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Unformatted text preview: mmm Wf / I xSm*^i?c?^n A TREATISE ON THE DIFFERENTIAL GEOMETRY OF PURVES AND SURFACES BY LUTHER PFAHLER EISENHART PROFESSOR OF MATHEMATICS IN PRINCETON UNIVERSITY GINN AND COMPANY BOSTON NEW YORK CHICAGO LONDON CvA COPYRIGHT, 1909, BY LUTHER PFAHLER EJSENHART ALL RIGHTS RESERVED 898 gftc SUftengum GINN AND COMPANY PKOPRILTORS BOSTON U.S.A. 6 A.ATH. . STAT. LIBRARY PEEFACE This book is a development from courses which I have given in Princeton for a number of years. During this time I have come to feel that more would be accomplished by my students if they had an introductory treatise written in English and otherwise adapted to the use of men beginning their graduate work. Chapter I is devoted to the theory of twisted curves, the method in general being that which is usually followed in discussions of this subject. But in addition I have introduced the idea of moving axes, and have derived the formulas pertaining thereto from the previously obtained FrenetSerret formulas. In this way the student is made familiar with a method which is similar to that used by Darboux in the first volume of his Lemons, and to that of Cesaro in his Geometria Intrinseca. This method is not only of great advantage in the treat ment of certain topics and in the solution of problems, but it is valu able in developing geometrical thinking. The remainder of the book may be divided into three parts. The first, consisting of Chapters IIYI, deals with the geometry of a sur face in the neighborhood of a point and the developments therefrom, such as curves and systems of curves defined by differential equa tions. To a large extent the method is that of Gauss, by which the properties of a surface are derived from the discussion of two quad ratic differential forms. However, little or no space is given to the algebraic treatment of differential forms and their invariants. In addition, the method of moving axes, as defined in the first chapter, has been extended so as to be applicable to an investigation of the properties of surfaces and groups of surfaces. The extent of the theory concerning ordinary points is so great that no attempt has been made to consider the exceptional problems. For a discussion of such questions as the existence of integrals of differential equa tions and boundary conditions the reader must consult the treatises which deal particularly with these subjects. In Chapters VII and VIII the theory previously developed is applied to several groups of surfaces, such as the quadrics, ruled surfaces, minimal surfaces, surfaces of constant total curvature, and surfaces with plane and spherical lines of curvature. iii iv PEEFACE The idea of applicability of surfaces is introduced in Chapter III as a particular case of conformal representation, and throughout the book attention is called to examples of applicable surfaces....
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