Proceedings of the London Mathematical Society

Abstract

PROF. CAYLEY contributes to this volume several memoirs bearing on the theory of attraction. References to some of his earlier papers on the subject are given in Tod-hunter's “History.” The titles of the present papers are “On the Potentials of Polygons and Polyhedra” “On the Potentials of the Ellipse and the Circle,” “Determination of the Attraction of an Ellipsoidal Shell on an Exterior Point,” “Note on a Point in the Theory of Attraction.” The order of the papers will indicate the direction of growth the subject took in the author's hands. Mehler has treated of the attraction of polyhedra, but Prof. Cayley's results “are exhibited under forms which are very different from his, and which give rise to further developments of the theory.” He finds general formulæ for the potentials of a cone and a shell, he then takes the case of a polyhedron or a polygon, obtains results for rectangular pyramid, rectangle, and cuboid, and verifies some of these results. The attraction of an indefinitely thin ellipsoidal shell was shown by Poisson to be in the direction of the axis of the circumscribed cone, this property was also demonstrated geometrically by Steiner. The geometrical investigation was subsequently completed by Prof. Adams so as to obtain from it the finite expression for the attraction of the shell, a result which had also been obtained analytically by Poisson. Prof. Cayley states the geometrical theorems, proves them, and obtains analytical expressions for the attraction of the shell and for the resolved attractions. The law of attraction throughout is that of the inverse square. The same writer also contributes a paper “On the Expression of the Co-ordinates of a Point of a Quartic Curve as Functions of a Parameter.” This last is the development of a process of Prof. Sylvester's. Dr. Hirst's remarks on “Correlation in Space” are a mere abstract of results, a fuller statement of which is reserved for a future communication. Prof. Wolstenholme contributes a neat piece of analysis called “A New View of the Porism of the In and circum-scribed Triangle.” Prof. Sylvester contributes two interesting notes from M. Mannheim with reference to Peaucellier's cells and their application. The Rev. W. H. Laverty supplies an “Extension of Peaucellier's Theorem.” Mr. Routh has a paper “On Laplace's Three Particles, with a Supplement on the Stability of Steady Motion;” Mr. Samuel Roberts contributes a paper “On a Simplified Method of obtaining the Order of Algebraical Conditions.” This method is illustrated by various geometrical applications. Further papers of an analytical character are “On the Solution of Linear Differential Equations in Series,” Mr. J. Hammond; “Note on some Relations between Certain Elliptic and Hyperbolic Functions,” Mr. J. Griffiths; “Notes on Laplace's Coefficients,” Mr. J. W. L. Glaisher. In mixed mathematics we have papers “On the Application of Hamilton's Characteristic Function to the Theory of an Optical Instrument symmetrical about its Axis,” and “On Hamilton's Characteristic Function for a Narrow Beam of Light” Prof. Clerk-Maxwell; “On the Vibrations of a Stretched Uniform Chain of Symmetrical Gyrostats,” Sir W. Thomson. The President (Prof. H. J. Smith) contributes papers “On the Higher Singularities of Plane-curves” and “On the Integration of Discontinuous Functions;” Major J, R. Campbell gives an account of “The Diagonal Scale Principle applied to Angular Measurement in the Circular Slide Rule.” Shorter papers are “On the Method of Reversion applied to the Transformation of Angles,” Rev. C. Taylor (the basis of the communication of which, an abstract only is given in the “Proceedings,” the full paper being printed in the Quarterly Journal of Mathematics, No. 53, is a work on Conic Sections, by G. Walker, 1794); “On some Proposed Forms of Slide Rule” and “On the Mechanical Description of Equipotential Lines,” Mr. G. H. Darwin; and “On the Mechanical Description of a Spheroconic” and “a Parallel Motion,” by Mr. Hart.

Proceedings of the London Mathematical Society.

Vol. VI. (London: Messrs. Hodgson, 1876.)