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XIII. On the Explanation of a Difficulty in Analysis noticed by Sir William Hamilton. By ARTHUR AUGUSTUS MOORE, ESQ. of Trinity College.

[Read May 1, 1837.]

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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-, 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- 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, à priori, as exceptions to the general principle was that in the demonstration

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* M. Cauchy remarks that this function and all it differential coefficients vanish for the particular value of the variable x = 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.

of this principle itself we commit the error of passing directly from finite to infinite states of functions and variables, instead of estimating and comparing their relations in the different stages of their convergency. I have attempted in what follows to give a rigorous demonstration of the principle in question, at the same time fixing the precise limits of its application, and enumerating the different classes of functions to which it necessarily does not apply.

1. To effect this object I shall begin by explaining what is understood by infinitesimals of different orders. If a function of a converges indefinitely towards zero along with x, in such a manner that, for a very small value of x, f(x) shall be less than any given magnitude, the function f(x) at the limit of those values of a which converge indefinitely towards zero is called an infinitely small quantity or infinitesimal. But as for similar decreasing values of x the ratio of convergency may be much higher in one function than in another, we are led naturally to consider indefinitely decreasing quantities of different degrees or orders of convergency. And having fixed upon some one function whose ratio of decrease we assume as the unit of convergency, we call a second function which for similar decreasing values of a decreases in m times as fast a ratio as the first, an indefinitely decreasing quantity of the mth order. We extend this definition to the infinitesimals which are the limits of these quantities, and call the infinitesimal which is the limit of the former of the quantities, an infinitesimal of the first, and the infinitesimal which is the limit of the latter, an infinitesimal of the mth order. Choosing x itself for the function whose ratio of decrease is taken as the unit of convergency, we see clearly that when x is less than unity Ax is an indefinitely decreasing quantity of the mth order, where m may be integer or fractional. From this we infer that Aa may represent an indefinitely decreasing quantity of any order, and that the limit of Aam for values of a which converge indefinitely towards zero may represent an infinitesimal of any order. This we shall designate by the notation limo (Ax"). A very wide generalization, which only suggests itself from the study of the different analytical functions, is given to this definition by defining limo {f(x)} to be an infinitesimal of the mth order, if any finite and positive value of m can

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