The Cambridge And Dublin Mathematical Journal Volume 7 9 PDF Download

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1848 edition. Excerpt: ...thus. Let T, U, V, TV, be four solutions which are of the form $ (a, x), $ (Jb, z), &c. Let b, c, e be a + 0, a + 20, a + 30, and throw AT + BV4 CV+ EW into the form AT + B&T + C"Aar+ E'A'T, AT being U-T, &c; and this again into the form whence the usual result is easily obtained. ON THE RELATION BETWEEN DIFFERENT CURVES AND CONES CONNECTED WITH A SERIES OF CONFOCAL ELLIPSOIDS. By John Y. Rtttledqe, B.A., Trinity College, Dublin. Thb object of the present paper is to shew the natural connexion that exists between several systems of curves traced upon the surface of an ellipsoid, which hitherto have been viewed as isolated objects of curiosity, but which, from the fact of their resulting from one and the same source, may be regarded in a still more interesting and instructive point of view. It is well known that all questions relating to conjugate diameters in central surfaces of the second order may be discussed as questions relating to rectangular axes, by the transformation where x, y, z, x1, y, ', x," y," z," are the coordinates of the extremities of the conjugate semidiameters, and a, 3 y a, /3', y', &c. are the angles made by the new rectangular axes of reference with the original axes of x, y, z. Suppose that we have a number of confocal ellipsoids: x2 y2, X y2 z2, . z y2 z2, . o c tr o c a o c In the first let us take three conjugate semidiameters, the respective coordinates of the extremities of which are x, y, z, x," y', z, and x," y," z." Let us take the corresponding points on the remaining ellipsoids, i.e. points such that? = =-&c a a a Then, if we connect the origin of coordinates with these points on the respective ellipsoids, it is easy to see...