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Todhunter (I.) — continued. The subject of this volume has high claims to consideration on account of the subtle problems which it involves, the valuable contributions to analysis which it has produced, its important practical applications, and the emi&shy;nence ''of those who have cultivated it. The subject claims all the interest'' which illustrious names can confer: nearly every great mathematician within the range of a century and a half comes up in the course of the ''history. The present work, though principally a history, may claim the'' title of a comprehensive treatise on the Theory of Probability, for it assumes in the reader only so much knowledge as can be gained from an elementary book on Algebra, and introduces him to almost every process and every ''species of problem which the literature of the subject can furnish. The'' author has been careful to reproduce the essential elements of the original works which he has analysed, and to corroborate his statements by exact quotations from the originals, in the languages in which they were published.

The subject of this Essay was prescribed in the following terms by the Examiners: — “A determination of the circumstances under which dis&shy;continuity of any kind presents itself in the solution of a problem of maximum or minimum in the Calculus of Variations, and applications to ''particular instances. It is expected that the discussion of the instances'' should be exemplified as far as possible geometrically, and that attention be especially directed to cases of real or supposed failure of the Calculus.” The Essay, then, is mainly devoted to the consideration of discontinuous solutions; but incidentally various other questions in the Calculus of Variations are ''examined and elucidated. The author hopes that he has definitely contri&shy;buted'' to the extension and improvement of our knowledge of this refined department of analysis.

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