A Treatise on Differential Equations

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

writers, to Professor De Morgan. For methods and examples, a very large number of memoirs English and foreign have been consulted: these are, for the most part, acknowledged. At the same time it is right to add that, in almost every part of the work, I found it necessary to engage more or less in original investigation, and especially in those parts which relate to Riccati's equation, to integrating factors, to singular solutions, to the inverse problems of Geometry and Optics, to partial differential equations both of the first and second order, and, as has already been intimated, to symbolical methods. The demonstrations scattered through the work are also many of them new, at least in form.

In recent years much light has been thrown on certain classes of differential equations by the researches of Jacobi on the Calculus of Variations, and of the same great analyst, with Sir W. R. Hamilton and others, on Theoretical Dynamics. I have thought it more accordant with the design. of an elementary treatise to endeavour to prepare the way for this order of inquiries than to enter systematically upon them. This object has been kept in view in the writing of various portions of the following work, and more particularly of that which relates to partial differential equations of the first order.

GEORGE BOOLE.

QUEEN'S COLLEGE, CORK,
February, 1859.

PREFACE TO THE SECOND EDITION.

IN composing his Treatise on Differential Equations Professor Boole found himself deeply interested in the subject to which his first labours as an original investigator had been devoted. In consequence he determined soon after the publication of the volume to continue his studies and researches with the design of ultimately reconstructing the Treatise on a more extensive scale.. During the last six years of his life he worked steadily at this object; and he was about to send the first sheets of the new edition to the press when he was attacked by the illness which terminated in his sudden and lamented death.

His manuscripts were entrusted to me early in the present year. After careful consideration it seemed to me that the best plan to pursue was to reprint the original volume, and to collect into a supplementary volume the additional matter which had been prepared for enlarging the work. The propriety, I might almost say the necessity, of this course will be shewn more conveniently in the preface to the supplementary volume, which will soon be published.

The present volume then is a reprint of the original Treatise with changes and corrections, some of which were indicated in Professor Boole's interleaved copy, and some of which have been made on my own authority. The sheets have been carefully read by the Rev. J. Sephton, Fellow of St John's College, as well as by myself; and I trust that few misprints or errors will now be found in the volume.

ST JOHN'S COLLEGE, CAMBRIDGE,

October, 1865.

I. TODHUNTER.

CONTENTS.

CHAPTER I.

OF THE NATURE AND ORIGIN OF DIFFERENTIAL EQUATIONS

Definition, 2. Species, Order and Degree, 3. General Solution, Com-

plete primitive, 6. Genesis of Differential equations, 8. How

many of each order possible, how many independent, 15. Cri-

terion of derivation from a common primitive, 16. General form

of equations thus related, 17. Geometrical illustrations, 18.

Exercises, 20.