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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
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% Trinity College, 2000.

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\centerline{\Largebf ON A NEW SYSTEM OF TWO GENERAL}

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\centerline{\Largebf EQUATIONS OF CURVATURE}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Proceedings of the Royal Irish Academy, 9 (1867),
pp.\ 302--305.)}

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

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\centerline{\largerm 2000}

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\centerline{\Largebf NOTE ON THE TEXT}

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This edition is based on the original text published posthomously
in volume~9 of the {\it Proceedings of the Royal Irish Academy}.

The following obvious typographical errors have been corrected:---

\smallbreak

\item{}
before equation~(a), a full stop (period) has been changed to a
colon;

\smallbreak

\item{}
in equation~(k), `$(Z - x)$' has been corrected to `$(Z - z)$';

\smallbreak

\item{}
in equation~(o), `$C = e E'' - e' E$' has been corrected to
`$C = e E' - e' E$';

\smallbreak

\item{}
equation~(q) in the original text was given as
$$(e R^{-1} - e K^{-1}) (e'' R^{-1} - e'' K^{-1})
            = (e' R^{-1} - e K^{-1})^2;$$

\smallbreak

\item{}
in equation~(r), `$ = 0$' has been appended to the 
polynomial `$R^{-2} - F R^{-1} + G$';


\smallbreak

\item{}
in equation~(w), the equality sign $=$ has been added.

\bigbreak\bigskip

\line{\hfil David R. Wilkins}

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\line{\hfil Dublin, March 2000}

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\centerline{\sc On a New System of Two General Equations of Curvature,}

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\begingroup \advance\leftskip by 24pt
   \rightskip=\leftskip \parindent=-\leftskip
Including as easy consequences a new form of the Joint
Differential Equation of the Two Lines of Curvature, with a new
Proof of their General Rectangularity; and also a new Quadratic
for the Joint Determination of the Two Radii of Curvature: all
deduced by Gauss's Second Method, for discussing generally the
Properties of a Surface; and the latter being verified by a
Comparison of Expressions, for what is called by him the Measure
of Curvature.
\par\endgroup

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\centerline{\largerm Sir William Rowan Hamilton}

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\centerline{Communicated June 26, 1865.}

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\centerline{[{\it Proceedings of the Royal Irish Academy},
   vol.~ix (1867), pp.~302--305.]}

\bigbreak

1.
Notwithstanding the great beauty and importance of the
investigations of the illustrious {\sc Gauss}, contained in his
{\it Disquisitiones Generales circ\`{a} Superficies Curvas}, a
Memoir which was communicated to the {\it Royal Society of
G\"{o}ttingen\/} in October, 1827, and was printed in Tom.~vi. of
the {\it Commentationes Recentiores}, but of which a Latin
reprint has been since very judiciously given, near the beginning
of the Second Part (Deuxi\`{e}me Partie, Paris, 1850) of
{\sc Liouville's} {\it Edition\/}\footnote*{The foregoing dates,
or references, are taken from a note to page~505 of that
Edition.} {\it of\/} {\sc Monge}, it still appears that there is
room for some not useless Additions to the Theory of
{\it Lines\/} and {\it Radii of Curvature}, for {\it any given
Curved Surface}, when treated by what Gauss calls the {\it Second
Method\/} of discussing the {\it General Properties of Surfaces}.
In fact, the {\it Method\/} here alluded to, and which consists
chiefly in treating the {\it three\/} co-ordinates of the
{\it surface\/} as being so many {\it functions\/} of {\it two\/}
independent variables, does not seem to have been used {\it at
all\/} by Gauss, for the determination of the {\it Directions of
the Lines of Curvature\/}; and as regards the {\it Radii of
Curvature\/} of the {\it Normal Sections\/} which {\it touch\/}
these {\it Lines\/} of Curvature, he appears to have employed the
{\it Method, only for the Product}, and {\it not also\/} for the
{\it Sum}, of the {\it Reciprocals}, of those {\it Two Radii}.

\bigbreak

2.
As regards the {\it notations}, let $x$,~$y$,~$z$ be the
rectangular co-ordinates of a point~${\sc p}$ upon a surface
$(S)$, considered as {\it three\/} functions of {\it two\/}
independent variables, $t$ and $u$; and let the 15 partial
derivatives, or 15 partial differential coefficients, of
$x$,~$y$,~$z$ taken with respect to $t$ and $u$, be given by the
nine differential expressions:
$$\left\{
\eqalign{
dx &= x' \, dt + x_\prime \, du;\cr
dy &= y' \, dt + y_\prime \, du;\cr
dz &= z' \, dt + z_\prime \, du;\cr} \quad
\eqalign{
dx' &= x'' \, dt + x_\prime' \, du;\cr
dy' &= y'' \, dt + y_\prime' \, du;\cr
dz' &= z'' \, dt + z_\prime' \, du;\cr} \quad
\eqalign{
dx_\prime &= x_\prime' \, dt + x_{\prime\prime} \, du;\cr
dy_\prime &= y_\prime' \, dt + y_{\prime\prime} \, du;\cr
dz_\prime &= z_\prime' \, dt + z_{\prime\prime} \, du.\cr}
   \right.
   \leqno {\rm (a)} \, . \, .$$

\bigbreak

3.
Writing also, for abridgment,
$$e   = x'^2 + y'^2 + z'^2;\quad
  e'  = x' x_\prime + y' y_\prime + z' z_\prime;\quad
  e'' = x_\prime^2 + y_\prime^2 + z_\prime^2
   \leqno {\rm (b)} \, . \, .$$
we shall have
$$e e'' - e'^2 = K^2,
   \leqno {\rm (c)} \, . \, .$$
if
$$K^2  = L^2 + M^2 + N^2,
   \leqno {\rm (d)} \, . \, .$$
and
$$L = y' z_\prime - z' y_\prime;\quad
  M = z' x_\prime - x' z_\prime;\quad
  N = x' y_\prime - y' x_\prime;
   \leqno {\rm (e)} \, . \, .$$
so that
$$L x' + M y' + N z' = 0,\quad
  L x_\prime + M y_\prime + N z_\prime = 0.
   \leqno {\rm (f)} \, . \, .$$

Hence $K^{-1} L$, $K^{-1} M$, $K^{-1} N$ are the {\it
direction-cosines\/} of the {\it normal\/} to the surface $(S)$
at ${\sc p}$; and if $x$,~$y$,~$z$ be the co-ordinates of any
{\it other\/} point~${\sc q}$ of the same normal,we shall have
the equations
$$K (X - x) = LR;\quad
  K (Y - y) = MR;\quad
  K (Z - z) = NR;
   \leqno {\rm (g)} \, . \, .$$
with
$$R^2 = (X - x)^2 + (Y - y)^2 + (Z - z)^2;
   \leqno {\rm (h)} \, . \, .$$
where $R$ denotes the normal line ${\sc p} {\sc q}$, considered
as changing sign in passing through zero.

\bigbreak

4.
The following, however, is for some purposes a more convenient
{\it form\/} (comp.~(f)) of the {\it Equations of the Normal\/};
$$(X - x) x' + (Y - y) y' + (Z - z) z' = 0;
   \leqno {\rm (i)} \, . \, .$$
$$(X - x) x_\prime + (Y - y) y_\prime + (Z - z) z_\prime = 0.
   \leqno {\rm (j)} \, . \, .$$

Differentiating these, as if $X$, $Y$, $Z$ were constant, that
is, treating the point~${\sc q}$ as an intersection of two
consecutive normals, we obtain these two other equations,
$$\left\{ \eqalign{
(X - x) \, dx' + (Y - y) \, dy' + (Z - z) \, dz'
   &= x' \, dx + y' \, dy + z' \, dz;\cr
(X - x) \, dx_\prime + (Y - y) \, dy_\prime + (Z - z) \, dz_\prime
   &= x_\prime \, dx + y_\prime \, dy + z_\prime dz.\cr}
   \right.
   \leqno {\rm (k)} \, . \, .$$

If, then, we write, for abridgment,
$$\left\{ \eqalign{
v     &= du : dt;\cr
E'    &= L x_\prime' + M y_\prime' + N z_\prime';\cr}
   \quad \eqalign{
E     &= L x'' + M y'' + N z'';\cr
E''   &= L x_{\prime\prime} + M y_{\prime\prime} + N z_{\prime\prime};\cr}
   \right.
   \leqno {\rm (l)} \, . \, .$$
we shall have, by (a) (b) (g), the two important formul{\ae}:
$$R (E + E' v) = K (e + e' v);\quad
  R (E' + E'' v) = K (e' + e'' v);
   \leqno {\rm (m)} \, . \, .$$
which we propose to call the two general {\it Equations of Curvature}.

\bigbreak

5.
In fact, by elimination of $R$, these equations (m) conduct to a
{\it quadratic in\/} $v$, of which the roots may be denoted by
$v_1$ and $v_2$, which first presents itself under the form,
$$(e + e' v) (E' + E'' v) = (e' + e'' v) (E + E' v),
   \leqno {\rm (n)} \, . \, .$$
but may easily be thus transformed,
$$\left\{ \eqalign{
   &  Av^2 - Bv + C = 0, \hbox{ or }
      A \, du^2 - B \, dt \, du + C \, dt^2 = 0,\cr
   &  \hbox{with }
      A = e'  E'' - e'' E',\quad
      B = e'' E   - e   E'',\quad
      C = e   E'  - e'  E;\cr}
   \right.
   \leqno {\rm (o)} \, . \, .$$
so that we have the following {\it general relation},
$$e A + e' B + e'' C = 0,
   \leqno {\rm (p)} \, . \, .$$
(of which we shall shortly see the geometrical signification),
between the {\it coefficients}, $A$, $B$, $C$, of the {\it joint
differential equation\/} of the system of the two {\it Lines of
Curvature\/} on the surface.

\bigbreak

6.
The root $v_1$ of the quadratic (o) determines the
{\it direction\/} of what may be called the {\it First Line of
Curvature}, through the point~${\sc p}$ of that surface; and the
{\it First Radius of Curvature}, for the same point~${\sc p}$, or
the radius $R_1$ of curvature of the {\it normal section\/} of
the surface which {\it touches\/} that {\it first line}, may be
obtained from {\it either\/} of the two equations (m), as the
value of $R$ which corresponds in that equation to the
value~$v_1$ of $v$.  And in like manner, the {\it Second Radius
of Curvature\/} of the same surface at the same point has the
value $R_2$, which answers to the value~$v_2$ of $v$, in each of
the same two {\it Equations of Curvature\/} (m).  We see, then,
that this {\it name\/} for those two equations is justified by
observing that when the two independent variables $t$ and $u$ are
given or known; and therefore also the seven functions of them,
above denoted by $e$,~$e'$,~$e''$, $E$,~$E'$,~$E''$, and $K$.
The equations (m) are satisfied by {\it two\/} (but {\it only
two\/}) {\it systems of values}, $v_1$,~$R_1$, and $v_2$,~$R_2$,
of (I.) the {\it differential quotient\/}~$v$, or
$\displaystyle {du \over dt}$,
which determines the {\it direction\/} of a {\it line of
curvature\/} on the surface; and (II.) the symbol~$R$, which
determines (comp.\ No.~4) at once the {\it length\/} and the {\it
direction}, of the {\it radius of curvature}, corresponding to
that {\it line}.

\bigbreak

7.
Instead of eliminating $R$ between the two equations (m), we may
{\it begin\/} by eliminating~$v$; a process which gives the
following quadratic in $R^{-1}$ (the curvature):---
$$(e R^{-1} - E K^{-1}) (e'' R^{-1} - E'' K^{-1})
   = (e' R^{-1} - E' K^{-1})^2;
   \leqno \phantom{\hbox{or}\quad} {\rm (q)} \, . \, .$$
$$R^{-2} - F R^{-1} + G = 0; \hbox{ where (because $e e'' - e'^2 = K^2$),}
   \leqno \hbox{or}\quad {\rm (r)} \, . \, .$$
$$F = R_1^{-1} + R_2^{-1} = (e E'' - 2 e' E' + e'' E) K^{-3}, \hbox{ and}
   \leqno \phantom{\hbox{or}\quad} {\rm (s)} \, . \, .$$
$$G = R_1^{-1} R_2^{-1} = (E E'' - E'^2) K^{-4}.
   \leqno \phantom{\hbox{or}\quad} {\rm (t)} \, . \, .$$
We ought, therefore, as a {\it First General Verification}, to
find that this last expression, which may be thus written,
$$G   = R_1^{-1} R_2^{-1}
      = {E E'' - E' E' \over (L^2 + M^2 + N^2)^2},
   \leqno {\rm (u)} \, . \, .$$
agrees with that reprinted in page~521 of Liouville's Monge, for
what Gauss calls the {\it Measure of Curvature\/} ($k$) of a
{\it Surface\/}; namely,
$$k = {D D'' - D' D' \over (AA + BB + CC)^2};
   \leqno {\rm (v)} \, . \, .$$
which accordingly it evidently does, because our symbols
$L$~$M$~$N$ $A$~$B$~$C$ represent the combinations which he
denotes by $A$~$B$~$C$~$D$~$D'$~$D''$.

\bigbreak

8.
As a {\it Second General Verification}, we may observe that if
$I$ be the {\it inclination\/} of any {\it linear element},
$du = v \, dt$, to the {\it element\/} $du = 0$, at the
point~${\sc p}$, then
$$\tan I = {Kv \over e + e' v};
   \leqno {\rm (w)} \, . \, .$$
and therefore, that if $H$ be the {\it angle\/} at which the {\it
second crosses the first}, of {\it any two lines\/} represented
{\it jointly\/} by such an equation as
$$A v^2 - B v + C = 0, \hbox{ with $v_1$ and $v_2$ for roots, then}
   \leqno {\rm (x)} \, . \, .$$
$$\tan H = \tan (I_2 - I_1)
	= {K (B^2 - 4AC)^{1 \over 2} \over e A + e' B + e'' C};
   \leqno {\rm (y)} \, . \, .$$
so that the {\it Condition of Rectangularity\/} ($\cos H = 0$),
for any {\it two\/} such lines, may be thus written:
$$e A + e' B + e'' C = 0.
   \leqno {\rm (z)} \, . \, .$$
But this {\it condition\/} (z) had already occurred in No.~5, as
an equation (p) which is satisfied generally by the {\it Lines of
Curvature\/}; we see therefore anew, by this analysis, that those
{\it lines\/} on {\it any surface\/} are in general (as is indeed
well known) {\it orthogonal\/} to each other.

\bigbreak

9.
Finally, as a {\it Third General Verification}, we may assume $x$
and $y$ {\it themselves\/} (instead of $t$ and $u$), as the two
independent variables of the problem, and then, if we use
{\it Monge's Notation\/} of $p$, $q$, $r$, $s$, $t$, we shall
easily recover all his leading results respecting {\it Curvatures
of Surfaces}, but by transformations on which we cannot here
delay.

\bye
