% 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 SOME INVESTIGATIONS CONNECTED WITH}

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\centerline{\Largebf THE CALCULUS OF PROBABILITIES}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (British Association Report, 1843,
   Part~II, pp.\ 3--4.)}

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

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\centerline{\largerm 2000}

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{\largeit\noindent
On some investigations connected with the Calculus of Probabilities.
By\/}
{\largerm Sir} {\largesc William Rowan Hamilton}.

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\centerline{[{\it Report of the Thirteenth Meeting of the British
Association for the Advancement of}}
\centerline{{\it Science; held at Cork in August 1843}.}
\centerline{(John Murray, London, 1844), Part~II, pp.\ 3--4.]}

\bigbreak

Many questions in the mathematical theory of probabilities
conduct to approximate expressions of the form
$$p = {2 \over v \pi} \int_0^t dt \, e^{-t^2},
   \quad\hbox{that is,}\quad
  p = \Theta(t),$$
$\Theta$ being the characteristic of a certain function which has
been tabulated by Encke in a memoir on the method of least
squares, translated in vol.~ii. part~7. of Taylor's Scientific
Memoirs, $p$ being the probability sought, and $t$ an auxiliary
variable. Sir William Hamilton proposes to treat the equation
$p = \Theta(t)$ as being, in all cases, rigorous, by suitably
determining the auxiliary variable~$t$, which variable he
proposes to call the {\it argument of probability}, because it is
the argument with which Encke's Table should be entered, in order
to obtain, from that table, the numerical value of the
probability~$p$.  He shows how to improve several of Laplace's
approximate expressions for this argument~$t$, and uses in many
such questions a transformation of a certain double definite
integral of the form
$${4 s^{1 \over 2} \over \pi} \int_0^r dr \,
      \int_0^\infty du \,
      e^{-s u^2} {\rm U} \cos (2 r s^{1 \over 2} u {\rm V})
   =  \Theta(r + \nu_1 r s^{-1} + \nu_2 r s^{-2} \ldots ),$$
in which
$${\rm U} = 1 + \alpha_1 u^2 + \alpha_2 u^4 \ldots,\quad
  {\rm V} = 1 + \beta_1  u^2 + \beta_2  u^4 \ldots,$$
which $\nu_1, \nu_2,\ldots$ depend on
$\alpha_1 \, \ldots \, \beta_1 \, \ldots$ and $r$; thus
$\nu_1 = {1 \over 2} \alpha_1 - \beta_1 r^2$.
The function~$\Theta$ has the same form as before, so that if,
for sufficiently large values of the number~$s$ (which
represents, in many questions, the number of observations or
events to be combined) a probability~$p$ can be expressed,
exactly or nearly, by the foregoing double definite integral,
then the {\it argument\/}~$t$, of this probability~$p$, will be
expressed nearly by the formula
$$t = r (1 + \nu_1 s^{-1} + \nu_2 s^{-2}).$$

Numerical examples were given, in which the approximations thus
obtained appeared to be very close.  For instance, if a common
die (supposed to be perfectly fair) be thrown six times, the
probability that the sum of the six numbers which turn up in
these six throws shall not be less than 18, nor more than 24, is
represented rigorously by the integral
$$p = {2 \over \pi} \int_0^{\pi \over 2} dx \,
         {\sin 7x \over \sin x}
         \left( {\sin 6x \over 6 \sin x} \right)^6,
   \quad\hbox{or by the fraction }
      {27448 \over 46656};$$
while the approximate formula, deduced by the foregoing method,
gives 27449 for the numerator of this fraction, or for the
product $6^6 p$; the error of the resulting probability being
therefore in this case only $6^{-6}$.  The advantage of the
method is that what has here been called the {\it argument of
probability\/} depends, in general, more simply than the
probability itself on the conditions of a question; while the
introduction of this new conception and nomenclature allows some
of the most important known results respecting the mean results
of many observations to be enunciated in a simple and elegant
manner.

\bye