% 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 THEOREMS RELATING TO SURFACES,}

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\centerline{\Largebf OBTAINED BY THE METHOD OF QUATERNIONS}

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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,
   4 (1850), pp.\ 306--308.)}

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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 Theorems relating to Surfaces, obtained
by the Method of Quaternions.}

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\centerline{{\largeit By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}

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\centerline{Communicated February~26, 1849.}

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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~4 (1850), pp.\ 306--308.]}

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The following letter from Sir William R. Hamilton was read,
giving some general expressions of theorems relating to surfaces,
obtained by his method of quaternions:

``The equation of curved surface being put under the form
$$f(\rho) = \hbox{const.}:$$
while its {\it tangent plane\/} may be represented by the equation,
$$df(\rho) = 0,$$
or
$${\rm S} \mathbin{.} \nu \, d \rho = 0,$$
if $d\rho$ be the vector drawn to a point of that plane, from the
point of contact; the equation of {\it an osculating surface of
the second order\/} (having complete contact of the second order
with the proposed surface at the proposed point) may be thus
written:
$$0 = df(\rho) + {\textstyle {1 \over 2}} d^2 f(\rho);$$
(by the extension of Taylor's series to quaternions); or thus,
$$0 = 2 {\rm S} \mathbin{.} \nu \, d \rho
      + {\rm S} \mathbin{.} d \nu \, d \rho,$$
if
$$df(\rho) = 2 {\rm S} \mathbin{.} \nu \, d \rho.$$

``The {\it sphere, which osculates in a given direction}, may be
represented by the equation
$$0 = 2 {\rm S} {\nu \over \Delta \rho}
       + {\rm S} {d \nu \over  d \rho};$$
where $\Delta \rho$ is a chord of the sphere, drawn from the
point of osculation, and
$${\rm S} {d \nu \over d \rho}
   =  {{\rm S} \mathbin{.} d \nu \, d \rho \over d \rho^2}
   =  {d^2 f(\rho) \over 2 \, d \rho^2}$$
is a scalar function of the versor ${\rm U} \, d \rho$, which
determines the direction of osculation.  Hence the important
formula:
$${\nu \over \rho - \sigma} = {\rm S} {d \nu \over d \rho};$$
where $\sigma$ is the vector of the centre of the sphere which
osculates in the direction answering to ${\rm U} \, d \rho$.

``By combining this with the expression formerly given by me for
a normal to the ellipsoid, namely
$$(\kappa^2 - \iota^2)^2 \nu
   =  (\iota^2 + \kappa^2) \rho
       + \iota \rho \kappa + \kappa \rho \iota,$$
the known value of the curvature of a normal section of that
surface may easily be obtained.  And for {\it any\/} curved
surface, the formula will be found to give easily this general
theorem, which was perceived by me in 1824; that if, on a normal
plane ${\sc o} {\sc p} {\sc p}'$, which is drawn through a given
normal ${\sc p} {\sc o}$, and through any linear
element~${\sc p} {\sc p}'$ of the surface, we project the
infinitely near normal~${\sc p}' {\sc o}'$, which is erected to
the same surface at the end of the element~${\sc p} {\sc p}'$;
the projection of the near normal will cross the given normal in
the centre~${\sc o}$ of the same sphere which osculates to the
given surface at the given point~${\sc p}$, in the direction of
the given element~${\sc p} {\sc p}'$.

``I am able to shew that the formula
$$0 = \delta {\rm S} {d \nu \over d \rho},$$
which follows from the above, for determining the directions of
osculation of the greatest and least osculating spheres, agrees
with my formerly published formula,
$$0 = {\rm S} \mathbin{.} \nu \, d \nu \, d \rho,$$
for the directions of the lines of curvature.

``And I can deduce Gauss's {\it general\/} properties of geodetic
lines by showing that if $\sigma_1$,~$\sigma_2$ be the two
extreme values of the vector~$\sigma$, then
$${- 1 \over (\rho - \sigma_1) (\rho - \sigma_2)}
   =  \hbox{measure of curvature of surface}
   =  {1 \over R_1 R_2}
   =  {d^2 {\rm T} \, \delta \rho
         \over {\rm T} \, \delta \rho \mathbin{.} d \rho^2};$$
where $d$ answers to motion along a normal section, and $\delta$
to the passage from one near (normal) section to another; while
${\rm S}$, ${\rm T}$, and ${\rm U}$, are the characteristics of
the operations of taking the scalar, tensor and versor of a
quaternion: and the variation $\delta v$ of the inclination~$v$
of a given geodetic line to a variable normal section, obtained
by passing from one such section to a near one, without changing
the geodetic line, is expressed by the analogous formula,
$$\delta v
   = - {d {\rm T} \, \delta \rho \over {\rm T} \, d \rho}.\hbox{''}$$

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