A Treatise on the Calculus of Variations Arranged with the Purpose of Introducing, as Well as Illustrating, Its Principles to the Reader by Means of Problems, and Designed to Present in All Important Particulars a Complete View of the Present State...

TREATISE ON THE CALCULUS OF VARIATIONS. ARRANGED WITH THE PURPOSE OF INTRODUCING, AS WELL AS ILLUS TRATING, ITS PRINCIPLES TO THE READER BY MEANS OF PROBLEMS, AND DESIGNED TO PRESENT IN ALL IMPORTANT PARTICULARS A COMPLETE VIEW OF THE PRESENT STATE OF THE SCIENCE, BY LEWIS BUFFETT CARLL, A. M. NEW YORK JOHN WILEY SONS, 15 ASTOR PLACE. 1881. PREFACE. THIRTY years have now elapsed since the appearance of the treatise on the Calculus of Variations by Prof. Jellett, which, although it had been preceded by the smaller work of Woodhouse in 1810, and of Abbatt in 1837, s justly deemed the only complete treatise which has ever appeared in Eng lish. But all the works named have long since been out of print, and are now so rare as not to be found in the majority of the college libraries of the United States. Moreover, even Prof. Jelletts treatise can no longer be regarded as complete, since its author had not read the memoirs of Sarrus and Cauchy relative to multiple integrals, while the contributions of Hesse, Moigno and Lindelof, and Todhunter were subse quent to the publication of his work. It should be added, also, that all the memoirs and contributions just named are contained in works which are likewise out of print, and are now almost as difficult of access to the general reader as is that of Prof. Jellett. These considerations first led the author to undertake the preparation of the present treatise, in which he has endeav ored to present, in as simple a manner as he could command, everything of importance which is at present known concern ing this abstruse department of analysis. In the execution of this design the following method has, so far as possible, been pursued When a newprinciple is to be introduced for the first time, a simple problem involving it is first proposed, and the principle is established when tre PREFACE. quired in the discussion of this problem. This having been followed by other problems of the same class, the general theory of the subject is finally given and illustrated by one or two of the most difficult problems obtainable after which another principle is introduced in like manner. Although the view taken of a variation is that of Profs. Airy and Todhunter, and the methods of varying functions are those of Jellett and Strauch, still all the other leading conceptions and methods have, it is hoped, been explained with sufficient fulness to enable the reader to follow them when they occur in other works. The history of the subject is also briefly given in the last chapter, it being believed that the proper time for the presen tation of the history of any science is after the reader has become familiar with its principles, as it can then, by the use of some technical terms, be accomplished more fully within a given space. To aid the non-classical reader, the use of Greek letters has, with the exception of two, whose use is now universal, and which are explained, been avoided, except in references, or in such passages as may be omitted without serious loss. Attention is also called to the words bracliistochrone and parallelcpipcdon, which are in this work spelled according to their derivation. The correct orthography of the former had been previously adopted by Moigno and Todhunter, and it is hoped that it may be sufficient to call the attention of Greek scholars to the latter. One of the great obstacles to the preparation of the pres enttreatise has been the difficulty of procuring the author ities which it was necessary to consult and the author would here return his thanks to the officers of his Alma Mater, Columbia College to Dr. Noah Porter, the President, and Mr. A. Van Name, the Librarian, of Yale College and to Mr. Walter M. Ferris, of Bay Ridge, L. I., for the extended loan of rare works which could not be found in other libraries, or PREFACE. V if found, could not be had at home for that careful study which they in many cases demanded. The author is also greatly indebted to Lieut. Fred. V...