On a class of particular solutions of the problem of four bodies

Introduction. In 1772 LAGRANGE published a celebrated meinoiron the Problem of Three Bodies, which contained all the solutions in which the ratios of the mutual distances of the bodies are constants. He found two distinct configurations. In the one, the three bodies always lie in a straight line; in the other, they are always at the vertices of an equilateral triangle. Their distribution upon the line depends upon their masses, being explicitly defined by the real positive root of a certain quintic equation. The equilateral triangular configuration is possible for all distributions of the masses. In both cases the three bodies move in the same plane, in conic sections with respect to each other or with respect to their common center of gravity, and in such a manner that the law of areas is true for each body considered separately. No other periodic solutions of the motion of three or more bodies were discovered for more than a century, although many splendid papers appeared on the Problem of Three Bodies. A new impetus was given to the subject by the celebrated memoir of Dr. HILL on the Lunar Theory,t in which he discussed a new species of per iodic solutions. He made the restrictions that one body should be infinitesimal, and that the finite bodies should describe circles around their center of gravity. For the purposes of the applications to the Lunar Theory he neglected the ratio of the distances of the iiioon and the sun, thus introducing errors in the numerical results which in certain cases became important. Professor DARWIN has more recently taken up the subjectt by a method not essentially different from that employed by HILL, and discussed a great many orbits in detail, not neglecting the solar parallax. He has considered only cases in which the infinitesimal body moves in the plane of the finite bodies with direct motion. To these must be added the masterful researches ? of POINCARP, who has