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 Wilbraham. 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 delivered 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.] CONTENTS. XXI. ADDITIONS TO CHAPTER VIII. XXII. ADDITIONS TO CHAPTER IX. XXIII. ADDITIONS TO CHAPTER X. XXIV. ADDITIONS TO CHAPTER XIV. XXV. ON SYSTEMS OF SIMULTANEOUS LINEAR PARTIAL DIFFER- ENTIAL EQUATIONS OF THE FIRST ORDER, AND ON Asso. XXVIII. PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 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 and let (Ay' + C) dx' + (A'x' + C') dy' = 0......(1) x' = x+m1y, y' = x+m2y. 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 m1, m, are roots of the quadratic am2 − (b + a') m + b' = 0. The quantities A, A', C, C' are determined by the system of equations A+ A' = a, C+C'= c, C'm1+ C'm2 = c', 1 Am ̧+ A'm2 = a', B. D.E. II. Now (1) gives on dividing by (A'x' + C') (Ay' + C) and integrating or 1 A' 1 ¡, log (A'x' + C') + log (Ay' + C') = const., A 1 (A'x' + C') 1⁄4 (Ay' + C)1 = const., which on substitution and reduction gives {(am,— a') (x + m ̧y) + cm, — c'}um,—a′ 2. Under certain circumstances the general solutions of differential equations of the first order fail. This happens in the above example if m2 = m1, the solution then reducing to 2 1 = const. The theory of the deduction of the true limiting form of the solution in such cases requires a distinct statement. Let the supposed general solution be represented by u = C being the arbitrary constant and u a function of x, y, and constants which are not arbitrary. Suppose too that when one of these constants k assumes a particular value, the function u reduces to a constant v. Then we have Now the second member being a function of an arbitrary constant is equivalent to an arbitrary constant and may be replaced by C. The first member is a vanishing fraction, the limiting value of which is (du), the brackets being used to dk denote that after the differentiation k is to be made equal to K. Hence the solution becomes In applying this theory to the reduction of the general solution (2) in the case in which m1 = m2, it must be observed that the numerator of the first member is the same function of m1, x, y, as the denominator is of m。, x, y; or attending solely to their functional character with respect to m1, m,, we may affirm that the numerator is the same function of m, as the denominator is of m. Representing these functions by (m), (m) respectively, we have But m,, m, being roots of a quadratic equation may be represented in the form |