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% School of Mathematics, Trinity College, Dublin 2, Ireland
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\centerline{\Largebf ON THE NATURE AND PROPERTIES OF THE}
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\centerline{\Largebf ACONIC FUNCTION OF SIX VECTORS}
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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,
5 (1853), p.\ 177--186.)}
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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 the Nature and Properties of the Aconic
Function of Six Vectors.}
\vskip 6pt
\centerline{{\largeit By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}
\bigskip
\centerline{Communicated June~23, 1851.}
\bigskip
\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~5 (1853), p.\ 177--186.]}
\bigskip
Sir William Rowan Hamilton entered into some explanatory details
respecting the nature and properties of that {\sc aconic function}
of six vectors, of which he had spoken in a recent communication
with reference to a certain generalization or extension of
Pascal's theorem, conducting to a relation between ten points on
a surface of the second order.
In the Proceedings of the Royal Irish Academy for July 20, 1846,
it was remarked by Sir W. Rowan Hamilton, that the theorem of
Pascal might, in the calculus of quaternions, be expressed by the
following general equation of cones of the second degree:
$${\rm S} \mathbin{.} \beta \beta' \beta'' = 0,$$
where
$$\eqalign{
\beta
&= {\rm V} (
{\rm V} \mathbin{.}
\alpha \alpha^{\rm I} \mathbin{.}
{\rm V} \mathbin{.}
\alpha^{\rm III} \alpha^{\rm IV} ),\cr
\beta'
&= {\rm V} (
{\rm V} \mathbin{.}
\alpha^{\rm I} \alpha^{\rm II} \mathbin{.}
{\rm V} \mathbin{.}
\alpha^{\rm IV} \alpha^{\rm V} ),\cr
\beta''
&= {\rm V} (
{\rm V} \mathbin{.}
\alpha^{\rm II} \alpha^{\rm III} \mathbin{.}
{\rm V} \mathbin{.}
\alpha^{\rm V} \alpha );\cr}$$
$\alpha$, $\alpha^{\rm I}$, $\alpha^{\rm II}$,
$\alpha^{\rm III}$, $\alpha^{\rm IV}$, $\alpha^{\rm V}$
being any six homoconic vectors, and the letters ${\rm S}$ and
${\rm V}$ being the characteristics of the operations of taking
respectively the scalar and vector parts of a quaternion. Now it
is precisely {\it that function\/} of six vectors
$\alpha \, \ldots \, \alpha^{\rm V}$,
which was denoted in that communication of 1846, by
${\rm S} \mathbin{.} \beta \beta' \beta''$, to which it has since
appeared to Sir W. Rowan Hamilton convenient to give the name of
the {\sc aconic} (or {\it heteroconic\/}) {\it function\/} of
those six vectors; because in the more general case, when they
are {\it not\/} sides of any common cone of the second degree,
this function no longer {\it vanishes}, but acquires some
positive or negative value.
One of the most important properties of this {\it aconic\/}
function is, that it {\it changes its sign without otherwise
changing its value}, whenever {\it any two\/} of the {\it six\/}
vectors on which it depends {\it change places\/} among
themselves. Admitting this property, which there are many ways
of easily proving by the general rules of quaternions, and
observing that the following function of {\it four\/} vectors,
$\alpha^{\rm VI}$,~$\alpha^{\rm VII}$,~$\alpha^{\rm VIII}$,~$\alpha^{\rm IX}$,
namely
$${\rm S} \mathbin{.}
(\alpha^{\rm VI} - \alpha^{\rm VII})
(\alpha^{\rm VII} - \alpha^{\rm VIII})
(\alpha^{\rm VIII} - \alpha^{\rm IX}),$$
can be shewn to change sign in like manner, for any binary
interchange among the vectors on which it depends, and to vanish
when any two of them are equal; denoting also, for conciseness,
the former function by $012345$, the latter by $6789$, and their
product by
$$012345 \mathbin{.} 6789;$$
Sir W.~Rowan Hamilton proceeds to form, by binary transpositions
of these figures, or of the vectors which they denote, from one
factor of each product to the other, accompanied with a change of
the algebraic sign prefixed to each such product as a term, for
every such binary interchange, a system of 210 terms, namely
$$\eqalign{
& + 012345 \mathbin{.} 6789 - 012346 \mathbin{.} 5789 \cr
& + 012347 \mathbin{.} 5689 - 012348 \mathbin{.} 5679 \cr
& + 012349 \mathbin{.} 5678 - 012359 \mathbin{.} 4678 \cr
& + 012358 \mathbin{.} 4679 - 012357 \mathbin{.} 4689 \cr
& + 012356 \mathbin{.} 4789 - 012376 \mathbin{.} 4589 \cr
& + (\hbox{a hundred other products})
- (\hbox{a hundred other products});\cr}$$
these remaining terms being easily formed in succession,
according to the lately mentioned law. And to the algebraic sum
of all these 210 terms, of which each separately is a positive or
negative number,---its positive or negative character depending of
course not alone on the prefixed sign $+$ or $-$, but also on the
positive or negative characters of the {\it factors\/} of the
product, which enters with that sign prefixed into the term,---Sir
W.~Rowan Hamilton proposes to give the name of the
{\it heterodeuteric}, or (more shortly) the {\sc adeuteric
function} of the ten vectors
$\alpha \, \ldots \, \alpha^{\rm IX}$,
for a reason which will presently appear.
To make the formation of this function of {\it ten\/} vectors
more completely clear, it may be observed, that the function of
{\it four\/} vectors, which has been above denoted by the symbol
$6789$, is easily found to represent the sextupled volume of the
{\it pyramid}, whose corners are the terminations of the four
vectors (all drawn from one common origin); this form being
regarded as positive or negative, according to the character (as
right handed or left handed) of a certain {\it rotation\/}; which
character or direction is {\it reversed\/} when {\it any two\/}
of the four vectors, and therefore, also, their terminations, are
made to change places with each other. On this account the
lately mentioned function of four vectors may be called their
{\sc pyramidal function}; and then the foregoing {\it rule\/} for
the composition of the {\it adeuteric function\/} may be
expressed in words as follows:---Starting with {\it any one
set\/} of {\it four\/} vectors, form {\it their\/} pyramidal
function, and multiply it by the aconic function of the remaining
{\it six}, out of the proposed {\it ten\/} vectors, arranging the
vectors of each set in any one selected {\it order}. Choose any
vector of the four, and any other of the six, and interchange
these {\it two\/} vectors, without altering the arrangement of
the rest, so as to form a new group of four vectors, and another
new group of six; and multiply the pyramidal function of the
former group by the aconic function of the latter. Proceeding
thus, we can gradually and successively form all the 210 possible
groups or sets of four vectors, accompanied each with another set
of six; and the four or the six vectors in each set will have an
arrangement among themselves, determined by the foregoing
process; so that the 210 pyramidal and the 210 aconic functions
have each a determined value, {\it including\/} a known positive
or negative sign or character. Each of the 210 {\it products},
thus obtained, is therefore itself also {\it determinate}, as
being equal to some one positive or negative number, of which the
{\it sign\/} as well as the absolute {\it value\/} can be
definitely found, and may be considered as being {\it known,
before\/} we introduce or employ any rule for {\it combining\/}
or incorporating these various products among themselves, by any
{\it additions\/} or {\it subtractions}. But if we {\it now\/}
employ, for such incorporation, the rule that all those products
which have been formed by any {\it even\/} number of binary
interchanges, from the product first assumed, which we may still
suppose to be
$$012345 \mathbin{.} 6789,$$
are to be {\it algebraically added\/} thereto; while, on the
contrary, all which are formed from that original product by any
{\it odd\/} number of binary interchanges are to be
{\it algebraically subtracted\/} from it: we shall complete (as
was before more briefly stated) the determination of that
{\it function of\/} {\sc ten} {\it vectors}, $0$ to $9$, which
was lately called the {\sc adeuteric}.
Indeed, it may for a moment still appear that this function is in
some degree {\it indeterminate}, because there may be many
different ways of passing, by successive binary interchanges,
from one given set of six, and a companion set of four vectors,
to a second given set of six, with its own companion set of four.
For example, we passed from the first to the tenth of the
products already written, by a succession of {\it nine\/} binary
interchanges, which may be indicated thus:
$$56,\quad 67,\quad 78,\quad 89,\quad 45,\quad 98,\quad
87,\quad 76,\quad 57.$$
But we might also have passed from the same first product,
$$+ 012345 \mathbin{.} 6789$$
by the {\it two\/} binary interchanges $47$, $56$, to this other
product and sign,
$$+ 012376 \mathbin{.} 5489,$$
where the sign~$+$ is prefixed, on account of their being now an
{\it even\/} number (two) of such changes. On the other hand,
the {\it odd\/} number (nine), of binary interchanges above
described, had given the term
$$- 012376 \mathbin{.} 4589.$$
But because, by the properties of the pyramidal function of four
vectors above referred to, we have
$$+ 5489 = - 4589,$$
the two terms thus obtained differ only in appearance from each
other. And similar reductions will in every other case hold
good, in virtue of the properties of the pyramidal and aconic
functions, combined with a principle respecting transpositions of
symbols (which probably is well known): namely, that if a set of
$n$ symbols (as here the ten figures from $0$ to $9$) be brought
in any two different ways, by any two numbers~$l$ and $m$ of
binary interchanges, to any one other arrangement, the
{\it difference\/} $m - l$ of these two {\it numbers\/} is
{\it even}.
The {\sc value} (including sign) of the foregoing
{\it adeuteric\/} function, of any ten determined vectors, is
therefore itself completely {\it determined}, if we fix (as
before) the {\it arrangement\/} of the ten vectors in the
{\it first\/} of the 210 terms from which the others are to be
derived: because the {\it value\/} of {\it each\/} separate term
becomes then fixed, although the {\it forms\/} of these various
terms may undergo considerable variations, by interchanges
conducted as above. If then we choose {\it any two\/} of the ten
vectors, suppose those numbered~$4$ and $7$, we may
{\it prepare\/} the expression of the {\it adeuteric\/} function
as follows. We may first collect into one group the 70 terms in
which these two vectors both enter into one common aconic
function; and may call the sum of all these terms, Polynome~I.
We may next collect into a second group all those other terms, in
number 28, for each of which the two selected vectors both enter
into the composition of one common pyramidal function; and may
call the sum of these 28 terms, Polynome~II. And finally, we may
arrange (after certain permitted transpositions) the remaining
112 terms into 56 pairs, such as
$$+ 012345 \mathbin{.} 6789 - 012375 \mathbin{.} 6489,$$
and
$$- 012346 \mathbin{.} 5789 + 012376 \mathbin{.} 5489.$$
and may call the sum of these 56 pairs of terms, Polynome~III;
the rule of pairing being here, that the two selected vectors (in
the present case $4$ and $7$) shall be interchanged in passing
from any one term of the pair to the other, with a change of sign
as before. But when the expression of the {\it adeuteric\/} has
been thus prepared, it becomes clear that {\it each\/} of its
{\it three\/} partial polynomes is changed to its own
{\it negative\/}, when the two selected vectors are interchanged.
In fact, {\it each term\/} of the first polynome changes sign, by
this interchange, in virtue of the properties of the
{\it aconic\/} function of six vectors. Again, {\it each\/} term
of the {\it second\/} polynome in like manner changes sign, on
account of the properties of the {\it pyramidal\/} function of
four vectors. And finally, {\it each pair\/} of terms in the
third polynome changes sign, from the manner in which that pair
is composed. On the whole then we must infer, that the sum of
these three polynomes, or the function above is called the
{\sc adeuteric, changes sign} {\it without otherwise changing
value, when\/} {\sc any two} {\it of the} {\sc ten} {\it vectors
on which it depends are made to\/} {\sc change places} {\it with
each other\/}: whence it is very easy to infer, that {\it this
adeuteric function\/} {\sc vanishes}, {\it when any two of its
ten vectors become\/} {\sc equal}.
Now the aconic function is of the {\it second\/} degree, with
respect to each of the six vectors on which it depends; while the
pyramidal function is easily shewn to be only of the
{\it first\/} degree, with respect to each of the four other
vectors which enter into its composition. Hence each of the 210
terms of the adeuteric rises no higher than the {\it second
degree}; and {\it if we equate this adeuteric function to zero,
we thereby oblige any one of the ten vectors to terminate on a
given surface of the second order, if the other nine vectors be
given}. But it has been seen, that the adeuteric vanishes, when
{\it any two\/} of its ten vectors are made equal to each other;
the surface which is thus the {\it locus\/} of the extremity of
the {\it tenth\/} vector, must, therefore, pass {\it through the
nine points\/} in which the {\it nine other\/} vectors
respectively terminate. On this account the ten vectors, or
their extremities, may be said to be, under this condition,
{\sc homodeuteric}, as belonging all to {\it one common surface
of the second order}. And thus we at once justify, by contrast,
the foregoing appellation of the {\sc adeuteric} function, and
also see that to equate (as above) this adeuteric to zero, is to
establish what may be called the {\sc equation of homodeuterism},
as in fact it was so called in a recent communication to the
Academy; while, as an abbreviation of the recent notation, we may
now write that equation as follows:
$$\Sigma ( \pm 012345 \mathbin{.} 6789 ) = 0;$$
where the sum in the left-hand member represents the adeuteric
function.
What has been shewn respecting the composition of this {\it adeuteric},
may naturally produce a wish to possess some {\it geometrical
rule for constructing the aconic function\/} $(012345)$, of any
{\it six\/} given vectors; and the {\it quaternion expression\/}
for that function enables us easily to assign such a {\it rule}.
For this purpose, let
${\sc a}$,~${\sc b}$,~${\sc c}$,~${\sc d}$,~${\sc e}$,~${\sc f}$
be the six points at which the six vectors lately numbered as
$0$,~$1$,~$2$,~$3$,~$4$,~$5$ terminate, being supposed to be all
drawn from some assumed and common {\it origin\/}~${\sc o}$;
while ${\sc g}$,~${\sc h}$,~${\sc i}$,~${\sc k}$ may denote the
four other points, through which the surface of the second order
passes, when the equation of homodeuterism is satisfied, and
which are the terminations of the four other vectors above
numbered as $6$, $7$, $8$, $9$. The aconic function, above
denoted by $012345$, of the six vectors
${\sc o} {\sc a}$,
${\sc o} {\sc b}$,
${\sc o} {\sc c}$,
${\sc o} {\sc d}$,
${\sc o} {\sc e}$,
${\sc o} {\sc f}$,
which terminate generally at the six corners of a gauche hexagon
${\sc a} {\sc b} {\sc c} {\sc d} {\sc e} {\sc f}$, may now be
concisely expressed by the symbol
$${\sc o} \mathbin{.} {\sc a} {\sc b} {\sc c} {\sc d} {\sc e} {\sc f};$$
or even simply by
${\sc a} {\sc b} {\sc c} {\sc d} {\sc e} {\sc f}$,
the reference to an origin being understood. To construct it,
Sir W.~Rowan Hamilton constructs first the six vectors
$${\rm V} \mathbin{.} \alpha \alpha^{\rm I},\quad
{\rm V} \mathbin{.} \alpha^{\rm I} \alpha^{\rm II},\quad
{\rm V} \mathbin{.} \alpha^{\rm II} \alpha^{\rm III},\quad
{\rm V} \mathbin{.} \alpha^{\rm III} \alpha^{\rm IV},\quad
{\rm V} \mathbin{.} \alpha^{\rm IV} \alpha^{\rm V},\quad
{\rm V} \mathbin{.} \alpha^{\rm V} \alpha,$$
and then the three other vectors $\beta$,~$\beta'$,~$\beta''$,
which depend on these, in order to form thence that scalar
${\sc S} \mathbin{.} \beta \beta' \beta''$, which, by what was
stated near the commencement of the present Abstract, is the
{\it aconic\/} function required. It will be seen that all the
steps of the following construction of that function are in this
way obvious consequences from the quaternion expression above
given. The construction itself was communicated to a few
scientific friends of his about the end of August and beginning
of September, 1849, and has since been publicly stated at the
Edinburgh Meeting of the British Association in 1850, although it
has not hitherto been printed.
Regarding the given and gauche hexagon,
${\sc a} {\sc b} {\sc c} {\sc d} {\sc e} {\sc f}$,
as a sort of {\it base\/} of a {\it hexahedral angle}, of which
the {\it vertex\/} is the assumed point~${\sc o}$, Sir W.~Rowan
Hamilton {\it represents\/} the {\it doubled areas\/} of the six
plane and triangular faces of this angle, namely,
$${\sc a} {\sc o} {\sc b},\quad
{\sc b} {\sc o} {\sc c},\quad
{\sc c} {\sc o} {\sc d},\quad
{\sc d} {\sc o} {\sc e},\quad
{\sc e} {\sc o} {\sc f},\quad
{\sc f} {\sc o} {\sc a},$$
by {\it six right lines\/} from the vertex,
$${\sc o} {\sc l},\quad
{\sc o} {\sc m},\quad
{\sc o} {\sc n},\quad
{\sc o} {\sc l}',\quad
{\sc o} {\sc m}',\quad
{\sc o} {\sc n}',$$
which are respectively {\it normals\/} to the six faces, and are
distinguished from their own opposites by a simple and uniform
rule of {\it rotation\/}: for example, the line~${\sc o} {\sc l}$
contains as many linear units as the doubled area of the
triangle~${\sc a} {\sc o} {\sc b}$ (to the plane of which it is
perpendicular) contains units of area; and the notation round
${\sc o} {\sc l}$ from ${\sc o} {\sc a}$ to ${\sc o} {\sc b}$ is
right-handed. The doubled areas of the three new triangles,
$${\sc l} {\sc o} {\sc l}',\quad
{\sc m} {\sc o} {\sc m}',\quad
{\sc n} {\sc o} {\sc n}',$$
are next to be {\it represented}, on the same general plan, by
{\it three new lines\/} from the vertex,
$${\sc o} {\sc l}'',\quad
{\sc o} {\sc m}'',\quad
{\sc o} {\sc n}'';$$
which three lines will thus be the intersection of the three
pairs of opposite faces of the hexahedral angle, and consequently
will, by Pascal's theorem, be situated in one common plane, if
the given hexagon
${\sc a} {\sc b} {\sc c} {\sc d} {\sc e} {\sc f}$
can be inscribed in a cone of the second degree, with the
point~${\sc o}$ for its vertex. But in the more {\it general\/}
case, when the given hexagon {\it cannot\/} be so inscribed, in
any such cone with that assumed point for vertex, we can
construct a parallelepipedon with the three last lines,
${\sc o} {\sc l}''$, ${\sc o} {\sc m}''$, ${\sc o} {\sc n}''$,
for three adjacent edges: and the {\it volume of this solid\/} is
the geometrical representation which Sir W.~Rowan Hamilton's
method assigns for what he calls (as above) the {\it aconic
function\/} of the six given vectors, or of the six given points
${\sc a}$, ${\sc b}$, ${\sc c}$, ${\sc d}$, ${\sc e}$, ${\sc f}$,
in which those vectors terminate, or of the (generally gauche)
hexagon of which those points are corners. And with respect to
the {\it sign\/} of this function, it is to be regarded as being
positive or negative, according as the rotation round
${\sc o} {\sc n}''$, from ${\sc o} {\sc m}''$ towards
${\sc o} {\sc l}''$, is to the right hand or to the left.
Such then is the construction of the {\it aconic\/} function,
$012345$, or ${\sc a} {\sc b} {\sc c} {\sc d} {\sc e} {\sc f}$;
and it is still more easy to construct the {\it pyramidal\/}
function~$6789$, which may also be denoted by the symbol
${\sc g} {\sc h} {\sc i} {\sc k}$; since the absolute value of
this function is constructed (as above remarked) by the
{\it sextuple volume of the pyramid}, which has the four points
${\sc g}$,~${\sc h}$,~${\sc i}$,~${\sc k}$ for corners, or by the
volume of the {\it parallelepipedon\/} which has
${\sc g} {\sc h}$, ${\sc g} {\sc i}$, ${\sc g} {\sc k}$,
for edges; while the quaternion expression assigned near the
commencement of this Abstract, admits of being thus written,
$${\rm S} \mathbin{.}
(\alpha^{\rm IX} - \alpha^{\rm VI})
(\alpha^{\rm VIII} - \alpha^{\rm VI})
(\alpha^{\rm VII} - \alpha^{\rm VI}),$$
and conducts to the regarding this volume, or the
function~$6789$, or ${\sc g} {\sc h} {\sc i} {\sc k}$, as being
positive when the rotation round ${\sc g} {\sc h}$ from
${\sc g} {\sc i}$ towards ${\sc g} {\sc k}$ is right-handed, but
negative in the contrary case. And the aconic and pyramidal
functions having thus been {\it separately\/} constructed, they
have only to be {\it combined\/} with each other, according to
the law already stated, in order to assign a
{\it geometrical signification\/} to each term of the
{\it adeuteric function}, namely, the sum,
$$\Sigma (\pm {\sc a} {\sc b} {\sc c} {\sc d} {\sc e} {\sc f}
\mathbin{.} {\sc g} {\sc h} {\sc i} {\sc k} );$$
and also to the {\it equation of homodeuterism}, which may now be
written thus (as in a recent communication to the Academy),
$$\Sigma (\pm {\sc a} {\sc b} {\sc c} {\sc d} {\sc e} {\sc f}
\mathbin{.} {\sc g} {\sc h} {\sc i} {\sc k} )
= 0,$$
and which expresses that the {\it ten points},
${\sc a}, {\sc b},\ldots, {\sc k}$,
are situated {\it upon one common surface of the second order}.
And if we place the arbitrary origin~${\sc o}$ at one of the ten
points, the {\it number of terms\/} in the adeuteric function, or
in the equation of homodeuterism, is easily seen to
{\it reduce\/} itself, then, from 210 to 84.
If the thirty {\it co-ordinates\/} of the ten points were
substituted in the function above called the {\it adeuteric}, the
resulting expression could doubtless only differ by some
numerical coefficient from that {\it determinant\/} which might
otherwise be found, as the result of the elimination of the nine
coefficients $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, $I$, between
the equations,
$$\eqalign{
A x_0^2 + B y_0^2 + C z_0^2 + D y_0 z_0 + E z_0 x_0 + F x_0 y_0
+ G x_0 + H y_0 + I z_0 + 1 &= 0,\cr
\noalign{\hbox{$\ldots\ldots\ldots$}}
A x_9^2 + B y_9^2 + C z_9^2 + D y_9 z_9 + E z_9 x_9 + F x_9 y_9
+ G x_9 + H y_9 + I z_9 + 1 &= 0.\cr}$$
And Sir W.~Rowan Hamilton has much pleasure in referring to a
paper by Mr.~Cayley, printed near the commencement of the Fourth
Volume of the Cambridge Mathematical Journal, on Pascal's Theorem
considered in connexion with determinants, which paper had not
been noticed by the present writer till his attention was called
to it by a friend to whom he had communicated the above-stated
construction. But while gladly acknowledging the great
mathematical learning and originality exhibited in that and every
paper by Mr.~Cayley, Sir W.~Rowan Hamilton thinks it right to
state, that he was led to his own results, respecting the
{\it relation\/} (above assigned) between {\it ten points on the
surface of the second order}, not by any system of
{\it co-ordinates}, but by the considerations of {\it vectors},
and by seeking to extend to {\it ellipsoids\/} the results
respecting {\it cones}, which he had submitted to the Academy in
July, 1846, and had also published in the Philosophical Magazine
for the following month, as derived from the Calculus of
Quaternions.
\bye