% This paper has been transcribed in Plain TeX by
% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
%
% Trinity College, 1st June 1999.

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\centerline{\Largebf NOTE TO A PAPER ON THE ERROR OF A}

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\centerline{\Largebf RECEIVED PRINCIPLE OF ANALYSIS}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Transactions of the Royal Irish Academy,
   vol.~16, part~2, (1831), pp. 129--130.)}

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\centerline{\largerm Edited by David R. Wilkins}

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\centerline{\largerm 1999}

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\centerline{\largeit
Note to a Paper on the Error of a received Principle of Analysis.}

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\centerline{\largeit
By {\largerm WILLIAM R. HAMILTON}, \&c.}

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\centerline{Read April~18, 1831.}

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\centerline{[{\it Transactions of the Royal Irish Academy},
vol.~16, part~2, (1831), pp. 129--130.]}

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The Royal Irish Academy having done me the honour to publish, in
the First Part of the Sixteenth Volume of their Transactions, a
short Paper, in which I brought forward a certain exponential
function as an example of the Error of a received Principle
respecting Developments, I am desirous to mention that I have
since seen (within the last few days) an earlier Memoir by a
profound French Mathematician, in which the same function is
employed to prove the fallacy of another usual principle.  In the
French Memoir, (tom.~{\sc v}. of the Royal Academy of Sciences,
at page~13, of the History of the Academy,) the exponential
$\left( e^{- {1 \over x^2}} \right)$
is given by {\sc M.~Cauchy}, as an example of the vanishing of a
function and of all its differential coefficients, for a
particular value of the variable ($x$), without the function
vanishing for other values of the variable.  In my Paper the same
exponential is given as an example of a function, which vanishes
with its variable, and yet cannot be represented by any
development according to powers of that variable, with constant
positive exponents, integer or fractional.  There is therefore a
difference between the purposes for which this function has been
employed in the two Memoirs, although there is also a sufficient
resemblance to induce me to wish, that at the time of publishing
my Paper, I had been acquainted with the earlier remarks of
{\sc M.~Cauchy}, in order to have noticed their existence.

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{\sc Observatory},

{\it April}~16, 1831.

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