% 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 OF CENTRAL FORCES}
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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,
3 (1847), pp.\ 308--309.)}
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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 of Central Forces.}
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\centerline{{\largeit By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}
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\centerline{Communicated November 30, 1846.}
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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~3 (1847), pp.\ 308--309.]}
\bigskip
Sir William R. Hamilton stated the following theorems of central
forces, which he had proved by his calculus of quaternions, but
which, as he remarked, might be also deduced from principles more
elementary.
If a body be attracted to a fixed point, with a force which
varies directly as the distance from that point, and inversely as
the cube of the distance from a fixed plane, the body will
describe a conic section, of which the plane intersects the fixed
plane in a straight line, which is the polar of the fixed point
with respect to the conic section.
And in like manner, if a material point be obliged to remain upon
the surface of a given sphere, and be acted on by a force, of
which the tangential component is constantly directed (along the
surface) towards a fixed point or pole upon that surface, and
varies directly as the sine of the arcual distance from that
pole, and inversely as the cube of the sine of the arcual
distance from a fixed great circle; then the material point will
describe a spherical conic, with respect to which the fixed great
circle will be the polar of the fixed point.
Thus, a spherical conic would be described by a heavy point upon
a sphere, if the vertical accelerating force were to vary
inversely as the cube of the perpendicular and linear distance
from a fixed plane passing through the centre.
The first theorem had been suggested to Sir W. Hamilton by a
recently resumed study of a part of Sir Isaac Newton's
Principia; and he had been encouraged to seek for the
second theorem, by a recollection of a result respecting motion
in a spherical conic, which was stated some years ago to the
Academy by the Rev.~C.~Graves. In that result of Mr.~Graves, the
fixed pole was a focus of the conic, and the polar was therefore
the director arc; consequently, the sine of the distance from the
polar was proportional to the sine of the distance from the pole,
and, instead of the law now mentioned to the Academy, there was
the simpler law of proportionality to the inverse square of the
sine of the distance from the fixed pole or focus.
\bye