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\centerline{\Largebf ON A GENERAL CENTRE OF APPLIED 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, 8 (1864),
p.~394.)}

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

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\centerline{\largerm 2000}

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\centerline{{\sc On a general Centre of applied Forces}}

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\centerline{\largerm Sir William Rowan Hamilton}
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\centerline{[Read June 22nd, 1863.]}
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\centerline{[{\it Proceedings of the Royal Irish Academy},
   vol.~viii (1864), p.~394.]}

\bigbreak

Sir W.~R. Hamilton wishes a note to be preserved in the
Proceedings of the Royal Irish Academy, that on recently
reconsidering an application of Quaternions to the Statics of a
Solid Body, some account of which was laid before the Academy
many years ago (see the Proceedings for December 1845), he has
been led to perceive the {\it theoretical\/} (and to suspect the
{\it practical\/}) existence of a certain {\it Central Point\/}
for {\it every system of applied forces}, not reducible to a
{\it couple}, nor to {\it zero\/}: which {\it generally new
point}, for the case of {\it parallel forces}, coincides with
their well-known {\it centre}.

An {\it applied force\/} $A B$, acting at a
point~$A$, being said to have a {\it quaternion moment},
equal to the quaternion {\it product\/}
$O A \mathbin{.} A B$,
with respect to any assumed point~$O$, the {\it sum\/} of
all {\it such\/} moments, or the quaternion,
${\rm Q}
   =  \Sigma ( O A \mathbin{.} A B )
   =     O A \mathbin{.} A B
       + O A' \mathbin{.} A' B'
       + \hbox{\&c.}$,
is called the {\it total quaternion moment\/} of the applied
system with respect to the same point~$O$.

This {\it total moment~${\rm Q}$ varies\/} generally with the
{\it point\/} to which it is referred; and there is {\it one\/}
point~$C$, or {\it one position\/} of $O$, for which
the condition
$${\rm T} {\rm Q} = \hbox{\it a minimum},$$
is satisfied, with the exceptions (of {\it couple\/} and
{\it equilibrium}) above alluded to.

It is {\it this point\/}~$C$, which Sir W.~R.~H. proposes
to call {\it generally\/} the {\it Centre of a System of Applied
Forces}.

In the most general case of such a system, he finds it to be
situated {\it on the Central Axis}, the {\it minimum\/}
${\rm T} {\rm Q}$ representing then what was called by Poinsot
the {\it Energy of the Central Couple}.

For the less general case of an {\it unique resultant force}, the
quaternion~${\rm Q}$ reduces itself to {\it zero\/} at the new
Central Point~$C$, which is now situated {\it on the
resultant}, and determines its {\it line of application}.

\bye