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We classify the set of central configurations lying on a common circle in the Newtonian four-body problem. Using mutual distances as coordinates, we show that the set of four-body co-circular central configurations with positive masses is a two-dimensional surface, a graph over two of the exterior side-lengths. Two symmetric families, the kite and isosceles trapezoid, are investigated extensively. We also prove that a co-circular central configuration requires a specific ordering of the masses and find explicit bounds on the mutual distances. In contrast to the general four-body case, we show that if any two masses of a four-body co-circular central configuration are equal, then the configuration has a line of symmetry.MSC Classifications: 70F10, 70F15, 37N05

In this paper we consider both the dynamical and parameter planes for the complex exponential family E λ (z) = λe z where the parameter λ is complex. We show that there are infinitely many curves or "hairs" in the dynamical plane that contain points whose orbits under E λ tend to infinity and hence are in the Julia set. We also show that there are similar hairs in the λ-plane. In this case, the hairs contain λ-values for which the orbit of 0 tends to infinity under the corresponding exponential. In this case it is known that the Julia set of E λ is the entire complex plane.

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This paper concerns the linear stability of the well-known periodic orbits of Lagrange in the three-body problem. Given any three masses, there exists a family of periodic solutions for which each body is at the vertex of an equilateral triangle and travels along an elliptic Kepler orbit. Reductions are performed to derive equations which determine the linear stability of the periodic solutions. These equations depend on two parameters -the eccentricity e of the orbit and the mass parameterA combination of numerical and analytic methods is used to find the regions of stability in the be-plane. In particular, using perturbation techniques it is rigorously proven that there are mass values where the truly elliptic orbits are linearly stable even though the circular orbits are not. # 2002 Elsevier Science (USA)