Nonr-285(46). Reproduction in whole or in part is permitted for any purpose of the United States Government. 51 7 578 J . MOSER -LEMMA 1 (PoincarC). If w, w, are square integrable in Q and then Received February, 1961.

Lectures on Celestial MechanicsCarl Ludwig Siegel was born on December 31, 1896 in Berlin. He studied mathematics and astronomy in Berlin and Göttingen and held chairs at the Universities of Frankfurt and Göttingen before moving to the Institute for Advanced Study in Princeton in 1940. He returned to Göttingen in 1951 and died there in 1981.Siegel was one of the leading mathematicians of the twentieth century, whose work, noted for its depth as weil as breadth, ranged over many different fields such as number theory from the analytic, algebraic and geometrical points of view, automorphic functions of several complex variables, symplectic geometry, celestial mechanics.Jürgen Moser was bornon July 4, 1928 in Königsberg, then Germany. After the war he studied in Göttingen, where he received his doctoral degree in 19; 2 and subsequently was assistant to C. L. Siegel. In 1955 he emigrated to the USA. He held positions at M.I. T., Cambridge and primarily at the Courant Institute of Mathematical Seiences in New York; from 1967 to 1970 he was Director of this institute. In 1980 he moved to the ETH in Zürich where he now is Director of the Mathematical Research Institute.Moser has worked in va rious areas of analysis. Besides celestial mechanics and KAM theory (presented in Chapter 3 of this book) he contributed to spectral theory, partial differential equations and complex analysis.

Abstract. Discrete versions of several classical integrable systems are investigated, such as a discrete analogue of the higher dimensional force-free spinning top (Euler-Arnold equations), the Heisenberg chain with classical spins and a new discrete system on the Stiefel manifold. The integrability is shown with the help of a Lax-pair representation which is found via a factorization of certain matrix polynomials. The complete description of the dynamics is given in terms of Abelian functions; the flow becomes linear on a Prym variety corresponding to a spectral curve. The approach is also applied to the billiard problem in the interior of an N-dimensional ellipsoid.

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