% 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 A NEW METHOD OF INVESTIGATING}
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\centerline{\Largebf THE RELATIONS OF SURFACES TO THEIR}
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\centerline{\Largebf NORMALS, WITH RESULTS RESPECTING}
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\centerline{\Largebf THE CURVATURES OF ELLIPSOIDS}
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\centerline{\Largebf By}
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\centerline{\Largebf William Rowan Hamilton}
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\centerline{\largerm (Dublin University Review, July 1833, p.~583--584.)}
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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 a New Method of investigating the relations of Surfaces to
their Normals, with results respecting the Curvatures of
Ellipsoids.}
{\largerm By William R. Hamilton, Royal Astronomer of Ireland.}
\bigbreak
\centerline{[{\it Dublin University Review}, July 1833, p.~583--584.]}
\bigbreak
The important researches of Clairaut, Euler, Monge, and Dupin, on
the curvatures of surfaces, leave much still to be discovered;
and it appears to me that a new method of research, founded on
certain new forms of equations of surfaces and of their normals,
may be introduced with advantage into the subject. The method
which I propose, consists in expressing, for any surface, the
perpendicular distance $W$ of an assumed origin from any tangent
plane of the surface as a homogeneous function of the first
dimension of the cosines $\alpha$,~$\beta$,~$\gamma$ of the
inclinations of this perpendicular~$W$ to three rectangular axes
of coordinates $x$,~$y$,~$z$; and in then employing the following
equations of a normal:
$$x = V \alpha + {\delta W \over \delta \alpha},\quad
y = V \beta + {\delta W \over \delta \beta},\quad
y = V \gamma + {\delta W \over \delta \gamma},$$
in which $V$ is the length of the normal, from the surface to the
point $x$,~$y$,~$z$: so that the conditions of intersection of two
near normals are found by making $\alpha$,~$\beta$,~$\gamma$
vary, subject to the relation
$$\alpha \, \delta \alpha + \beta \, \delta \beta
+ \gamma \, \delta \gamma = 0,$$
and leaving $x$,~$y$,~$z$, and $V$, unchanged.
Thus for an Ellipsoid referred to its centre and axes, I use the
equation
$$W = \sqrt{ a^2 \alpha^2 + b^2 \beta^2 + c^2 \gamma^2 },$$
and easily obtain the following new results.
1st. The normals to an ellipsoid, for any line of one curvature
have a constant sum, and for any line of the other curvature they
have a constant difference, of inclinations to the normals of the
two planes of circular section.
2d. The difference of the two radii of curvature at any variable
point of the ellipsoid, is proportional to the product of the
sines of the inclinations of the variable normal to the normals of
the two circular sections, divided by the cube of the
perpendicular distance of the centre from the tangent plane.
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{\it Observatory, June}, 1833.
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