By William R. Hamilton

Hamilton published a short paper entitled

which appeared in the Transactions of the Royal Irish Academy, volume 16, part 1 (1830), pp. 63-64.On the Error of a received Principle of Analysis, respecting Functions which vanish with their Variables.

This paper is available in the following formats:

In this paper, Hamilton observes that the function equal to the
exponential of -1/*x*^{2} for positive values of the
real variable *x* cannot be expanded as a power series in
*x*, despite the fact that the function tends to the limit
zero as *x* tends to zero.

Hamilton subsequently published a note on this paper,

which appeared in the Transactions of the Royal Irish Academy, volume 16, part 2 (1831), pp. 129-130.Note to a Paper on the Error of a received Principle of Analysis.

This note is available in the following formats:

In this note Hamilton cites a paper by Cauchy, which had just
come to his attention, in which the function he had considered,
equal to the exponential of -1/*x*^{2} for positive
values of the real variable *x*, was given as an example of
a non-zero function with the property that the function and all
its derivatives vanish at zero.

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D.R. Wilkins(

School of Mathematics

Trinity College, Dublin