Abstract
American Journal of Mathematics, vol. ix. No. 4 (Baltimore, June 1887).—The number opens with a further instalment of Prof. Sylvester's lectures on the “Theory of Reciprocants” (pp. 297–352), which grow in interest as we approach their close—promised in a subsequent number. Lectures xxv. to xxxii. are reported as before by Mr. Hammond, and are accompanied by the lecturer's notes.—M. Maurice d'Ocagne (pp. 354–80) in a paper “Sur une Classe de Nombres remarquables,” discusses properties of the numbers symbolically represented by Kmp. Form a table of squares, as in the case of Pascal's arithmetical triangle, putting in the top left corner K, and in the vertical and horizontal lines the successive numbers 1, 2, 3 … The K-numbers will then be, first row 1, second row, 11, third row 131, fourth row, 1761, fifth row, 11525101, and so on; the law of formation being, “Multiply the number of the pth column of the qth row by the number of the column, and add to the result the number in the p−1th column of the qth row to get the number in the pth column of the q+1th row”: thus, in the above, 15=2.7+1, 25=3.6+7, 10=4.1+6. These numbers, like those of Bernoulli and Euler, frequently occur in analysis. Many curious results are obtained. —We next have “Extraits. de Deux Lettres addressées à M. Craig par M. Hermite” (pp. 381–88). These notes are upon a definite integral formula of Fourier, upon a formula due to Gauss, and upon a formula first given by Weierstrass (an expression for the sine by a product of prime factors).—The volume closes with a notelet by Prof. Franklin, entitled “Two Proofs of Cauchy's Theorem.”
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Scientific Serials . Nature 36, 262 (1887). https://doi.org/10.1038/036262a0
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DOI: https://doi.org/10.1038/036262a0