% 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 ON DIFFERENCES AND DIFFERENTIALS OF}

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\centerline{\Largebf FUNCTIONS OF ZERO}

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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,
   17 (1837), pp.\ 235--236.)}

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

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\centerline{\largerm 1999}

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{\largeit\noindent
On Differences and Differentials of Functions of Zero.
\hskip 0pt plus1em minus0pt
By {\largerm WILLIAM R. HAMILTON}, Royal Astronomer of Ireland.}

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\centerline{Read June~13, 1831.}

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\centerline{[{\it Transactions of the Royal Irish Academy},
vol.~17 (1837), pp.\ 235--236.]}

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The first important researches on the differences of powers of
zero, appear to be those which Dr.~{\sc Brinkley} published in
the Philosophical Transactions for the year 1807.  The subject
was resumed by Mr.~{\sc Herschel} in the Philosophical
Transactions for 1816; and in a collection of Examples on the
Calculus of Finite Differences, published a few years afterwards
at Cambridge.  In the latter work, a remarkable theorem is given,
for the development of any function of a neperian exponential, by
means of differences of powers of zero.  In meditating upon this
theorem of Mr.~{\sc Herschel}, I have been led to one more
general, which is now submitted to the Academy.  It contains
three arbitrary functions, by making one of which a power and
another a neperian exponential, the theorem of Mr.~{\sc Herschel}
may be obtained.

Mr.~{\sc Herschel's} Theorem is the following:
$$f(e^t) = f(1) + t f(1 + \Delta) o^1
      + {t^2 \over 1 \mathbin{.} 2} f(1 + \Delta) o^2
      + \hbox{\&c.}
   \eqno {\rm (A)}$$
$f(1 + \Delta)$ denoting any function which admits of being
developed according to positive integer powers of $\Delta$, and
every product of the form $\Delta^m o^n$ being interpreted, as in
Dr.~{\sc Brinkley's} notation, as a difference of a power of
zero.

The theorem which I offer as a more general one may be thus
written:
$$\phi(1 + \Delta) f \psi(o)
   = f(1 + \Delta') \phi(1 + \Delta) ( \psi(o) )^{o'};
   \eqno {\rm (B)}$$
or thus
$$F(D) f \psi(o) = f(1 + \Delta') F(D) (\psi(o) )^{o'}.
   \eqno {\rm (C)}$$

In these equations, $f$, $\phi$, $F$, $\psi$, are arbitrary
functions, such however that $f(1 + \Delta')$, $\phi(1 +
\Delta)$, $F(D)$, can be developed according to positive integer
powers of $\Delta'$ $\Delta$ $D$; and after this development
$\Delta'$ $\Delta$ are considered as marks of differencing,
referred to the variables $o'$ $o$, which vanish after the
operations, and $D$ as a mark of derivation by differentials,
referred to the variable~$o$.  And if in the form (C) we
particularise the functions $F$, $\psi$, by making $F$ a power,
and $\psi$ a neperian exponential, we deduce the following
corollary:
$$D^x f(e^o) = f(1 + \Delta') D^x e^{o'}
   = f(1 + \Delta') o'^x;$$
that is, the coefficient of
$\displaystyle {t^x \over 1 \mathbin{.} 2 \, \ldots \, x}$
in the development of $f(e^t)$ may be represented by
$f(1 + \Delta) o^x$; which is the theorem (A) of
Mr.~{\sc Herschel}.

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June~13, 1831.

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\centerline{ADDITION.}

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The two forms (B) (C) may be included in the following:
$$\nabla' f \psi(o') = f(1 + \Delta) \nabla' ( \psi(o') )^o.
   \eqno {\rm (D)}$$
To explain and prove this equation, I observe that in
{\sc Maclaurin's} series,
$$f(x)
   =  f(o) + {D f(o) \over 1} x
       + {D^2 f(o) \over 1 \mathbin{.} 2} x^2
       + \cdots
       + {D^n f(o) \over 1 \mathbin{.} 2 \, \ldots \, n} x^n
       + \cdots $$
we may put $x = (1 + \Delta) x^o$ and therefore may put the
series itself under the form
$$f(x)
   =  f(o) + {D f(o) \over 1} (1 + \Delta) x^o
       + {D^2 f(o) \over 1 \mathbin{.} 2} (1 + \Delta)^2 x^o
       + \hbox{\&c.}$$
or more concisely thus
$$f(x) = f(1 + \Delta) x^o:
   \eqno {\rm (E)}$$
which latter expression is true even when {\sc Maclaurin's}
series fails, and which gives, by considering $x$ as a
function~$\psi$ of a new variable $o'$ and performing any
operation $\nabla'$ with reference to the latter variable,
$$\nabla' f \psi(o') = \nabla' f(1 + \Delta) ( \psi(o') )^o.
   \eqno {\rm (F)}$$

If now the operation $\nabla'$ consist in any combination of
differencings and differentiatings, as in the equations (B) and
(C), and generally if we may transpose the symbols of operation
$\nabla'$ and $f(1 + \Delta)$, which happens for an infinite
variety of forms of $\nabla'$, we obtain the theorem (D).  It is
evident that this theorem may be extended to functions of several
variables.

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June~20, 1831.

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