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, 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. 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. PAGE Integrating factors always exist, 56. General form of integrating factors, 57. Special determinations for homogeneous equations and for equations of the form F1 (xy) ydx+F2 (xy) xdy =0, 59—62. ON THE GENERAL DETERMINATION OF THE INTEGRATING FACTORS OF THE EQUATION Mdx + Ndy = 0. Partial differential equation for integrating factors, 69. Solution when the integrating factor is a function of x, 70—of y, 71—of xy, 72. When homogeneous, 74–76. More general application, 82. So- lution of P1dx+P2dy + Q(xdy − ydx)=0,84. Jacobi's equation, 85. Primary definition-positive and negative marks, 139, 140. Deriva- tion of the singular solution from the complete primitive, 141. General theorem, 148. Geometrical interpretation, 152. Deri- vation of the singular solution from the differential equation, 153. More general definition of singular solution, 163. Distinction of species, ib. General theorem, 164. Review of prior methods, 174. Relation to complete primitive, 187. Solution by development, 189. equations, 204. Analogy with Algebraic equations, 206. Exer- 139 187 |