% 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, 2000.

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\centerline{\Largebf ON THE CELEBRATED THEOREM OF DUPIN}

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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,
   6 (1858), pp.\ 86--88.)}

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

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\centerline{\largerm 2000}

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\centerline{\largeit On the celebrated Theorem of Dupin.}

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\centerline{Sir William Rowan Hamilton.}

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\centerline{Communicated May 8th, 1854.}
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\centerline{[{\it Proceedings of the Royal Irish Academy},
   vol.~vi (1858), pp.\ 86--88.]}

\bigbreak

Sir W.~R. Hamilton, having been lately induced to consider, in
connexion with the Calculus of Quaternions, the celebrated
theorem of Dupin, respecting the character of the intersection
lines of three systems of orthogonal surfaces, as lines of
curvature thereon, stated that he had thus been led to perceive
some symbolical results which he supposed to be new, and which
seemed to him to be of sufficient interest to be submitted to the
Academy.

As long ago as 1846, he had proposed the notation,
$$\vecd = i {d \over dx} + j {d \over dy} + k {d \over dz};$$
and had pointed out a theorem, differing only slightly in its
expression from the following:
$${\rm V} \mathbin{.} \alpha \, {\rm V} \beta \gamma
   =    \gamma \, {\rm S} \mathbin{.} \alpha \beta
      - \beta  \, {\rm S} \mathbin{.} \alpha \gamma;$$
which may also be thus written,
$${\rm V} \mathbin{.} \alpha \, ({\rm V} \mathbin{.} \beta \gamma)
   =    {\rm S} \mathbin{.} \alpha \beta \mathbin{.} \gamma
      - \beta  \, {\rm S} \mathbin{.} \alpha \gamma,$$
or thus,
$${\rm V} \mathbin{.} \alpha \, ({\rm V} \mathbin{.} \beta \gamma)
   =    {\rm S} \mathbin{.} \beta \alpha \mathbin{.} \gamma
      - \beta  \, {\rm S} \mathbin{.} \alpha \gamma.$$
The recent results just referred to have a remarkable symbolical
resemblance to those comparatively old ones, since they admit of
being written thus:
$${\rm V} \mathbin{.} \alpha ( {\rm V} \mathbin{.} \vecd \nu )
   =     {\rm S} \mathbin{.} \alpha \vecd \mathbin{.} \nu
       - \vecd  \, {\rm S} \mathbin{.} \alpha \nu;
   \leqno \hskip 4em \llap{I.}$$
$${\rm V} \mathbin{.} \vecd ( {\rm V} \mathbin{.} \beta \nu )
   =     {\rm S} \mathbin{.} \beta  \vecd \mathbin{.} \nu
       - \beta  \, {\rm S} \mathbin{.} \vecd  \nu;
   \leqno \hskip 4em \llap{II.}$$
where $\vecd$ is {\it not\/} an ordinary {\it vector}, but a
certain {\it symbol of operation, analogous to a vector}, in its
combinations with other symbols, and defined by a foregoing
formula: while $\alpha$ and $\beta$ are constant vectors, and
$\nu$ is a variable vector, regarded as a function of
$x \, y \, z$, or of $\rho = ix + jy + kz$, and subject as such
to the operations $\vecd$, ${\rm S} \mathbin{.} \alpha \vecd$,
${\rm S} \mathbin{.} \beta \vecd$; where
$${\rm S} \mathbin{.} \alpha \vecd
   =   - \left(
            a {d \over dx}
          + b {d \over dy}
          + c {d \over dz}
         \right).$$
if $\alpha = i a + jb + kc$, and the symbol
${\rm S} \mathbin{.} \beta \vecd$ is similarly interpreted.

These were among the chief elements of calculation employed, in
proving by quaternions the theorem above mentioned of Dupin; of
which one expression, in the quaternion calculus, is the
following:---

``If the three differential equations,
$${\rm S} \mathbin{.} \nu  \, d \rho = 0,\quad
  {\rm S} \mathbin{.} \nu' \, d \rho = 0,\quad
  {\rm S} \mathbin{.} \nu \nu' \, d \rho = 0,$$
be integrable, and if ${\rm S} \mathbin{.} \nu \nu' = 0$, then
the supposition ${\rm V} \mathbin{.} \nu' \, d \rho = 0$ conducts
to the equation
$${\rm S} \mathbin{.} \nu \nu' d \nu = 0.\hbox{''}$$

Another expression of the same theorem is as follows:

``If
$${\rm S} \mathbin{.} \nu   \vecd \nu   = 0,\quad
  {\rm S} \mathbin{.} \nu'  \vecd \nu'  = 0,\quad
  {\rm S} \mathbin{.} \nu'' \vecd \nu'' = 0,$$
and
$${\rm V} \mathbin{.} \nu \nu' \nu'' = 0,$$
then
$${\rm S} \mathbin{.} \nu'' ( {\rm S} \nu' \vecd \mathbin{.} \nu )
   = 0.\hbox{''}$$

In this last formula, the symbol
${\rm S} \mathbin{.} \nu' \vecd \mathbin{.} \nu$
denotes a vector having the direction of $- d\nu$, if $d\rho$
have the direction of $\nu'$; and the equation expresses, that if
we thus move a little along the first surface in the direction of
the normal to the second surface, the new or near normal to that
first surface will be contained in the tangent plane to the third
surface, and therefore will intersect the old normal to the first
surface: which is a form of the theorem of Dupin.

Although not very closely connected with that well-known theorem,
Sir W.~R.~H. wishes to add that another old form of his, for any
three vectors, namely,
$${\rm V} ( {\rm V} \mathbin{.} \gamma \beta \mathbin{.} \alpha)
   =     \gamma \, {\rm S} \mathbin{.} \beta  \alpha
       - \beta  \, {\rm S} \mathbin{.} \gamma \alpha,$$
has suggested to him this new symbolical result,
$${\rm V} ( {\rm V} \mathbin{.} \gamma \, \vecd \mathbin{.} \nu )
   =     \gamma \, {\rm S} \mathbin{.} \vecd  \nu
       - \vecd  \, {\rm S} \mathbin{.} \gamma \nu;
   \leqno \hskip 4em \llap{III.}$$
and that each of the three general theorems, expressed by the
formul{\ae} I.~II.~III.\ of this Abstract, can be proved to
continue to be true, when his old signification of the
symbol~$\vecd$, to which Mr.~Carmichael's researches have lately
given an additional interest, is changed to this other and more
extensive signification,
$$\vecd = i \, \delta_1 + j \, \delta_2 + k \, \delta_3;
   \leqno \hskip 4em \llap{IV.}$$
where $\delta_1 \, \delta_2 \, \delta_3$ are three new
distributive symbols, operating on functions of $x \, y \, z$,
and commutative (in order) not only with any ordinary and scalar
constants, but also with $i \, j \, k$.

\bye
