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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
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% Trinity College, 2000.
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\centerline{\Largebf RESEARCHES RESPECTING VIBRATION,}
\vskip12pt
\centerline{\Largebf CONNECTED WITH THE THEORY OF LIGHT}
\vskip24pt
\centerline{\Largebf By}
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\centerline{\Largebf William Rowan Hamilton}
\vskip24pt
\centerline{\largerm (Proceedings of the Royal Irish Academy,
1 (1841), pp.\ 341--349.)}
\vskip36pt
\vfill
\centerline{\largerm Edited by David R. Wilkins}
\vskip 12pt
\centerline{\largerm 2000}
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\null\vskip36pt
\centerline{\largeit Researches respecting Vibration, connected
with the Theory of Light.}
\vskip 6pt
\centerline{{\largeit By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}
\bigskip
\centerline{Communicated June~24, 1839.}
\bigskip
\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~1 (1841), pp.\ 341--349.]}
\bigskip
The President concluded his account of his First Series of
Researches respecting Vibration, connected with the Theory of
Light. The following is an outline of one of the investigations
which are contained in the Series referred to.
It is proposed to integrate the system of equations in mixed
differences,
$${\sc d}_t^2 \, \delta x_{g,h}
= \Sigma_{\Delta g} \delta
( {\sc r} \mathbin{.} \Delta_g x_{g,h} );
\eqno (1)$$
in which $h$ is any integer number from $1$ to $n$ inclusive;
$x_{g,h}$ is independent of $t$, but $\delta x_{g,h}$ is a
function of $t$ and of $x_{g,1},\ldots \, x_{g,n}$, the form of
which function it is the object of the problem to discover;
$${\sc r}
= m_{g + \Delta g} \phi
\left(
{\textstyle {1 \over 2}}
\Sigma_{(h) \,}{}_1^n ( \Delta_g x_{g,h} )^2
\right),
\eqno (2)$$
$\phi$ being any real function of the semi-sum which follows it,
and $m$ being any other real function of the index
$g + \Delta g$; while $g$ and $g + \Delta g$ represent any
integer numbers from negative to positive infinity. The
equations to be integrated may also be thus written:
$$\xi_{g,h,t}''
= \Sigma_{\Delta g}
\left(
{\sc r} \, \Delta_g \xi_{g,h,t}
+ {\sc r}' \, \Delta_g x_{g,h}
\Sigma_{(h) \,}{}_1^n \Delta_g x_{g,h}
\Delta_g \xi_{g,h,t}
\right),
\eqno (1)'$$
in which
$${\sc r}'
= m_{g + \Delta g} \,
\phi'
\left(
{\textstyle {1 \over 2}}
\Sigma_{(h) \,}{}_1^n ( \Delta_g x_{g,h} )^2
\right),
\eqno (2)'$$
the functions to be found by integration are now those of the
form $\xi_{g,h,t}$, considered as depending on $t$ and on
$x_{g,1},\ldots \, x_{g,n}$; their initial values, and initial
rates of increase (relatively to $t$), namely $\xi_{g,h,0}$ and
$\xi'_{g,h,0}$, are regarded as arbitrary but given and real
functions of $x_{g,1},\ldots \, x_{g,n}$; it is also supposed, in
order to simplify the question, that all the sums of the forms
$$\Sigma_{\Delta g} {\sc r} ( \Delta_g x_{g,1} )^{\alpha_1}
\, \ldots \, ( \Delta_g x_{g,n} )^{\alpha_n},\quad
\Sigma_{\Delta g} {\sc r}' ( \Delta_g x_{g,1} )^{\alpha_1}
\, \ldots \, ( \Delta_g x_{g,n} )^{\alpha_n},
\eqno (3)$$
are independent of $g$, and are $= 0$ when any one of the
exponents $\alpha_1, \ldots \, \alpha_n$ is an odd number. These
equations are analogous to, and include, those which M.~Cauchy
has considered on his memoir on the Dispersion of Light, and may
be integrated by a similar analysis.
A particular integral system may in the first case be found by
assuming
$$\xi_{g,h,t}
= {\sc x}_r {\sc a}_{h,r}
\cos ( \epsilon_r + s_r t
- \Sigma_{(i) \,}{}_1^n u_i x_{g,i} );
\eqno (4)$$
$$\Sigma_{(h) \,}{}_1^n {\sc a}_{h,r}^2 = 1;
\eqno (5)$$
$$s_r^2 {\sc a}_{h,r}
= \Sigma_{(i) \,}{}_1^n {\sc h}_{h,i} {\sc a}_{i,r};
\eqno (6)$$
$${\sc h}_{h,h}
= \Sigma_{\Delta g}
\left( {\sc r} + {\sc r}' ( \Delta_g x_{g,h} )^2 \right)
\mathop{\rm vers}
\left( \Sigma_{(i) \,}{}_1^n u_i \, \Delta_g x_{g,i} \right),
\eqno (7)$$
$${\sc h}_{h,i}
= \Sigma_{\Delta g} {\sc r}' \,
\Delta_g x_{x,h} \, \Delta_g x_{g,i}
\mathop{\rm vers}
\left( \Sigma_{(i) \,}{}_1^n u_i \, \Delta_g x_{g,i} \right);
\eqno (7)'$$
the index~$r$ being any integer from $1$ to $n$, and being
introduced in order to distinguish among themselves the $n$
different (and in general real) systems of values of $s^2$, and
of the $n - 1$ ratios of
${\sc a}_1,\ldots \, {\sc a}_h,\ldots \, {\sc a}_n$,
which are obtained by resolving the system of the $n$ equations
of the form
$$s^2 {\sc a}_h = \Sigma_{(i) \,}{}_1^n {\sc h}_{h,i} {\sc a}_i,
\eqno (6)'$$
in which, by (7)${}'$,
$${\sc h}_{i,h} = {\sc h}_{h,i}.
\eqno (7)''$$
It is important to observe, that by the form of these
equations~(6)${}'$, (which occur in may researches,) we have the
relation
$$\Sigma_{(h) \,}{}_1^n {\sc a}_{h,q} {\sc a}_{h,r} = 0,
\eqno (5)'$$
if $q$ be different from $r$, and that, by (5) and (5)${}'$, we
have also the relations
$$\Sigma_{(r) \,}{}_1^n {\sc a}_{h,r}^2 = 1,
\eqno (8)$$
$$\Sigma_{(r) \,}{}_1^n {\sc a}_{h,r} {\sc a}_{i,r} = 0.
\eqno (8)'$$
In the particular integral~(4), we may consider
$u_1,\ldots \, u_n$ as arbitrary parameters, of which ${\sc x}_r$
and $\epsilon_r$ are real and arbitrary, while $s_r^2$ and
${\sc a}_{h,r}$ are real and determined functions; and hence, by
summations relatively to the index~$r$, and integrations
relatively to the parameters~$u_i$, employing also the relations
(5) (5)${}'$ (8) (8)${}'$, and Fourier's theorem extended to
several variables, we deduced this general integral, applying to
all arbitrary real values of the initial data:
$$\xi_{g,h,t}
= \left(
\Pi_{(i) \,}{}_1^n \int_{-\infty}^\infty du_i
\right)
\left(
{\sc e}_{h,t} \cos + {\sc f}_{h,t} \sin
\right)
\Sigma_{(i) \,}{}_1^n u_x x_{g,i};
\eqno (9)$$
in which
$$\Pi_{(i) \,}{}_1^n \int_{-\infty}^\infty du_i
= \int_{-\infty}^\infty du_1 \,
\int_{-\infty}^\infty du_2 \, \ldots \,
\int_{-\infty}^\infty du_n;
\eqno (10)$$
$$\left. \eqalign{
{\sc e}_{h,t}
= \Sigma_{(r) \,}{}_1^n {\sc a}_{h,r}
\left(
{\sc y}_r \cos t s_r + {\sc y}_r' s_r^{-1} \sin t s_r
\right),\cr
{\sc f}_{h,t}
= \Sigma_{(r) \,}{}_1^n {\sc a}_{h,r}
\left(
{\sc z}_r \cos t s_r + {\sc z}_r' s_r^{-1} \sin t s_r
\right);\cr}
\right\}
\eqno (11)$$
$$\left.
\eqalign{
{\sc y}_r
&= \Sigma_{(h) \,}{}_1^n {\sc a}_{h,r} {\sc e}_{h,0},\cr
{\sc z}_r
&= \Sigma_{(h) \,}{}_1^n {\sc a}_{h,r} {\sc f}_{h,0},\cr}
\quad
\eqalign{
{\sc y}_r'
&= \Sigma_{(h) \,}{}_1^n {\sc a}_{h,r} {\sc e}_{h,0}',\cr
{\sc z}_r'
&= \Sigma_{(h) \,}{}_1^n {\sc a}_{h,r} {\sc f}_{h,0}';\cr}
\right\}
\eqno (12)$$
$$\left. \eqalign{
{\sc e}_{h,0}
= \left( {1 \over 2 \pi} \right)^n
\left(
\Pi_{(i) \,}{}_1^n \int_{-\infty}^\infty dx_{g,i}
\right)
\xi_{g,h,0} \cos
\left( \Sigma_{(i) \,}{}_1^n u_i x_{g,i} \right),\cr
{\sc e}_{h,0}'
= \left( {1 \over 2 \pi} \right)^n
\left(
\Pi_{(i) \,}{}_1^n \int_{-\infty}^\infty dx_{g,i}
\right)
\xi_{g,h,0}' \cos
\left( \Sigma_{(i) \,}{}_1^n u_i x_{g,i} \right),\cr
{\sc f}_{h,0}
= \left( {1 \over 2 \pi} \right)^n
\left(
\Pi_{(i) \,}{}_1^n \int_{-\infty}^\infty dx_{g,i}
\right)
\xi_{g,h,0} \sin
\left( \Sigma_{(i) \,}{}_1^n u_i x_{g,i} \right),\cr
{\sc f}_{h,0}'
= \left( {1 \over 2 \pi} \right)^n
\left(
\Pi_{(i) \,}{}_1^n \int_{-\infty}^\infty dx_{g,i}
\right)
\xi_{g,h,0}' \sin
\left( \Sigma_{(i) \,}{}_1^n u_i x_{g,i} \right).\cr}
\right\}
\eqno (13)$$
This general solution involves multiple integrals, of the
order $2n$; but many particular suppositions, respecting the
initial data, conduct to simpler expressions, among which the
following appear worthy of remark.
Suppose that having assumed some particular set
$u_1^\backprime, \ldots \, u_n^\backprime$
of values of the $n$ arbitrary quantities $u_1, \ldots \, u_n$,
we deduce a corresponding set of coefficients
${\sc h}_{h,h}^\backprime$,~${\sc h}_{h,i}^\backprime$,
by the formul{\ae} (7) and (7)${}'$, and represent by
$s_1^{\backprime 2}$ and by
${\sc a}_{1,1}^\backprime, \ldots \,
{\sc a}_{h,1}^\backprime, \ldots \,
{\sc a}_{n,1}^\backprime$
some one corresponding system of quantities which satisfy the
equations
$$\Sigma_{(h) \,}{}_1^n {\sc a}_{h,1}^{\backprime 2}
= 1,
\eqno (5)^\backprime$$
$$s_1^{\backprime 2} {\sc a}_{h,1}^\backprime
= \Sigma_{(i) \,}{}_1^n
{\sc h}_{h,i}^\backprime {\sc a}_{i,1}^\backprime;
\eqno (6)^\backprime$$
we shall then have, as a particular integral system, that which
is thus denoted:
$$\xi_{g,h,t}
= {\sc x}_1^\backprime {\sc a}_{h,1}^\backprime
\cos ( \epsilon_1^\backprime + s_1^\backprime t
- \Sigma_{(i) \,}{}_1^n u_i^\backprime x_{g,i} );
\eqno (4)^\backprime$$
${\sc x}_1^\backprime$ and $\epsilon_1^\backprime$ denoting here
any arbitary real quantities. If therefore we suppose that the
initial data $\xi_{g,h,0}$ and $\xi_{g,h,0}'$ are all such as to
agree with this particular solution, that is, if we have for all
values of $g$ and $h$,
$$\xi_{g,h,0}
= {\sc x}_1^\backprime {\sc a}_{h,1}^\backprime
\cos ( \epsilon_1^\backprime
- \Sigma_{(i) \,}{}_1^n u_i^\backprime x_{g,i} ),
\eqno (14)$$
$$\xi_{g,h,0}'
= - s_1^\backprime
{\sc x}_1^\backprime {\sc a}_{h,1}^\backprime
\sin ( \epsilon_1^\backprime
- \Sigma_{(i) \,}{}_1^n u_i^\backprime x_{g,i} ),
\eqno (14)'$$
we see, {\it \`{a} priori}, that the multiple integrations ought
to admit of being all effected in finite terms, so as to reduce
the general expression~(9) to the particular
form~(4)${}^\backprime$; an expectation which the calculation,
accordindingly, {\it \`{a} posteriori}, proves to be correct. An
analogous but less simple reduction takes place, when we suppose
that the initial equations (14) and (14)${}'$ hold good, after
their second members have been multiplied by a discontinuous
factor such as
$${\textstyle {1 \over 2}}
\left(
1 - {2 \over \pi} \int_0^\infty
{\displaystyle \sin
\left(
k \Sigma_{(i) \,}{}_1^n u_i^\backprime x_{g,i}
\right)
\over k} \, dk
\right),
\eqno (15)$$
which is $= 1$, or $= {1 \over 2}$, or $= 0$, according as the
sum~$\displaystyle \Sigma_{(i) \,}{}_1^n u_i^\backprime x_{g,i}$
is $< 0$, $= 0$, or $> 0$. It is found that, in this case, the
$2n$ successive integrations (required for the general solution)
can in part be completely effected, and in the remaining part be
reduced to the calculation of a simple definite integral; in such
a manner that the expression~(9) now reduces itself rigorously to
the following:
$$\xi_{g,h,t}
= {\textstyle {1 \over 2}}
{\sc x}_1^\backprime {\sc a}_{h,1}^\backprime
\cos ( \epsilon_1^\backprime + t s_1^\backprime
- \Sigma_{(i) \,}{}_1^n u_i^\backprime x_{g,i} );
+ {1 \over \pi} {\sc x}_1^\backprime \int_0^\infty
{dk \over k^2 - k^{\backprime 2}}
( {\sc l}_t \cos \epsilon_1^\backprime
+ {\sc m}_t \sin \epsilon_1^\backprime );
\eqno (16)$$
in which
$$\left. \eqalign{
{\sc l}_t
&= {\sc p}_t k^\backprime \cos kx - {\sc q}_t k \sin kx,\cr
{\sc m}_t
&= {\sc p}_t k \sin kx + {\sc q}_t k^\backprime \cos kx,\cr}
\right\}
\eqno (17)$$
$$\left. \eqalign{
{\sc p}_t
&= s_1^\backprime \Sigma_{(r) \,}{}_1^n
( {\sc a}_{h,r} s_r^{-1} \sin ts_r \mathbin{.}
\Sigma_{(h) \,}{}_1^n
{\sc a}_{h,r}
{\sc a}_{h,1}^\backprime ),\cr
{\sc q}_t
&= \Sigma_{(r) \,}{}_1^n
( {\sc a}_{h,r} \cos ts_r \mathbin{.}
\Sigma_{(h) \,}{}_1^n
{\sc a}_{h,r}
{\sc a}_{h,1}^\backprime ),\cr}
\right\}
\eqno (18)$$
$$x = \Sigma_{(i) \,}{}_1^n a_i^\backprime x_{g,i},
\eqno (19)$$
$$k a_i^\backprime = u_i,\quad
k^\backprime a_i^\backprime = u_i^\backprime,\quad
k^{\backprime 2} = \Sigma_{(i) \,}{}_1^n u_i^{\backprime 2},
\eqno (20)$$
and $s_r$, ${\sc a}_{h,r}$ are the same functions as before of
$u_1,\ldots \, u_n$.
A remarkable conclusion may now be drawn from these expressions,
by supposing that all the quantities of the form $s_r^2$ are not
only real but positive, so that the functions $\cos t s_r$ and
$\sin t s_r$ are periodic. For in this case the functions
$\cos (t s_r \pm k x)$ and $\sin (t s_r \pm k x)$,
will vary rapidly, and pass often through all their fluctuations
of value, between the limits $1$ and $-1$, while $k$ and the
other functions of that variable remain almost unchanged,
provided that
$\displaystyle t {ds_r \over dk} \pm x$
is large, and that the denominator $k^2 - k^{\backprime 2}$ is
not extremely small. We may therefore in general confine
ourselves to the consideration of small values of this
denominator; and consequently may put it under the form
$2 k^\backprime (k - k^\backprime)$, making $k = k^\backprime$ in
the numerator, except under the periodical signs, and integrating
relatively to $k$ between any two limits which include
$k^\backprime$, for example between $- \infty$ and $+ \infty$.
And because
$$\Sigma_{(h) \,}{}_1^n
{\sc a}_{h,r}^\backprime {\sc a}_{h,1}^\backprime
= 1, \hbox{ or } = 0,$$
according as $r = 1$ or $> 1$, we may make
$${\sc p}_t = {\sc a}_{h,1}^\backprime \sin t s_1,\quad
{\sc q}_t = {\sc a}_{h,1}^\backprime \cos t s_1,$$
$${\sc l}_t
= k^\backprime {\sc a}_{h,1}^\backprime
\sin (t s_1 - k x),\quad
{\sc m}_t
= k^\backprime {\sc a}_{h,1}^\backprime
\cos (t s_1 - k x)$$
and
$$\xi_{g,h,t}
= {\textstyle {1 \over 2}}
{\sc x}_1^\backprime {\sc a}_{h,1}^\backprime
\left\{
\cos ( \epsilon_1^\backprime + t s_1^\backprime
- k^\backprime x)
+ \int_{-\infty}^\infty dk
{\sin ( \epsilon_1^\backprime + t s_1 - kx )
\over \pi (k - k^\backprime)}
\right\},
\eqno (21)$$
that is, nearly, if $x$ be considerably different from
$\displaystyle t {d s_1^\backprime \over d k^\backprime}$,
$$\xi_{g,h,t}
= {\textstyle {1 \over 2}}
{\sc x}_1^\backprime {\sc a}_{h,1}^\backprime
\cos ( \epsilon_1^\backprime + t s_1^\backprime
- k^\backprime x)
\left\{
1 + \int_{-\infty}^\infty {dk \over \pi k}
\left(
\left(
t {ds_1^\backprime \over dk^\backprime} - x
\right)
k
\right)
\right\}.
\eqno (21)'$$
We have therefore the approximate expressions:
$$\xi_{g,h,t}
= {\sc x}_1^\backprime {\sc a}_{h,1}^\backprime
\cos ( \epsilon_1^\backprime + t s_1^\backprime
- k^\backprime x),
\quad\hbox{if }
x < t {d s_1^\backprime \over dk^\backprime};
\eqno (22)$$
and
$$\xi_{g,h,t} = 0,
\quad\hbox{if }
x > t {d s_1^\backprime \over dk^\backprime};
\eqno (22)'$$
we have also nearly, in general,
$$\xi_{g,h,t}
= {\textstyle {1 \over 2}}
{\sc x}_1^\backprime {\sc a}_{h,1}^\backprime
\cos ( \epsilon_1^\backprime + t s_1^\backprime
- k^\backprime x),
\quad\hbox{if }
x = t {d s_1^\backprime \over dk^\backprime};
\eqno (22)''$$
but the discussion of the case when $x$ is nearly
$\displaystyle = t {d s_1^\backprime \over dk^\backprime}$
is too long to be cited here. The formula (22) for
$\xi_{g,h,t}$ coincides with the particular
integral~(4)${}^\backprime$; and the condition which it involves
with respect to $x$, expresses the law according to which this
particular integral comes to be (nearly) true for greater and
greater positive values of $x$ and $t$ (if
$\displaystyle {d s_1^\backprime \over dk^\backprime} > 0$,)
after having been true only for negative valus of $x$ when $t$
was $= 0$.
In the particular case $n = 3$, the foregoing formul{\ae} have an
immediate dynamical application, and correspond to the
propagation of vibratory motion through a system of mutually
attracting or repelling particles; and they conduct to this
remarkable result, that the velocity with which such vibration
spreads into those portions of the vibratory medium which were
previously undisturbed, is in general different from the velocity
of a passage of a given phase from one particle to another within
that portion of the medium which is already fully agitated; since
we have
$$\hbox{\it velocity of transmission of phase}
= {s \over k},
\eqno ({\rm A})$$
but
$$\hbox{\it velocity of propagation of vibratory motion}
= {ds \over dk},
\eqno ({\rm B})$$
if the rectangular components of the vibrations themselves be
represented by the formul{\ae}
$${\sc x} {\sc a}_1 \cos (\epsilon + st - kx),\quad
{\sc x} {\sc a}_2 \cos (\epsilon + st - kx),\quad
{\sc x} {\sc a}_3 \cos (\epsilon + st - kx),
\eqno ({\rm C})$$
$t$ being the time, and $x$ the perpendicular distance of the
vibrating point from some determined plane.
This result, which is believed to be new, includes as a
particular case that which was stated in a former communication
to the Academy, on the 11th of February last, (Proceedings,
No.~15, page~269,) respecting the propagation of transversal
vibration along a row of equal and equidistant particles, of
which each attracts the two that are immediately before and
behind it; in which particular question $s$ was
$\displaystyle = 2a \sin {k \over 2}$,
and the velocity of propagation of vibration was
$\displaystyle = a \cos {k \over 2}$.
Applied to the theory of light, it appears to show that if the
phase of vibration in an ordinary dispersive medium be
represented for some one colour by
$$\epsilon + {2 \pi \over \lambda}
\left( {t \over \mu} - x \right),
\eqno ({\rm C})'$$
so that $\lambda$ is the length of an undulation for that colour
and for that medium, and if it be permitted to represent
dispersion by developing the velocity
$\displaystyle {1 \over \mu}$ of the transmission of phase in a
series of the form
$${1 \over \mu}
= {\sc m}_0
- {\sc m}_1 \left( {2 \pi \over \lambda} \right)^2
+ {\sc m}_2 \left( {2 \pi \over \lambda} \right)^4
- \hbox{\&c.},
\eqno ({\rm A})'$$
then the {\it velocity wherewith light of this colour conquers
darkness}, in this dispersive medium, by the {\it spreading of
vibration into parts which were not vibrating before, is somewhat
less than\/} $\displaystyle {1 \over \mu}$, being represented by
this other series
$${\sc m}_0
- 3 {\sc m}_1 \left( {2 \pi \over \lambda} \right)^2
+ 5 {\sc m}_2 \left( {2 \pi \over \lambda} \right)^4
- \hbox{\&c.}
\eqno ({\rm B})'$$
For other details of this inquiry it is necessary to refer to the
memoir itself, which will be pubished in the Transactions of the
Academy, and will be found to contain many other investigations
respecting vibratory systems, with applications to the theory of
light.
\bye