American Journal of Mathematics, vol. xviii. No. 2, April.—The intermediary orbit, i.e. the Moon's periodic orbit relatively to the Sun obtained from the variation terms when all terms but those depending on the ratio of the mean motions only are omitted, has been considered in vol. i, by Dr. Hill, and subsequently in the Acta Manthematica (vol. viii.) the same writer obtained an expression for the motion of the Moon's perigee, so far as it depends on the ratio of the mean motions. These papers have been followed by others by Prof. E. W. Brown, in which the terms depending on the solar parallax and the lunar eccentricity are computed.—The object of the opening paper of the present number, on the inclinational terms in the Moon's coordinates, by P. H. Cowell, is to take into account, according to Dr. Hill's method, the inclination of the orbit, considering it as being the manifestation of a small oscillation about Dr. Hill's distorted circular orbit, which relatively to the Sun is a closed curve. The terms multiplied by the first power of the inclination have been calculated to the sixth order, and an expression for the part of the motion of the Moon's node, that depends upon the mean motions only, has been found as far as the eighth order, i.e. one term further than in Delaunay's series. The terms multiplied by the square of the inclination have been calculated to the fifth order, and the terms multiplied by the third power of the inclination to the fourth order in m. The notation adopted is that of the paper by Prof. Brown (Am. Journ. Math., vol. xvii.).—A short note by A. S. Chessin, on non-uniform convergence of infinite series, brings out more clearly a point in a previous note (vol. xviii. No. 1), which the writer says has been misunderstood.—On a certain class of equipotential surfaces, by B. O. Peirce, discusses the nature of such systems of plane curves as are at once the right sections of possible systems of equipotential cylindrical surfaces belonging to distributions of matter which attract, according to the law of nature, and the generating curves of possible systems of equipotential surfaces of revolution.—M. Petrovitch contributes “Remarques sur les équations de dynamique et sur le mouvement tautochrone.”—A note on C. S. Peirce's paper on a quincunclal projection of the sphere, by J. Pierpont, corrects an inaccuracy in that paper (vol. ii. p. 394). Mr. Pierpont, in a note on the invariance of the factors of composition of a substitution-group, gives a much simplified proof of this important theorem.—H. Maschke, in a long article (pp. 156-188) on the representation of finite groups, especially of the rotation—groups of the regular bodies of three-and four-dimensional space, by Cayley's colour diagrams, shows that Cayley's method (the theory of groups, graphical representation, Am. Journ., vol. i., and on the theory of groups, Am. Journ., vol xi.) can be readily applied to the construction and investigation of numerous groups of higher orders. In particular, the writer says, the colour diagrams for the rotation groups of the regular bodies can be arranged in such a way that they lend themselves much easier, at least in some respects, to a study of the groups concerned, than even the models of the regular bodies. Numerous diagrams of interest accompany the paper.
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