XII. On the theory of probabilities

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Abstract
This paper has for its object the investigation of the general analytical conditions of a Method for the solution of Questions in the Theory of Probabilities, which was proposed by me in a work entitled “An Investigation of the Laws of Thought” (London, Walton and Maberly, 1854). The application of this method to particular problems has been illustrated in the work referred to, and yet more fully in a ‘Memoir on the Combination of Testimonies and of Judgments’ published in the Transactions of the Royal Society of Edinburgh (vol. xxi. Part 4). Some observations, too, on the general character of the solutions to which the method leads, founded upon induction from particular cases, were contained in the original treatise, and the outlines, still in some measure conjectural, of their general theory were given in an Appendix to the Memoir. But the complete development of that theory was attended with analytical difficulties which I have only lately succeeded in overcoming. It involves discussions relating to the properties of a certain functional determinant, and to the possible solutions of a system of algebraic equations of peculiar form—discussions which will, I trust, be thought to possess a value, as contributions to Mathematical Analysis, independent of their present application.

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