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% School of Mathematics, Trinity College, Dublin 2, Ireland
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% Trinity College, 2000.

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\centerline{\Largebf CALCULUS OF PRINCIPAL RELATIONS}

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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, 1836,
   Part~II, pp.\ 41--44.)}

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

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\centerline{\largerm 2000}

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{\largeit\noindent
Calculus of Principal Relations.  By Professor\/}
{\largerm Sir} {\largesc W. R. Hamilton.}

\bigbreak

\centerline{[{\it Report of the Sixth Meeting of the British
Association for the Advancement of}}
\centerline{{\it Science; held at Bristol in August 1836}.}
\centerline{(John Murray, London, 1837), Part~II, pp. 41--44.]}

\bigbreak

The method of principal relations is an extension of that mode of
analysis which Sir William Hamilton has applied before to the
sciences of optics and dynamics; its nature and spirit may be
understood from the following sketch.

Let $x_1, x_2,\ldots, x_n$ be any number~$n$ of functions of any
one independent variable~$s$, with which they are connected by
any one given differential equation of the first order, but not
of the first degree,
$$0 = f(s, x_1,\ldots \, x_n, ds, dx_1,\ldots \, dx_n),
   \eqno (1)$$
and also by $n - 1$ other differential equations, of the second
order, to which the calculus of variations conducts, as
supplementary to the given equation (1), and which may be thus
denoted:
$${f'(x_1) - d f'(dx_1) \over f'(dx_1)}
   =  \cdots
   =  {f'(x_n) - d f'(dx_n) \over f'(dx_n)};
   \eqno (2)$$
Let, also, $a_1,\ldots \, a_n$ be the $n$ initial values of the
$n$ functions $x_1,\ldots \, x_n$, and let $a_1',\ldots \, a_n'$
be the initial values of their $n$ derived functions or
differential coefficients
$\displaystyle x_1' = {dx_1 \over ds},\ldots \,
  x_n' = {dx_n \over ds}$,
corresponding to any assumed initial value~$a$ of the independent
variable~$s$.  If we could integrate the system of the $n$
differential equations (1) and (2), we should thereby obtain $n$
expressions for the $n$ functions $x_1,\ldots \, x_n$, of the
forms
$$\left. \eqalign{
x_1 &= \phi_1(s, a, a_1,\ldots \, a_n, a_1',\ldots \, a_n'),\cr
\noalign{\hbox{$\cdots\cdots\cdots$}}
x_n &= \phi_n(s, a, a_1,\ldots \, a_n, a_1',\ldots \, a_n');\cr}
   \right\}
   \eqno (3)$$
and, by the help of the initial equation analogous to (1), might
then eliminate $a_1',\ldots \, a_n'$, and deduce a relation of
the form
$$0 = \psi(s, x_1,\ldots \, x_n, a, a_1,\ldots \, a_n);
   \eqno (4)$$
that is, a relation between the initial and final values of the
$n + 1$ connected variables $s, x_1,\ldots, x_n$.  Reciprocally,
the author has found that if this one relation (4) were known, it
would be possible thence to deduce expressions for the $n$ sought
integrals (3) of the system of the $n$ differential equations (1)
and (2), or for the $n$ sought relations between
$s, x_1,\ldots \, x_n$, and
$a, a_1,\ldots \, a_n, a_1',\ldots \, a_n'$,
however large the number~$n$ may be; in such a manner that all
these many relations (3) are implictly contained in the one
relation (4), which latter relation the author proposes to call
on this account the {\it principal integral relation}, or simply,
the {\sc principal relation}, of the problem.

For he has found that the $n$ following equations hold good,
$$    {f'(ds) \over \psi'(s)}
   =  {f'(dx_1) \over \psi'(x_1)}
   =  \cdots
   =  {f'(dx_n) \over \psi'(x_n)};
   \eqno (5)$$
which may be put under the forms
$$\left. \eqalign{
a_1 &= \phi_1(a, s, x_1,\ldots \, x_n, x_1',\ldots \, x_n'),\cr
\noalign{\hbox{$\cdots\cdots\cdots$}}
a_n &= \phi_n(a, s, x_1,\ldots \, x_n, x_1',\ldots \, x_n'),\cr}
   \right\}
   \eqno (6)$$
and are evidently transformations of the $n$ sought integrals
(3).  And with respect to the mode in which, without previously
effecting the integrations (3), it is possible to determine the
{\it principal relation\/} (4), or the {\it principal function\/}
which it introduces, when it is conceived to be resolved, as
follows, for the originally independent variable~$s$,
$$s = \phi(x_1,\ldots \, x_n, a_1,\ldots \, a_n),
   \eqno (7)$$
the author remarks that a partial differential equation of the
first order may be assigned, which this principal
function~$\phi$ must satisfy, and also an initial condition
adapted to remove the arbitrariness which otherwise would remain.
In fact the equations (5) may be thus written,
$${\delta d s \over \delta d x_1}
   =  {\delta s \over \delta x_1},\ldots \,
  {\delta d s \over \delta d x_n}
   =  {\delta s \over \delta x_n},
   \eqno (8)$$
in which
$${\delta ds \over \delta d x_i}
   =  - {f'(dx_i) \over f'(ds)},
   \quad\hbox{and}\quad
  {\delta s \over \delta x_i} = \phi'(x_i),
   \eqno (9)$$
and since, by (1), there subsists a known relation of the form
$$0 = F
      \left(
         s, x_1,\ldots \, x_n,
         {\delta ds \over \delta dx_1},\ldots \,
         {\delta ds \over \delta dx_n}
      \right),
   \eqno (10)$$
the following relation must also hold good,
$$0 = F
      \left(
         s, x_1,\ldots \, x_n,
         {\delta s \over \delta x_1},\ldots \,
         {\delta s \over \delta x_n}
      \right),
   \eqno (11)$$
that is, the principal function~$\phi$ must satisfy the following
partial differential equation of the first order,
$$0 = F(\phi, x_1,\ldots, x_n, \phi'(x_1),\ldots \, \phi'(x_n));
   \eqno (12)$$
it must also satisfy the following initial condition,
$$0 = \mathop{\rm lim.}\limits_{s=a}
      f(a, a_1,\ldots, a_n,
         \varphi - a, x_1 - a_1,\ldots \, x_n - a_n).
   \eqno (13)$$

Such are the most essential principles of the new method in
analysis which Sir William Hamilton has proposed to designate by
the name of the {\it Method of Principal Relations}, and of
which, perhaps, the simplest {\it type\/} is the formula
$${\delta ds \over \delta dx} = {\delta s \over \delta x},
   \eqno (14)$$
to be interpreted like the equations (8).

The simplest {\it example\/} which can be given, to illustrate
the meaning and application of these principles, is, perhaps,
that in which the differential equations are
$$0 =    \left( {dx_1 \over ds} \right)^2
       + \left( {dx_2 \over ds} \right)^2
       - 1,
   \eqno (1)'$$
and
$${d dx_1 \over dx_1} = {d dx_2 \over dx_2}.
   \eqno (2)'$$

Here, ordinary integration gives
$$x_1 = a_1 + a_1' (s - a),\quad
  x_2 = a_2 + a_2' (s - a);
   \eqno (3)'$$
and consequently conducts to the following relation, (in this
case the {\it principal\/} one,)
$$0 = (x_1 - a_1)^2 + (x_2 - a_2)^2 - (s - a)^2,
   \eqno (4)'$$
or
$$s = a + \sqrt{ (x_1 - a_1)^2 + (x_2 - a_2)^2 },
   \eqno (7)'$$
because, by (1)${}'$, we have
$$a_1'^2 + a_2'^2 = 1;$$
it enables us therefore to verify the relations (8) or (14), for
it gives
$${\delta s \over \delta x_1}
   =  {x_1 - a_1 \over s - a}
   =  {dx_1 \over ds}
   =  {\delta ds \over \delta d x_1},$$
and in like manner,
$${\delta s \over \delta x_2}
   =  {\delta ds \over \delta d x_2}.$$

Reciprocally, in this example, the following known relation,
deduced from (1)${}'$,
$$0 =    \left( {\delta d s \over \delta d x_1} \right)^2
       + \left( {\delta d s \over \delta d x_2} \right)^2
       - 1,
   \eqno (10)'$$
would have given, by the principles of the new method, this
partial differential equation of the first order,
$$0 =    \left( {\delta s \over \delta x_1} \right)^2
       + \left( {\delta s \over \delta x_2} \right)^2
       - 1,
   \eqno (11)'$$
which might have been used, in conjunction with the initial
condition
$$0 = \mathop{\rm lim.}\limits_{s=a}
         \left\{
            \left( {x_1 - a_1 \over s - a} \right)^2
          + \left( {x_2 - a_2 \over s - a} \right)^2
          - 1
         \right\},
   \eqno (13)'$$
to determine the form (7)${}'$ of the principal function~$s$; and
thence might have been deduced, by the same new principles, the
ordinary integrals (3)${}'$, under the forms
$$a_1 = x_1 + a_1' (a - s),\quad
  a_2 = x_2 + a_2' (a - s).
   \eqno (6)'$$

In so simple an instance as this, there would be no advantage in
using the new method; but in a great variety of questions,
including all those of mathematical optics, and mathematical
dynamics, (at least, as those sciences have been treated by the
author of this communication,) and in general all the problems in
which it is required to integrate those systems of ordinary
differential equations (whether of the second or of a higher
order) to which the calculus of variations conducts, the method
of principal relations assigns immediately a system of finite
expressions for the integrals of the proposed equations, an
object which can only very rarely be attained by any of the
methods known before.  It seems, for example, to be impossible,
by any other method, to express rigorously, in finite terms, the
integrals of the differential equations of motion of a system of
many points, attracting or repelling one another; which yet was
easily accomplished by a particular application of the general
principles that have been here
explained\footnote*{See {\it Philosophical Transactions\/} for
1834 and 1835; also, {\it Report\/} of Edinburgh Meeting of the
British Association.}.
The author hopes to present these principles in a still more
general form hereafter.

\bye
