Consider a non-holonomic dynamical system specified by the N coordinates qi, and kinetic energy defined by

where amn are functions of qi and the dot denotes differentiation with respect to the time t.

In an interesting appendix to a letter written by John Collins to James Gregory on August 3, 1675, but not published until a few months ago, appear some formulae, given without proof, for expressing the roots of an equation of any degree from the 2nd to the 9th in terms of the coefficients, under the assumption that these roots are in arithmetical progression. The formulae were discovered by the well known contemporary of Leibniz, Baron W. von Tschirnhaus. It is evident that in the case of an equation of degree n this particular assumption imposes n − 2 conditions on the coefficients; so that two of these coefficients can be chosen ad libitum. Tschirnhaus did not go to the trouble of obtaining these relations explicitly, in fact he makes no mention of them, but he gives expressions, in the cases indicated above, for the roots as functions of the first two coefficients of the equation in question, and these coefficients, as we have observed, are arbitrary. It is not known by what approach he arrived at his formulae; it seems likely to us, however, that he expressed the desired roots in terms of two arbitrary unknowns, that he evaluated the sum of these, and the sum of their products two at a time, and that, finally, he equated the results to the first two coefficients of the equation. In this way two equations are obtained, sufficient to determine the two auxiliary unknowns; and the problem can be considered as solved. Without seeming to imply that this procedure was the same as that adopted by the eminent German mathematician, we shall show that by its means one can not only derive his results, but also solve the question in the case of an algebraic equation of any degree.

Let xir (i = 1,…., q; r = 1, …., m; q ≦ m) be random variables which have an elementary probability law p(x11, …., xqm) Let

It is to be expected that a finite number of plane curves of order four should have seven given lines as bitangents, because the number of conditions imposed is equal to the number of effective free constants in the equation of such a curve, viz., 14. Aronhold made the interesting discovery that one curve could be determined in which no three of the given lines have their six points of contact on a conic. The method, due to Geiser, of obtaining the bitangents as projections of the lines of a cubic surface leads to a simple proof of the existence of this quartic.