Abstract
To Mr. Samuel Roberts (see Reprint of Educational Times, vol. x. p. 47) is due the credit of having been the first to show that a direct method of elimination properly conducted leads to the differential equation for a cubic curve: but he has not attempted to obtain the general formula for a curve of any order. By aid of a very simple idea explained in a paper intended to appear in the Comptes rendus of the Institute, I find without calculation the general form of this equation. The eft-hand member of it may be conveniently termed the differential criterion to the curve. One single matrix will then serve to express the criteria for all curves whose order does not exceed any prescribed number. For instance, suppose we wish to have the criteria for the orders 1, 2, 3, 4:—
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SYLVESTER, J. On the Differential Equation to a Curve of any Order . Nature 34, 365–366 (1886). https://doi.org/10.1038/034365b0
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DOI: https://doi.org/10.1038/034365b0