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AUTHOR'S PREFACE. IT is customary to divide the Infinitesimal Calculus, or Calculus of Continuous Functions, into three parts, under the heads Differential Calculus, Integral Calculus, and Differential Equations. The first corresponds, in the language of Newton, to the direct method of tangents, the other two to the inverse method of tangents ; while the questions which come under this last head he further divided into those involving the two fluxions and one fluent, and those involving the fluxions and both fluents. On account of the inverse character which thus attaches to the present subject, the differential equation must necessarily at first be viewed in connection with a primitive, from which it might have been obtained by the direct process, and the solution consists in the discovery, by tentative and more or less artificial methods, of such a primitive, when it exists; that is to say, when it is expressible in the elementary functions which constitute the original field with which the Differential Calculus has to do. It is the nature of an inverse process to enlarge the field of its operations, and the present is no exception; but the adequate handling of the new functions with which the field is thus enlarged requires the introduction of the complex variable, and is beyond the scope of a work of this size. But the theory of the nature and meaning of a differential equation between real variables possesses a great deal of interest. To this part of the subject I have endeavored to give a full treatment by means of extensive use of graphic representations in rectangular coordinates. If we ask what it is that satisfies an ordinary differential equation of the first order, the answer must be certain sets of simultaneous values of x, y, and p. The geometrical representation of such a set is a point in a plane associated with a direction, so to speak, an infinitesimal stroke, and the solution consists of the grouping together of these strokes into curves of which they form elements. The treatment of singular solutions, following Cayley, and a comparison with the methods previously in use, illustrates the great utility of this point of view. Again, in partial differential equations, the set of simultaneous values of x, y, z, p, and q which satisfies an equation of the first order is represented by a point in space associated with the direction of a plane, so to speak by a flake, and the mode in which these coalesce so as to form linear surface elements and continuous surfaces throws light upon the nature of general and complete integrals and of the characteristics. The expeditious symbolic methods of integration applicable to some forms of linear equations, and the subject of development of integrals in convergent series, have been treated as fully as space would allow. Examples selected to illustrate the principles developed in each section will be found at its close, and a full index of subjects at the end of the volume.

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