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XIII. On the Eaplanation of a Difficulty in Analysis noticed by Sir William Hamilton. By ARTHUR AUGUSTU's MooRE, Esq. of Trinity College.

[Read May 1, 1837.]

IN the Memoirs of the Royal Irish Society, Sir William Hamilton has made an important observation upon a general principle of Analysis, which has been used by La Grange as the basis of his Calculus of Functions. Sir W. H. remarks that there is a case in which this principle (which had till then been considered axiomatic and universally true) does not hold good. The case which Sir W. H. cites is the function e-à, which M. Cauchy had already in his Calcul Différentiel shown to be an exception to another generally received principle of analysis”. M. Cauchy seems to be of opinion that the existence of this anomaly is a sufficient reason for rejecting the mode of exposition of the Differential Calculus of which La Grange is the author, and which is certainly based upon the assumption of both these principles, the latter however of which is comprised in the former as a particular case. But the function e-à is only one of a general class of functions which with another constitute the only known exceptions to La Grange's principle. The latter class has no apparent analogy with the former, but on examination we shall find that both these apparent anomalies are immediate consequences of the fundamental conditions of analytical developement, and that the only reason why they were not at once recognized, a priori, as exceptions to the general principle was that in the demonstration

* M. Cauchy remarks that this function and all it differential coefficients vanish for the particular value of the variable r = 0, although the function itself does not vanish for any other value of the variable, thus constituting an exception to a generally received analytical principle.

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