The Collected Mathematical Papers of Arthur Cayley, Vol. 12 (Classic Reprint)

From the Quarterly Journal of Pure and Applied Mathematics, vol. xix. (1883), pp. 122.] THE present memoir is based upon Mr Warren sE xercises inC urvilinear and Normal Coordinates, Camb. Phil. Trans, t. xn. (1877), pp. 455 502, and has for a principal object the establishment of the six differential equations of the second order corresponding to his six equations for normal coordinates: but the notation is different; the results are more general, inasmuch as I use throughout general curvilinear coordinates instead of his normal coordinates; and as regards my six equations for general curvilinear coordinates, the terms containing differential coefficients of the first order are presented under a different form. If the position of a point in space is determined by the rectangular coordinates x, y, zthen p, q, rbeing each of them a given function of x, y, z, we have conversely x, y, z, each of them a given function of p, q, r, which are thus in effect coordinates serving to determine the position of the point, and are called curvilinear coordinates. But it is not in the first instance necessary to regard x, y, zas rectangular coordinates, or even as Cartesian coordinates at all; we are simply concerned with the two sets of variables x, y, zand p, q, r, each variable of the one set being a given function of the variables of the other set; and, in particular, the x, y, zare regarded as being each of them a given function of the p, q, r. Except as regards the symbols ,rj, presently mentioned, the suffixes 1, 2, 3refer to the variables p, q, rrespectively, and are used to denote differentiations in regard to these variables, viz. ynp Yy Yrp C. XII.
(Typographical errors above are due to OCR software and don't occur in the book.)

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