% 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, 2000.

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\centerline{\Largebf ON FLUCTUATING FUNCTIONS}

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\centerline{\Largebf (ABSTRACT)}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Proceedings of the Royal Irish Academy,
   1 (1841), pp.\ 475--477)}

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

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\centerline{\largerm 2000}

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\centerline{{\largeit On Fluctuating Functions.}  {\largerm [Abstract.]}}

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\centerline{\largerm Sir William Rowan Hamilton}
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\centerline{[June 22nd, 1840.]}
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\centerline{[{\it Proceedings of the Royal Irish Academy},
   vol.~i (1841), pp.~475--477.]}

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The President gave an account of some investigations respecting
{\it Fluctuating Functions}, from which the following are
extracts:---

``Let ${\sc p}_x$ denote any real function [of] $x$, continuous
or discontinuous, but such that its first and second integrals,
$$\int_0^x dx \, {\sc p}_x,\quad\hbox{and}\quad
  \left( \int_0^x dx \right)^2 {\sc p}_x,$$
are always comprised between given finite limits.  Let also the
equation
$$\left( \int_0^x dx \right)^2 {\sc p}_x = \mu,$$
in which $\mu$ is some given constant, have infinitely many real
roots, both positive and negative, which are not themselves
comprised between any finite limits, but are such that the
interval between one and the next greater root is never greater
than some given finite interval.  Then
$$\mathop{\rm lim.}_{t = \infty} \int_a^b dx \, \int_0^{tx} dy \,
         {\sc p}_y {\sc f}_x
   = 0,
   \eqno {\rm (A)}$$
if $a$ and $b$ are any finite values of $x$, between which the
function~${\sc f}_x$ is finite.

``Again, the same things being supposed, let the arbitrary
function ${\sc f}_x$ vary gradually between the same values of
$x$, and let ${\sc p}_x$ be finite and vary gradually when $x$ is
infinitely small; then
$${\sc f}_y = \varpi^{-1} \int_0^\infty dt \, \int_a^b dx \,
         {\sc p}_{tx - ty} {\sc f}_x,
   \quad \left( y \matrix{ > a \cr < b \cr} \right),
   \eqno {\rm (B)}$$
in which
$$\varpi = \int_{-\infty}^\infty dx \, \int_0^1 {\sc p}_{tx}.$$

``For the case $y = a$, we must change $\varpi$, in (B), to
$$\varpi^\backprime
   =  \int_0^\infty dx \, \int_0^1 dt \, {\sc p}_{tx};$$
and for the case $y = b$, we must change it to
$$\varpi^{\backprime\backprime}
   =  \int_{-\infty}^0 dx \, \int_0^1 dt \, {\sc p}_{tx}.$$

``For values of $y > b$, or $< a$, the second member of the
formula (B) vanishes.

``If ${\sc f}_x$, although finite, were to receive any sudden
change for some particular value of $y$ between $a$ and $b$, so
as to pass suddenly from the value
${\sc f}^{\backprime\backprime}$ to the value
${\sc f}^\backprime$, we should then have, for this value of $y$,
$$\int_0^\infty dt \, \int_a^b dx \, {\sc p}_{tx - ty} {\sc f}_x
   =  \varpi^\backprime {\sc f}^\backprime
       + \varpi^{\backprime\backprime} {\sc f}^{\backprime\backprime}.$$
By changing ${\sc p}_x$ to $\cos x$, we obtain from (B) the
celebrated theorem of Fourier.  Indeed, that great mathematician
appears to have possessed a clear conception of the
{\it principles} of fluctuating functions, although he is not
known to have deduced from them consequences so general as the
above.

``Again another celebrated theorem is comprised in the
following:---
$${\sc f}_y
   =  \varpi^{-1} {\sc p}_0
      \left(
         \int_a^b dx \, {\sc f}_x
       + \sum\nolimits_{(n) 1}^{\phantom{(n)} \infty}
            \int_a^b dx \, {\sc q}_{x - y,n} {\sc f}_x
      \right),
   \eqno {\rm (C)}$$
in which, the function ${\sc q}$ is defined by the conditions
$${\sc q}_{x,n} \int_0^x dx \, {\sc p}_x
   =  \int_{2nx - x}^{2nx + x} dx \, {\sc p}_x;$$
$y$ is $> a$, $< b$; and no real root of the equation
$$\int_0^\infty dx \, {\sc p}_x = 0,$$
except the root~$0$ is included between the negative number
$a - y$ and the positive number $b - y$, nor are those numbers
themselves supposed to be roots of that equation.  When these
conditions are not satisfied, the theorem (C) takes other forms,
which, with other analogous results, may be deduced from the same
principles.''

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