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A TREATISE ON THE ANALYTIC GEOMETRY OF THREE DIMENSIONS EDITORS PREFACE TO FIFTH EDITION. VOLUME I. IN order to avoid delay it has been thought advisable to publish this edition in two volumes. While pre serving the substance of the fourth edition, I have added some new matter generally enclosed in square brackets and in different type giving brief accounts of methods or points of view which appear to me to be of interest and to fit in with the rest of the work. The additions include illustrations of models of most of the different species of quadrics, with gener ators or lines of curvature Chapter V. articles or paragraphs on the analytical classification of real quadrics 880, on projection and Fiedlers projective coordinates 144e, on the non-Eucliclean theory of distance and angle 144, and on the expression of twisted cubics and quartics by rational or eUigtic parameters 3330, 3470, 348, 349.-In differential geometry my aim has been to form a closer connecting-link between Salmons book and the more extensive and more purely analytical methods used by Bianchi, Darboux and others. I have there fore added articles on the now well-known Frenet-Serret formulae, with some applications 3680, on vi EDITOES PEEPACE TO FIFTH EDITION. the intrinsic equations of a twisted curve 368J, on Bertrand curves 368c, and on the application of Gausss parametric method to conformal represent ation, geodesic curvature and geodesic torsion 3960, 3965. To the portion dealing with the differential geometry of curves on quadrics, I have added Staudes thread-construction for ellipsoids 4210, which is the three-dimensional analogue of Gravess theorem and his definitions of confocal quadrics 421ft by means ofbroken distances these are the analogues of the ordinary definitions of conies by means of focal radii. In the Golden Age of Euclidean geometry, analogues of these types were of great interest to men like Jacobi, MacCullagh, Chasles and M. Roberts, but Staudes constructions have virtually brought the subject to a conclusion. Staudes treatment is also an excellent illustration of the elementary and visible meaning of elliptic and hyper-elliptic in tegrals. New matter is also contained in Arts. 8Qa, 80, 880, 1590, 172, 173, 1760, 261, 304, 384, and in various paragraphs throughout the book. Most of these additions are of the nature of commentaries. About 100 examples are added many of them being solved in order to illustrate the principles of the articles to which they are appended. The numbering of the chapters and of the articles is the same as in the fourth edition, except in Chapter III., where the order has been somewhat changed, and in Arts. 172, 173. The present edition has been published by the EDITOBS PBEFACE TO FIFTH EDITION. vii direction of the Board of Trinity College, who appointed me as Editor in November, 1910. REGINALD A. P. ROGERS. TRINITY COLLEGE, DUBLIN, November, 1911. The Sixth Edition 1914 of Vol. I. is reprinted from the Fifth, with a few corrections, of which the most important are in Arts 88, 338 Ex., 344, 357 PREFACE TO THE THIRD EDITION. IN the preface to the second edition of my Higher Plane Cnrres, I have explained the circumstances under which I obtained Professor Cayleys valuable help in the preparation of that volume. I have now very gratefully to acknowledge that the same assistance has been continued to me in the re-editing of the present work. Thechanges from the preceding edition are not so numerous here as in the case of the Higher Plane Curves, partly because the book not having been so long out of print required less alteration, partly because the size to which the volume had already swelled made it necessary to be sparing in the addition of new matter. Prof...