the advantage of students, and those who had the charge of his manuscripts were naturally inclined to adopt a course of which they believed he would himself have approved.

The design of reconstructing the Treatise was therefore abandoned; and it was resolved that the original volume should be reprinted, and that the manuscripts should be collected and published separately. This plan has the obvious recommendation of enabling those who are already familiar with the original work to turn their attention readily to the new investigations. It will be seen that many of the Chapters of the present volume may be regarded as independent essays or memoirs which lose nothing by being separated from the other volume; and indeed no indications had been left by Professor Boole of the place which such Chapters were to occupy in the enlarged edition.

I have printed all the unpublished matter relating to Differential Equations which I found among Professor Boole's papers. In a few cases it will be seen that an investigation is incomplete; such investigations have however been included in the volume, because I was unwilling that anything should be lost which so great a mathematician had written on a subject he had long and carefully studied.

I trust that no serious error will be found in the volume, and that any faults which may be detected will be excused on account of the nature and difficulty of the task that had to be performed. Many of the manuscripts had not been finally revised; some of them were very obscure and had to be carefully and laboriously copied for the press. In general the equations were not numbered, and thus only

blanks occurred in place of references; this circumstance often caused great trouble and perplexity: I hope however that a satisfactory result has been finally attained.

I may state for the benefit of those who are conversant with the first edition of the original work that the theorem which in the present volume is cited as contained in Chap. II. Art. 1 will be found in Chap. IV. Art. 2 of the first edition: the change was made by the direction of Professor Boole's interleaved copy. It was judged convenient to number the Chapters in the present volume in continuation of those in the original work,

All additions of my own are enclosed within square brackets. The sheets have been read by the Rev. J. Sephton, Fellow of St John's College, as well as by myself, and the volume is much indebted to his care and accuracy. Obvious mistakes in the manuscripts were of course corrected; thus, for example, the table at the end of the volume was calculated by Mr Sephton, because the table in the manuscript was rendered erroneous by the use of a wrong sign in a formula.

ST JOHN'S COLLEGE,
November, 1865.

I. TODHUNTER.

LIST OF PROFESSOR BOOLE'S WRITINGS.

In the Philosophical Transactions.

On a General Method in Analysis, 1844, pages 225...282. On the Comparison of Transcendents, with certain applications to the Theory of Definite Integrals, 1857, pages 745...803.

On the Theory of Probabilities, 1862, pages 225...252.

On Simultaneous Differential Equations of the First Order in which the Number of the Variables exceeds by more than one the Number of the Equations, 1862, pages 437...454.

On the Differential Equations of Dynamics. A sequel to a Paper on Simultaneous Differential Equations, 1863, pages 485...501.

On the Differential Equations which determine the form of the Roots of Algebraic Equations, 1864, pages 733...755.

In the Transactions of the Royal Irish Academy.

On the Analysis of Discontinuous Functions. Vol. 21, 1848, pages 124...139.

On a certain Multiple Definite Integral. Same Vol., pages 140...149.

In the Transactions of the Royal Society of Edinburgh. On the Application of the Theory of Probabilities to the Question of the Combination of Testimonies or Judgments. Vol. 21, 1857, pages 597...653.

In the Bulletin de l'Académie...de St Pétersbourg.

Considérations sur la recherche des intégrales premières des équations différentielles partielles du second ordre, Vol. iv. 1862, pages 198...215. [See page 143 of the present volume.]

In Crelle's Journal für Mathematik.

Ueber die partielle Differentialgleichung zweiter Ordnung Rr + Ss + Tt + Ū (s2 − rt) = V. Vol. 61, pages 309...333.

In the Cambridge Mathematical Journal.

Researches on the Theory of Analytical Transformations, with a special application to the Reduction of the General Equation of the Second Order. Vol. 2, 1841, pages 64...73.

On Certain Theorems in the Calculus of Variations. Same Vol., pages 97...102.

On the Integration of Linear Differential Equations with Constant Coefficients. Same Vol., pages 114...119.

Analytical Geometry. Same Vol., pages 179...188.

Exposition of a General Theory of Linear Transformations. Vol. 3, 1843, pages 1...20, 106...119.

On the Transformation of Definite Integrals. Same Vol., pages 216...224.

Remarks on a Theorem of M. Catalan. Same Vol., pages 277...283.

On the Transformation of Multiple Integrals. Vol. 4, 1845, pages 20...28.

On the Inverse Calculus of Definite Integrals. Same Vol., pages 82...87.

Notes on Linear Transformations. Same Vol., pages 167...171.
On the Theory of Developments. Same Vol., pages 214...223.

In the Cambridge and Dublin Mathematical Journal.

On the Equation of Laplace's Functions. Vol. 1, 1846, pages 10...22.

On the Attraction of a Solid of Revolution on an External Point. Vol. 2, 1847, pages 1...7.

On a certain Symbolical Equation. Same Vol., pages 7...12. On a General Transformation of any Quantitative Function. Vol. 3, 1848, pages 112...116.

The Calculus of Logic. Same Vol., pages 183...198.

On a General Theorem of Definite Integration. Vol. 4, 1849, pages 14...20.

On the Theory of Linear Transformations. Vol. 6, 1851, pages 87...106.

On the Reduction of the General Equation of the nth Degree. Same Vol., pages 106...113.

Letter to the Editor of the Journal. Same Vol., pages 284, 285. Proposed Question in the Theory of Probabilities. Same Vol., page 286.

On Reciprocal Methods in the Differential Calculus. Vol. 7, 1852, pages 156...166, and Vol. 8, 1853, pages 1...24.

In the London, Edinburgh, and Dublin Philosophical Magazine... Third Series.

Remarks on the Rev. B. Bronwin's Method for Differential Equations. Vol. 30, 1847, pages 6...8.

Note on a Class of Differential Equations. Same Vol., pages 96, 97.

Remarks on a Paper by the Rev. Brice Bronwin, On the Solution of a particular Differential Equation. Vol. 32, 1848, pages 413...418.

Remarks on a Paper by the Rev. Brice Bronwin, On the Solution of a Particular Differential Equation. Vol. 33, 1848, page 211. Notes on Quaternions. Same Vol., pages 278...280.

In the Fourth Series of the same Magazine.

On the Theory of Probabilities, and in particular on Mitchell's Problem of the Distribution of the Fixed Stars. Vol. 1, 1851, pages 521...530.

Further Observations on the Theory of Probabilities. Vol. 2, 1851, pages 96...101.

An Account of the late John Walsh of Cork. In a letter from Professor Boole to Professor de Morgan. Same Vol., pages 348...358.

Solution of a Question in the Theory of Probabilities. Vol. 7, 1854, pages 29...32.

Reply to some Observations published by Mr Wilbraham in the Philosophical Magazine, Vol. 7, p. 465, on the Theory of Chances developed in Professor Boole's Laws of Thought.' Vol 8, 1854, pages 87...91.