% 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 THE FOCAL LENGTHS AND ABERRATIONS}

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\centerline{\Largebf OF A THIN LENS OF UNIAXAL CRYSTAL,}

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\centerline{\Largebf BOUNDED BY SURFACES WHICH ARE OF}

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\centerline{\Largebf REVOLUTION ABOUT ITS AXIS}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Philosophical Magazine, vol.~19 (1841), pp. 289--294.)}

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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 the Focal Lengths and Aberrations of a thin Lens of Uniaxal
Crystal, bounded by Surfaces which are of Revolution about its
Axis.  By\/}
{\largerm Sir} {\largesc William Rowan Hamilton,}
\hskip 0pt plus10pt minus0pt
{\largeit P.R.I.A., Member of several Scientific Societies at
Home and Abroad, Professor of Astronomy in the University of
Dublin, and Royal Astronomer of Ireland\/}\footnote*{Communicated
by the Author.}.

\bigbreak

\centerline{[{\it The London, Edinburgh and Dublin Philosophical Magazine
and Journal of Science},}
\centerline{vol.~xix (1841), pp. 289--294.]}

\bigbreak

The following short inverstigation may perhaps be not without
interest to the students of mathematical optics, as serving to
illustrate a general method, and to correct an important error
into which an eminent writer has fallen.

\bigbreak

1.
Let a ray of ordinary light, {\it in vacuo}, and in the plane of
$xz$, proceeding from or towards a given point on the axis of
$z$, be incident nearly perpendicularly on a given surface of
revolution, and there undergo extraordinary refraction at
entering a given uniaxal crystal, of when the optical axis
coincides with the axis of revolution and of $z$.  It is required
to determine the intersection of the refracted ray with the axis;
the distance of the point of incidence from the vertex, or the
semiaperture of the crystal, being given.

\bigbreak

2.
The law of this extraordinary refraction may (according to the
general methods of my `Theory of Systems of Rays,' published in
the Transactions of the Royal Irish Academy,) be thus
expressed:
$$(\sigma - \alpha') \, \delta \xi
      + (\upsilon - \gamma') \, \delta \zeta
   = 0.
   \eqno {\rm (1.)}$$

In this formula $\xi$, $\zeta$, are (in the plane of $xz$) the
rectangular coordinates of incidence, and are connected by the
equation of the meridional section of the given refracting
surface of revolution, which equation we may suppose to be (at
least nearly enough for our present purpose) developed under the
form
$$\zeta = {r \over 2} \xi^2 + {s \over 4} \zeta^4,
   \eqno {\rm (2.)}$$
$r$ being the curvature at the vertex, and $s$ being another
constant, which in the case of a spheric surface is half of the
cube of that curvature; $\alpha'$, $\gamma'$ are the cosines of
the inclinations of the incident ray to the positive semiaxes of
$xz$, so that they are connected by the relation
$$\alpha'^2 + \gamma'^2 = 1;
   \eqno {\rm (3.)}$$
and $\sigma$, $\upsilon$ are the components of normal slowness of
the extraordinary wave within the crystal, so that, if $\mu$ be
the ordinary and $\nu$ the extraordinary index, these components
are connected by the relation
$$\mu^2 \sigma^2 + \nu^2 \upsilon^2 = \mu^2 \nu^2.
   \eqno {\rm (4.)}$$

And these quantities, $\alpha'$ $\gamma'$ $\sigma$ $\upsilon$,
are such, that if $x'$~$z'$ be the coordinates of any point on
the incident ray, we have
$$(\xi - x') \, \delta \alpha'
      + (\zeta - z') \, \delta \gamma' = 0;
   \eqno {\rm (5.)}$$
and if $x$~$z$ be the coordinates of any point on the refracted
ray, we have
$$(x - \xi) \, \delta \sigma + (z - \zeta) \, \delta \upsilon
   = 0.
   \eqno {\rm (6.)}$$

\bigbreak

3.
Making then
$$T = \xi ( \sigma - \alpha') + \zeta (\upsilon - \gamma'),
   \eqno {\rm (7.)}$$
we have first the equation
$${\delta T \over \delta \xi} = 0,
   \eqno {\rm (8.)}$$
which contains the law of extraordinary refraction, and by which
$\xi$ can be eliminated from the expression of $T$, so as to
leave that quantity~$T$ expressed as a function of $\alpha'$ and
$\sigma$ only; $\zeta$ being here treated as a known function of
$\xi$, $\gamma'$ of $\alpha'$, and $\upsilon$ of $\sigma$.  And
since, by (1.) (5.) (6.),
$$\delta T = x \, \delta \sigma + x \, \delta \upsilon
      - x' \, \delta \alpha' - z' \, \delta \gamma'.
   \eqno {\rm (9.)}$$
we have, by (3.) and (4.), the following equations, for the
incident and refracted ray respectively:
$$-x' + {\alpha' \over \gamma'} z'
   = {\delta T \over \delta \alpha'};
   \eqno {\rm (10.)}$$
$$x - {\mu^2 \sigma \over \nu^2 \upsilon} z
   = {\delta T \over \delta \sigma}.
   \eqno {\rm (11.)}$$

\bigbreak

4.
The approximate equation (2.) of the section of the refracting
surface, which gives $\zeta$ as an explicit function of $\xi$, is
now to be combined with the following analogous expressions for
the functions $\gamma'$ and $\upsilon$, deduced from the
relations (3.) and (4.):
$$\gamma' = 1 - {\alpha'^2 \over 2};\quad
  \upsilon = \mu - {\mu \sigma^2 \over 2 \nu^2};
   \eqno {\rm (12.)}$$
and thus the expression (7.) for $T$ becomes, if we neglect
terms which are small of the sixth dimension with respect to
$\alpha'$, $\sigma$, $\xi$:
$$T = T^{(2)} + T^{(4)};
   \eqno {\rm (13.)}$$
$$T^{(2)} = \xi (\sigma - \alpha')
      + {\textstyle {1 \over 2}} r \xi^2 (\mu - 1);
   \eqno {\rm (14.)}$$
$$T^{(4)} = {\textstyle {1 \over 4}} s \xi^4 (\mu - 1)
      + {\textstyle {1 \over 4}} r \xi^2
         \left( {\mu \sigma^2 \over \nu^2} - \alpha'^2 \right).
   \eqno {\rm (15.)}$$
And to eliminate $\xi$, it is sufficient to employ the equation
(8.) under the approximate form
$$0 = {\delta T^{(2)} \over \delta \xi}
   = \sigma - \alpha' + r \xi (\mu - 1);
   \eqno {\rm (16.)}$$
for although the complete expression for the abscissa $\xi$ of
incidence contains terms of the third and higher dimensions with
respect to $\alpha'$ and $\sigma$, yet the introduction of these
terms of $\xi$ would only introduce terms of the sixth and higher
dimensions, in the expression for the function~$T$.

\bigbreak

5.
Retaining therefore $\xi$ as an auxiliary symbol, of which the
meaning is determined by the formula (16.), and making the
absciss{\ae} $x'$ and $x$ to vanish in the equations of the two
rays, (10.) and (11.), in order to discover the relation between
the ordinates $z'$ and $z$ of the intersections of those two rays
with the axis, we find
$$+ \alpha' z' \left( 1 + {\alpha'^2 \over 2} \right)
   = - \xi + {\delta T^{(4)} \over \delta \alpha'};
   \eqno {\rm (17.)}$$
$$- {\mu \sigma z \over \nu^2}
      \left( 1 + {\sigma^2 \over 2 \nu^2} \right)
   = + \xi + {\delta T^{(4)} \over \delta \sigma};
   \eqno {\rm (18.)}$$
and therefore, taking the reciprocals,
$$- {1 \over z'} = {\alpha' \over \xi}
      \left( 1 + {\alpha'^2 \over 2} \right)
      + {\alpha' \over \xi^2} {\delta T^{(4)} \over \delta \alpha'};
   \eqno {\rm (19.)}$$
$${\nu^2 \over \mu z} = - {\sigma \over \xi}
      \left( 1 + {\sigma^2 \over 2 \nu^2} \right)
      + {\sigma \over \xi^2} {\delta T^{(4)} \over \delta \sigma}.
   \eqno {\rm (20.)}$$
Adding these last two equations, attending to the value (16.) of
$\xi$, and observing that $T^{(4)}$, after substitution of that
value, becomes a homogeneous function of the fourth dimension of
$\alpha'$ and $\sigma$, so that
$$\alpha' {\delta T^{(4)} \over \delta \alpha'}
      + \sigma {\delta T^{(4)} \over \delta \sigma}
   = 4 T^{(4)},
   \eqno {\rm (21.)}$$
we find this relation:
$${\nu^2 \over \mu z} - {1 \over z'} - (\mu - 1) r
   = {1 \over 2\xi}
         \left( \alpha'^3 - {\sigma^2 \over \nu^2} \right)
      + {4 \over \xi^2} T^{(4)}.
   \eqno {\rm (22.)}$$
And changing, in the second member, $\alpha'$ and $\sigma$ to
their approximate values given by (19.) and (20.), namely,
$$\alpha' = - {\xi \over z'},\quad
  \sigma = - {\nu^2 \xi \over \mu z},
   \eqno {\rm (23.)}$$
we find
$${\nu^2 \over \mu z} - {1 \over z'} - (\mu - 1) r
   = {\xi^2 \over 2}
      \left\{
         {\nu^4 \over \mu^3 z^3}
       - {1 \over z'^3}
       + 2 (\mu - 1) s
       - 2r \left( {\nu^2 \over \mu z^2} - {1 \over z'^2} \right)
      \right\}.
   \eqno {\rm (24.)}$$
By suppressing the second member, we get the intersection of the
axis with a refracted ray infinitely near it, or the
extraordinary focus of the central rays; by taking account of
that member we get the longitudinal aberration.

\bigbreak

6.
As a verification, we may consider the case of an ordinary
refraction at a spheric surface as being included in the
foregoing, and the formula for that case must result from the
equation (24.), by making therein
$$\nu = \mu,\quad s = {\textstyle{1 \over 2}} r^3.
   \eqno {\rm (25.)}$$
Accordingly these values give
$${\mu \over z} - {1 \over z'} - (\mu - 1) r
   =  {\xi^2 \over 2}
      \left\{
         \mu \left( {1 \over z^3} - {2r \over z^2} + r^3 \right)
       - \left( {1 \over z'^3} - {2r \over z'^2} + r^3 \right)
      \right\};$$
in the second member of which we have
$$\mu \left( {1 \over z^3} - {2r \over z^2} + r^3 \right)
   =  \mu \left( {1 \over z} - r \right)
         \left( {1 \over z^2} - {r \over z} - r^2 \right),$$
$${1 \over z'^3} - {2r \over z'^2} + r^3
   = \left( {1 \over z'} - r \right)
         \left( {1 \over z'^2} - {r \over z'} - r^2 \right),$$
and may write
$$\mu \left( {1 \over z} - r \right)
   = {1 \over z'} - r
   = {\mu \over \mu - 1} \left( {1 \over z'} - {1 \over z} \right);$$
the formula for an ordinary spheric refraction is therefore thus
found to be
$${\mu \over z} - {1 \over z'} - (\mu - 1) r
   = {\mu \over \mu - 1}
         \left( r - {1 \over z} - {1 \over z'} \right)
         \left( {1 \over z} - {1 \over z'} \right)^2
         {\xi^2 \over 2},
   \eqno {\rm (26.)}$$
in which it may be remarked that
$\displaystyle \left( {1 \over z} - {1 \over z'} \right)^2 \xi^2$
is the square of the angular deviation, and which is easily seen
to agree with known results.

\bigbreak

7.
Returning to the crystal, let it be bounded by a second surface
of revolution, infinitely near to the former, and about the same
axis; and let the light emerge at this second surface into a
vacuum again.  The equation of the second surface being
$$\zeta' = {\textstyle {1 \over 2}} r' \xi'^2
      + {\textstyle {1 \over 4}} s' \xi'^4,$$
and the ordinate of the intersection of the emergent ray with the
axis being $z''$, the formula (24.) will apply to this new case
by merely changing $r$, $s$, $z'$, $z$ respectively to
$-r'$, $-s'$, $-z''$, $-z$, without changing $\mu$, $\nu$, $\xi$;
and we have
$$- {\nu^2 \over \mu z} + {1 \over z''} + (\mu - 1) r'
   = {\xi^2 \over 2}
      \left\{
       - {\nu^4 \over \mu^3 z^3}
       + {1 \over z''^3}
       - 2 (\mu - 1) s'
       + 2r' \left( {\nu^2 \over \mu z^2} - {1 \over z''^2} \right)
      \right\}.
   \eqno {\rm (28.)}$$
And, adding the two equations (24.) and (28.), we find
$${1 \over z''} - {1 \over z'} - (\mu - 1)(r - r')
   =  {\xi^2 \over 2}
      \left\{
         {1 \over z''^3}
       - {1 \over z'^3}
       + 2 (\mu - 1) (s - s')
       + {2 (r' - r) \nu^2 \over \mu z^2}
       - {2 r' \over z''^2}
       + {2r \over z'^2}
      \right\};
   \eqno {\rm (29.)}$$
a formula for the focal lengths and aberrations of a lens of
uniaxal crystal bounded by any two infinitely near surfaces,
which have the optical axis of the crystal for their common axis
of revolution: all cases of cusps or other singular curvatures at
the common vertex being set aside.

\bigbreak

8.
We shall content ourselves at present with drawing two
conclusions from this formula.  First, that because the
extraordinary index~$\nu$ disappears from the part unaffected
with the small factor $\xi^2$, the central focus of the ordinary
rays, after emerging from the thin crystalline lens, coincides
with the central focus of the ordinary rays which emerge from the
same lens, wherever in the axis the focus of the incident rays
may be: whereas Malus,\footnote*{Perhaps it may not be improper
to mention here, as I am not aware that the correction of Malus's
result, respecting the extraordinary focal length of a lens of
uniaxal crystal, has been hitherto published by any other person,
that this correction occurred to me many years ago, in treating
the question by my own methods, which made it scarcely possible
to fall into the same error.  The discrepance between my
conclusion and his was, however, so great, that I was at first
perplexed to account for it, until I traced the error of sign in
his calculations, to which allusion has been made above.}
misled by an error of sign in a radical which he employed for
expressing the law of extraordinary refraction in the case of an
uniaxal crystal, thought that these foci, ordinary and
extraordinary, might differ widely from each other.  And second,
that on account of the presence of $\nu$ in the term
$${(r' - r) \nu^2 \xi^2 \over \mu z^2},
   \eqno {\rm (30.)}$$
the ordinary and extraordinary aberrations cannot be exactly the
same (the power
$$(\mu - 1) (r - r')$$
of the lens being supposed to be different from $0$), unless the
$z$ in the denominator of this term become infinite; that is,
unless, wherever the focus of the incident rays may be, the lens
is so placed as to allow the rays within the crystal to be
exactly parallel to the axis.  If then a lens of this sort be
used for the object-glass of a telescope, it seems to be
desirable that its anterior surface, or that on which the
parallel rays fall, should be plane, and that the correction of
the aberration of figure should be effected entirely by another
lens, composed of an uncrystallized material.

\nobreak\bigskip

\line{\hfill W.~R.~H.}

\nobreak\bigskip

Observatory of Trinity College, Dublin,

\qquad September 14, 1841.

\bye

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