I HAVE endeavoured, in the following Treatise, to convey as complete an account of the present state of knowledge on the subject of Differential Equations, as was consistent with the idea of a work intended, primarily, for elementary instruction. It was my object, first of all, to meet the wants of those who had no previous acquaintance with the subject, but I also desired not quite to disappoint others who might seek for more advanced information. These distinct, but not inconsistent aims determined the plan of composition. The earlier sections of each chapter contain that kind of matter which has usually been thought suitable for the beginner, while the latter ones are devoted either to an account of recent discovery, or to the discussion of such deeper questions of principle as are likely to present themselves to the reflective student in connexion with the methods and processes of his previous course. An appendix to the table of contents will shew what portions of the work are regarded as sufficient for the less complete, but still not unconnected study of the subject. The principles which I have kept in view in carrying out the above design, are the following: 1st, In the exposition of methods I have adhered as closely as possible to the historical order of their development. I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in a certain order, or that that order has been, to a great extent, necessary-being determined, either by steps of logical deduction, or by the successive introduction of new ideas and conceptions, when the time for their evolution had 395432 arrived. And these are causes which operate in perfect harmony. Each new scientific conception gives occasion to new applications of deductive reasoning; but those applications may be only possible through the methods and the processes which belong to an earlier stage. Thus, to take an illustration from the subject of the following work, the solution of ordinary simultaneous differential equations properly precedes that of linear partial differential equations of the first order; and this, again, properly precedes that of partial differential equations of the first order which are not linear. And in this natural order were the theories of these subjects developed. Again, there exist large and very important classes of differential equations the solution of which depends on some process of successive reduction. Now such reduction seems to have been effected at first by a repeated change of variables; afterwards, and with greater generality, by a combination of such transformations with others involving differentiation; last of all, and with greatest generality, by symbolical methods. I think it necessary to direct attention to instances like these, because the indications which they afford appear to me to have been, in some works of great ability, overlooked, and because I wish to explain my motives for departing from the precedent thus set. Now there is this reason for grounding the order of exposition upon the historical sequence of discovery, that by so doing we are most likely to present each new form of truth to the mind, precisely at that stage at which the mind is most fitted to receive it, or even, like that of the discoverer, to go forth to meet it. Of the many forms of false culture, a premature converse with abstractions is perhaps the most likely to prove fatal to the growth of a masculine vigour of intellect. In accordance with the above principles I have reserved the exposition, and, with one unimportant exception, the application of symbolical methods to the end of the work. The propriety of this course appears to me to be confirmed by an examination of the actual processes to which symbolical methods, as applied to differential equations, lead. Generally speaking, these methods present the solution of the proposed equation as dependent upon the performance of certain inverse operations. I have endeavoured to shew in Chap. XVI., that the expressions by which these inverse operations are symbolized are in reality a species of interrogations, admitting of answers, legitimate, but differing in species and character according to the nature of the transformations to which the expressions from which they are derived have been subjected. The solutions thus obtained may be particular or general,they may be defective, wholly or partially, or complete or redundant, in those elements of a solution which are termed arbitrary. If defective, the question arises how the defect is to be supplied; if redundant, the more difficult question whether the redundancy is real or apparent, and in either case how it is to be dealt with, must be considered. And here the necessity of some prior acquaintance with the things themselves, rather than with the symbolic forms of their expression, must become apparent. The most accomplished in the use of symbols must sometimes throw aside his abstractions and resort to homelier methods for trial and verification -not doubting, in so doing, the truth which lies at the bottom of his symbolism, but distrusting his own powers. The question of the true value and proper place of symbolical methods is undoubtedly of great importance. Their convenient simplicity-their condensed power-must ever constitute their first claim upon attention. I believe however that, in order to form a just estimate, we must consider them in another aspect, viz. as in some sort the visible manifestation of truths relating to the intimate and vital connexion of language with thought-truths of which it may be presumed that we do not yet see the entire scheme and con nexion. But, while this consideration vindicates to them a high position, it seems to me clearly to define that position. As discussions about words can never remove the difficulties that exist in things, so no skill in the use of those aids to thought which language furnishes can relieve us from the necessity of a prior and more direct study of the things which are the subjects of our reasonings. And the more exact, and the more complete, that study of things has been, the more likely shall we be to employ with advantage all instrumental aids and appliances. But although I have, for the reasons above mentioned, treated of symbolical methods only in the latter chapters of the work, I trust that the exposition of them which is there given will repay the attention of the student. I have endeavoured to supply what appeared to me to be serious defects in their logic, and I have collected under them a large number of equations, nearly all of which are important,-from their connexion with physical science or for other reasons. 2ndly, I have endeavoured, more perhaps than it has been usual to do, to found the methods of solution of differential equations upon the study of the modes of their formation. In principle, this course is justified by a consideration of the real nature of inverse processes, the laws of which must be ultimately derived from those of the direct processes to which they stand related; in point of expediency it is recommended by the greater simplicity, and even in some instances by the greater generality, of the demonstrations to which it leads. I would refer particularly to the demonstration of Monge's method for the solution of partial differential equations of the second order given in Chap. XV. With respect to the sources from which information has been drawn, it is proper to mention that, on questions relating to the theory of differential equations, my obligations are greatest to Lagrange, Jacobi, Cauchy, and, of living |