In the Cambridge Mathematical Journal, Researches on the Theory of Analytical Transformations, with & 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. 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. a In the Fourth Series of the same Magazine. a 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. On the Conditions by which the Solutions of Questions in the Theory of Probabilities are limited. Same Vol., pages 91...98. Further Observations relating to the Theory of Probabilities in reply to Mr Wilbrabam. Same Vol., pages 175, 176. On a General Method in the Theory of Probabilities. Same Vol, pages 431...444. On certain Propositions in Algebra connected with the Theory of Probabilities. Vol. 9, 1855, pages 165...179. . On a Question in the Theory of Probabilities. By A. Cayley, Esq. [This paper embodies some observations by Professor Boole.] Vol. 23, 1862, pages 361...365. On a Question in the Theory of Probabilities. Vol. 24, 1862, p. 80. Separate Publications. An Address on the Genius and Discoveries of Sir Isaac Newton. Lincoln, 1835. The Right Use of Leisure. London, 1847. The Mathematical Analysis of Logic, being an Essay towards & Calculus of Deductive Reasoning. Cambridge, 1847. The Claims of Science. London, 1851. An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities. London, 1854. The Social Aspect of Intellectual Culture. An Address dolivered in the Cork Athenæum.... Cork, 1855. A Treatise on Differential Equations. Cambridge, 1859. , A Treatise on the Calculus of Finite Differences. Cambridge, 1860. [This list contains all Professor Boole's writings which have fallen under the notice of the editor; it is possible that there may be a few omissions.] XXV. ON SYSTEMS OF SIMULTANEOUS LINEAR PARTIAL DIFFER- XXIX. ON THE SOLUTION OF THE PARTIAL DIFFERENTIAL EQUATION 31 CHAPTER XIX. ADDITIONS TO CHAPTER II. 1. [In Chapter II. Art. 9, two methods are given for solving the differential equation (ax + by + c) dx + (a'x + b'y + c) dy=0.] But there exists another transformation by which the equation may be reduced to, (because it may be constructed from), an equation in which the variables are separated. Assume as this equation (Ay' + C) dac' + (A'ac' + C') dy' = 0...... (1) and let ac' = x + my, y'= 2 + m y. x It will be seen that in these equations united we have as many constants as in the original equation. Now on substituting in the assumed equation the values of x' and y', and comparing with the equation given, we deduce a system of relations equivalent to the following, viz.: The quantities my, m, are roots of the quadratic am - (b + a') m + b = 0. The quantities A, A', C, C" are determined by the system of equations A+A'= a, C +C"=c, 1 2 2 B.D.E. II. |