% 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.
\magnification=\magstep1
\vsize=227 true mm \hsize=170 true mm
\voffset=-0.4 true mm \hoffset=-5.4 true mm
\def\folio{\ifnum\pageno>0 \number\pageno \else\fi}
\font\Largebf=cmbx10 scaled \magstep2
\font\largerm=cmr12
\font\largeit=cmti12
\font\tensc=cmcsc10
\font\sevensc=cmcsc10 scaled 700
\newfam\scfam \def\sc{\fam\scfam\tensc}
\textfont\scfam=\tensc \scriptfont\scfam=\sevensc
\font\largesc=cmcsc10 scaled \magstep1
\pageno=0
\null\vskip72pt
\centerline{\Largebf ON THEOREMS RELATING TO SURFACES,}
\vskip12pt
\centerline{\Largebf OBTAINED BY THE METHOD OF QUATERNIONS}
\vskip24pt
\centerline{\Largebf By}
\vskip24pt
\centerline{\Largebf William Rowan Hamilton}
\vskip24pt
\centerline{\largerm (Proceedings of the Royal Irish Academy,
4 (1850), pp.\ 306--308.)}
\vskip36pt
\vfill
\centerline{\largerm Edited by David R. Wilkins}
\vskip 12pt
\centerline{\largerm 2000}
\vskip36pt\eject
\null\vskip36pt
\centerline{\largeit On Theorems relating to Surfaces, obtained
by the Method of Quaternions.}
\vskip 6pt
\centerline{{\largeit By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}
\bigskip
\centerline{Communicated February~26, 1849.}
\bigskip
\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~4 (1850), pp.\ 306--308.]}
\bigskip
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,
$$\delta v
= - {d {\rm T} \, \delta \rho \over {\rm T} \, d \rho}.\hbox{''}$$
\bye