Societies and Academies

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    LONDON: Mathematical Society, December 14.—Lord Rayleigh, F.R.S., president, in the chair.—Mr. R. F. Davis, B.A., and Mr. H. Weston Eve, M.A., head-master of University College School, were elected members.—Prof. H. J. S. Smith, F.R.S., read a paper on the conditions of perpendicularity in a parallelopipedal system (the subject was of interest to crystallographers as well as to mathematicians).—Mr. Glaisher, F.R.S., gave an account of a paper by Prof. Cayley, F.R.S., on the condition for the existence of a surface cutting at right angles a given set of lines. "In a congraency or doubly infinite system of right lines, the direction-cosines α, β, γ of the line through any point x, y, z, are expressible as functions of x, y, z, and it was shown by Sir W. R. Hamilton in a very elegant manner that in order to the existence of a surface (or what is the same thing, a set of parallel surfaces) cutting the lines at right angles, αdx + βdy + γdz must be an exact differential; when this is so, writing V = ∫(αdx + βdy + γdz) we have V = c, the equation of the system of parallel surfaces, each cutting the given lines at right angles."The author obtains his results from the analytical equations of a congruency, viz., x = mz + p, y = nz + q, where m, n, p, q are functions of two parameters, and m, n, are given functions of p, q. The condition he gets for the existence of the set of surfaces is—

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    Societies and Academies . Nature 15, 191–192 (1876) doi:10.1038/015191b0

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