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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
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% Trinity College, 2000.
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\centerline{\Largebf ON A PROOF OF PASCAL'S THEOREM}
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\centerline{\Largebf BY MEANS OF QUATERNIONS; AND ON}
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\centerline{\Largebf SOME OTHER CONNECTED SUBJECTS}
\vskip24pt
\centerline{\Largebf By}
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\centerline{\Largebf William Rowan Hamilton}
\vskip24pt
\centerline{\largerm (Proceedings of the Royal Irish Academy,
3 (1847), pp.\ 273--292.)}
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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 a proof of Pascal's Theorem by means of
Quaternions; and on some}
\vskip 3pt
\centerline{\largeit other connected Subjects.}
\vskip 6pt
\centerline{{\largeit By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}
\bigskip
\centerline{Communicated July~20, 1846.}
\bigskip
\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~3 (1847), pp.\ 273--292.]}
\bigskip
Sir William R. Hamilton read a paper on the expression and
proof of Pascal's theorem by means of quaternions; and on some
other connected subjects.
This proof of the theorem of Pascal depends on the following form
of the general equation of cones of the second degree:
$${\sc s} \mathbin{.} \beta \beta' \beta'' = 0;
\eqno (1)$$
in which
$$\left. \eqalign{
\beta
&= {\sc v} (
{\sc v} \mathbin{.} \alpha \alpha' \mathbin{.}
{\sc v} \mathbin{.} \alpha''' \alpha^{\rm IV} ),\cr
\beta'
&= {\sc v} (
{\sc v} \mathbin{.} \alpha' \alpha'' \mathbin{.}
{\sc v} \mathbin{.} \alpha^{\rm IV} \alpha^{\rm V} ),\cr
\beta''
&= {\sc v} (
{\sc v} \mathbin{.} \alpha'' \alpha''' \mathbin{.}
{\sc v} \mathbin{.} \alpha^{\rm V} \alpha ),\cr}
\right\}
\eqno (2)$$
$\alpha$, $\alpha'$, $\alpha''$, $\alpha'''$, $\alpha^{\rm IV}$,
$\alpha^{\rm V}$, being any six homoconic vectors, and
${\sc s}$,~${\sc v}$, being characteristics of the operations of
taking separately the scalar and vector parts of a quaternion.
In all these geometrical applications of quaternions, it is to be
remembered that {\it the product of two opposite vectors is a
positive number}, namely the product of the numbers expressing
the lengths of the two factors; and that the {\it product of two
rectangular vectors is a third vector rectangular to both}, and
such that the rotation round it, from the multiplier to the
multiplicand, is positive. These conceptions, or definitions, of
geometrical multiplication, are essential in the theory of
quaternions, and are hitherto (so far as Sir William Hamilton
knows) peculiar to it. If they be adopted, they oblige us to
regard the {\it product\/} (or the quotient) {\it of two inclined
vectors\/} (neither parallel nor perpendicular to each other), as
being {\it partly a number and partly a line\/}; on which account
a quaternion, generally, as being always, in its geometrical
aspect, a product (or quotient) of two lines, may perhaps not
improperly be also called a {\sc grammarithm} (by a combination
of the two Greek words
$\gamma \rho \alpha \mu \mu \acute{\eta}$ and
$\overcomma{\alpha} \rho \iota \theta \mu \acute{o} \varsigma$,
which signify respectively a {\it line\/} and a {\it number\/}).
In this phraseology, the scalar part of a quaternion would be the
arithmic part of a grammarithm; and the vector part of a
quaternion would be the grammic part of a grammarithm. In the
form given above, of the general equation of cones of the second
degree, the six symbols, $\alpha,\ldots, \alpha^{\rm V}$ denote
six edges of a hexahedral angle inscribed in such a cone; the six
binary products
$\alpha \alpha',\ldots, \alpha^{\rm V} \alpha$,
of those lines taken in their order, are grammarithms, of which
the symbols ${\sc v} \mathbin{.} \alpha \alpha'$, \&c., denote
the grammic parts, namely, certain lines perpendicular
respectively to the six plane faces of the angle; the three
products
$$ {\sc v} \mathbin{.} \alpha \alpha' \mathbin{.}
{\sc v} \mathbin{.} \alpha''' \alpha^{\rm IV},
\quad\hbox{\&c.},$$
of normals to opposite faces, are again grammarithms, of which
the grammic parts are the three lines
$\beta$,~$\beta'$,~$\beta''$, situated respectively in the
intersections of the three pairs of opposite faces of the angle
inscribed in the cone; and the equation~(1) of that cone, which
expresses that the arithmic part of the product of these three
lines vanishes, shows also, by the principles of this theory,
that these lines themselves are {\it coplanar\/}; which is a form
of the theorem of Pascal.
The rules of this calculus of grammarithms, or of quaternions,
give, generally, for the arithmic or scalar part of the product
of the vector parts of the three products of any six lines or
vectors $\alpha \alpha'$, $\beta \beta'$, $\gamma \gamma'$,
taken two by two, the following transformed expression:
$${\sc s} (
{\sc v} \mathbin{.} \alpha \alpha' \mathbin{.}
{\sc v} \mathbin{.} \beta \beta' \mathbin{.}
{\sc v} \mathbin{.} \gamma \gamma' )
= {\sc s} \mathbin{.} \alpha \gamma \gamma' \mathbin{.}
{\sc s} \mathbin{.} \alpha' \beta \beta'
- {\sc s} \mathbin{.} \alpha' \gamma \gamma' \mathbin{.}
{\sc s} \mathbin{.} \alpha \beta \beta';
\eqno (3)$$
and by applying this general transformation to the recent
results, we find easily, that the equation~(1), under the
conditions~(2), may be put under the form:
$$ {{\sc s} \mathbin{.}
\alpha \alpha' \alpha''
\over {\sc s} \mathbin{.}
\alpha \alpha''' \alpha''}
\mathbin{.}
{{\sc s} \mathbin{.}
\alpha'' \alpha''' \alpha^{\rm IV}
\over {\sc s} \mathbin{.}
\alpha'' \alpha' \alpha^{\rm IV}}
= {{\sc s} \mathbin{.}
\alpha \alpha' \alpha^{\rm V}
\over {\sc s} \mathbin{.}
\alpha \alpha''' \alpha^{\rm V}}
\mathbin{.}
{{\sc s} \mathbin{.}
\alpha^{\rm V} \alpha''' \alpha^{\rm IV}
\over {\sc s} \mathbin{.}
\alpha^{\rm V} \alpha' \alpha^{\rm IV}};
\eqno (4)$$
which is another mode of expressing by quaternions the general
condition required, in order that six vectors
$\alpha,\ldots, \alpha^{\rm V}$, diverging from one common
origin, may all be sides of one common cone of the second degree.
The summit of this cone, or the common initial point of each of
these six vectors, being called~${\sc o}$, let the six final
points be
${\sc a} \, {\sc b} \, {\sc c} \, {\sc d} \, {\sc e} \, {\sc c}'$:
the transformed {\it equation of homoconicism\/} (4), expresses
that the {\it ratio compounded of the two ratios of the two
pyramids
${\sc o} {\sc a} {\sc b} {\sc c}$,
${\sc o} {\sc c} {\sc d} {\sc e}$,
to the two other pyramids
${\sc o} {\sc a} {\sc d} {\sc c}$,
${\sc o} {\sc c} {\sc b} {\sc e}$,
does not change when we pass from the point~${\sc c}$ to any
other point~${\sc c}'$ on the same cone of the second degree\/}:
which is a form of the theorem of M.~Chasles, respecting the
constancy of the anharmonic ratio. An intimate connexion between
this theorem and that of Pascal is thus exhibited, by this
symbolical process of transformation.
As the equation~(1) expresses that the {\it three\/} vectors
$\beta \, \beta' \, \beta''$ are {\it coplanar}, or that they are
contained on one common plane, if they diverge from one common
origin, and as the equation~(4) expresses that the {\it six\/}
vectors $\alpha,\ldots, \alpha^{\rm V}$ are {\it homoconic}, so
does this other equation,
$${\sc s} \mathbin{.} \rho (\rho - \gamma) (\gamma - \beta)
(\beta - \alpha) \alpha = 0,
\eqno (5)$$
express that the {\it four\/} vectors
$\alpha$,~$\beta$,~$\gamma$,~$\rho$ are {\it homosph{\ae}ric}, or
that they may be regarded as representing, in length and in
direction, {\it four diverging chords of one common sphere}.
Thus, the {\it arithmic part of the continued product of the five
successive sides of any rectilinear\/} (but not necessarily
plane) {\it pentagon, inscribed in a sphere, is zero\/}; and
conversely, if in any investigation respecting any rectilinear,
but, generally, uneven, pentagon
${\sc a} {\sc b} {\sc c} {\sc d} {\sc e}$
in space, the product
${\sc a} {\sc b} \times {\sc b} {\sc c} \times {\sc c} {\sc d}
\times {\sc d} {\sc e} \times {\sc e} {\sc a}$
of five successive sides, when determined by the rules of the
present calculus, is found to be a pure vector, or can be
entirely constructed by a {\it line}, so that in a notation
already submitted to the Academy (see account of the
communication made in last December) the equation
$${\sc s} \mathbin{.}
{\sc a} {\sc b} {\sc c} {\sc d} {\sc e} {\sc a}
= 0,
\eqno (6)$$
is found to be satisfied, we may then infer that the five
corners, ${\sc a}$,~${\sc b}$,~${\sc c}$,~${\sc d}$,~${\sc e}$,
of this pentagon, are situated on the surface of one common
sphere. This {\it equation of homosph{\ae}ricism\/} (5) or (6),
appears to the present author to be very fertile in its
consequences. To leave no doubt respecting its {\it meaning},
and to present it under a form under which it may be easily
understood by those who have not yet made themselves masters of
the whole of the theory, it may be stated thus: if we write for
abridgment,
$$\left. \eqalign{
\alpha_1 &= i (x_1 - x_2) + j (y_1 - y_2) + k (z_1 - z_2),\cr
\alpha_2 &= i (x_2 - x_3) + j (y_2 - y_3) + k (z_2 - z_3),\cr
\alpha_3 &= i (x_3 - x_4) + j (y_3 - y_4) + k (z_3 - z_4),\cr
\alpha_4 &= i (x_4 - x_5) + j (y_4 - y_5) + k (z_4 - z_5),\cr
\alpha_5 &= i (x_5 - x_1) + j (y_5 - y_1) + k (z_5 - z_1),\cr}
\right\}
\eqno (7)$$
and then develop the continued product of these five expressions,
{\it using the distributive, but not\/} (so far as relates to
$i$~$j$~$k$) {\it the commutative property of multiplication},
and reducing the result to the form of a quaternion,
$$\alpha_1 \alpha_2 \alpha_3 \alpha_4 \alpha_5
= w + ix + jy + kz,
\eqno (8)$$
by the fundamental symbolical relations between the {\it three
coordinate characteristics\/} $i$~$j$~$k$, which were
communicated to the Academy by Sir William Hamilton in November,
1843, and which may be thus concisely stated:\footnote*{These
fundamental equations between the author's symbols $i$,~$j$,~$k$,
appeared, under a slightly more developed form, in the number of
the {\it London, Edinburgh and Dublin Philosophical Magazine\/}
for July, 1844; in which Magazine the author has continued to
publish, from time to time, some articles of a Paper on
Quaternions; reserving, however, for the Transactions of the
Royal Irish Academy, a more complete and systematic account of
his researches on this extensive subject.}
$$i^2 = j^2 = k^2 = ijk = -1;
\eqno ({\rm A})$$
and if we find, as the result of this calculation, that the
term~$w$, or the part of the quaternion~(8) which is independent
of the characteristics $i \, j \, k$, vanishes, so that we have
the following equation, which is entirely freed from those
symbolic factors,
$$w = 0,
\eqno (9)$$
we shall then know that the points, of which the rectangular
coordinates are respectively
$(x_1 \, y_1 \, z_1)$
$(x_2 \, y_2 \, z_2)$
$(x_3 \, y_3 \, z_3)$
$(x_4 \, y_4 \, z_4)$
$(x_5 \, y_5 \, z_5)$,
are {\it five homosph{\ae}ric points}, or that one common spheric
surface will contain them all.
The actual process of this multiplication and reduction would be
tedious, nor is it offered as the easiest, but only as
{\it one\/} way of forming the equation in rectangular
coordinates, which is here denoted by (9). A much easier way
would be to prepare the equation~(5) by a previous development,
so as to put it under the following form:
$$\rho^2 \, {\sc s} \mathbin{.} \alpha \beta \gamma
= \alpha^2 \, {\sc s} \mathbin{.} \beta \gamma \rho
+ \beta^2 \, {\sc s} \mathbin{.} \gamma \alpha \rho
+ \gamma^2 \, {\sc s} \mathbin{.} \alpha \beta \rho;
\eqno (10)$$
which also admits of a simple geometrical interpretation. For,
by comparing it with the following equation, which is in this
calculus an {\it identical\/} one, or is satisfied by {\it any\/}
four vectors, $\alpha$,~$\beta$,~$\gamma$,~$\rho$:
$$\rho \, {\sc s} \mathbin{.} \alpha \beta \gamma
= \alpha \, {\sc s} \mathbin{.} \beta \gamma \rho
+ \beta \, {\sc s} \mathbin{.} \gamma \alpha \rho
+ \gamma \, {\sc s} \mathbin{.} \alpha \beta \rho,
\eqno (11)$$
we find that the form (10) gives
$$\rho^2 = \alpha \alpha' + \beta \beta' + \gamma \gamma',
\eqno (12)$$
if $\alpha'$,~$\beta'$,~$\gamma'$ denote three diverging edges of
a parallelepiped, of which the intermediate diagonal (or their
symbolic sum) is the chord~$\rho$ of a sphere, while
$\alpha \, \beta \, \gamma$ are three other chords of the same
sphere, in the directions of the three edges, and coinitial with
them and with~$\rho$; so that {\it the square on the
diagonal~$\rho$ is equal to the sum of the three rectangles under
the three edges $\alpha' \, \beta' \, \gamma'$ and the three
chords\/} $\alpha \, \beta \, \gamma$, with which, in direction,
those edges respectively coincide. This theorem is only
mentioned here, as a simple example of the {\it interpretation\/}
of the formulae to which the present method conducts; since the
same result may be obtained very simply from a more ordinary form
of the equation of the sphere, referred to the edges
$\alpha' \, \beta' \, \gamma'$ as oblique coordinates; and
doubtless, has been already obtained in that or in some other
way. An analogous theorem for the ellipsoid may be obtained with
little difficulty.
If we suppose in the formula~(6), that the point~${\sc e}$ of the
pentagon approaches to the point~${\sc a}$, the
side~${\sc e} {\sc a}$ tends to become an infinitely small
tangent to the sphere; and thus we find that
${\sc v} \mathbin{.} {\sc a} {\sc b} {\sc c} {\sc d} {\sc a}$,
or that {\it the vector part of the continued product
${\sc a} {\sc b} \times {\sc b} {\sc c} \times {\sc c} {\sc d}
\times {\sc d} {\sc a}$,
of the four sides of an uneven\/} (or {\it gauche\/})
{\it quadrilateral\/} ${\sc a} {\sc b} {\sc c} {\sc d}$, if
determined by the rules of multiplication proper to this
calculus, {\it is normal to the circumscribed sphere\/} at the
point~${\sc a}$, where the first and fourth sides are supposed to
meet. By the non-commutative character of quaternion
multiplication, we should get a different product, if we took the
factors in the order
${\sc b} {\sc c} \times {\sc c} {\sc d} \times {\sc d} {\sc a}
\times {\sc a} {\sc b}$;
and accordingly the vector or grammic part
${\sc v} \mathbin{.} {\sc b} {\sc c} {\sc d} {\sc a} {\sc b}$
of this new quaternion product would represent a new line in
space, namely, a normal to the same sphere at ${\sc b}$: and
similarly may the normals be found at the two other corners of
the quadrilateral, by two other arrangements of the four sides as
factors. To determine the lengths of the normal lines thus
assigned, we may observe that if
${\sc a}'$~${\sc b}'$,~${\sc c}'$,~${\sc d}'$
be the four points on the same sphere, which are diametrically
opposite to the four given points
${\sc a}$~${\sc b}$,~${\sc c}$,~${\sc d}$,
then the four diameters
${\sc a}' {\sc a}$, ${\sc b}' {\sc b}$,
${\sc c}' {\sc c}$, ${\sc d}' {\sc d}$,
are given by four expressions, of which it may be sufficient to
write one, namely
$${\sc a}' {\sc a}
= {{\sc v} \mathbin{.}
{\sc a} {\sc b} {\sc c} {\sc d} {\sc a}
\over {\sc s} \mathbin{.}
{\sc a} {\sc b} {\sc c} {\sc d}}.
\eqno (13)$$
The denominator of this expression denotes (as was remarked in a
former communication) the sextuple volume of the pyramid, or
tetrahedron, ${\sc a} {\sc b} {\sc c} {\sc d}$; it vanishes,
therefore, when the four points
${\sc a}$,~${\sc b}$,~${\sc c}$,~${\sc d}$
are in one plane: so that we have for {\it any plane
quadrilateral\/} the equation,
$${\sc s} \mathbin{.} {\sc a} {\sc b} {\sc c} {\sc d}
= 0.
\eqno (14)$$
If the sphere is then to become only {\it indeterminate}, and not
necessarily infinite, we must suppose that the numerator of the
same expression~(13) also vanishes, that is, we must have in this
case the condition
$${\sc v} \mathbin{.} {\sc a} {\sc b} {\sc c} {\sc d} {\sc a}
= 0.
\eqno (15)$$
In words, as the product of the five successive sides of an
uneven but rectilinear pentagon inscribed in a sphere, has been
seen to be {\it purely a line}, so we now see that {\it the
product of the four successive sides of a quadrilateral inscribed
in a circle is\/} (in this system) {\it purely a number\/}:
whereas, for {\it every other\/} rectilinear quadrilateral, {\it
whether plane or gauche}, the grammarithm obtained as the product
of four successive sides {\it involves a grammic part}, which
does not vanish. This condition (15), for a quadrilateral
inscribable in a circle, could not be always satisfied, when
${\sc d}$ approached to ${\sc a}$, and tended to coincide with
it, unless the following theorem were also true, which can
accordingly be otherwise proved: {\it the product\/}
${\sc a} {\sc b} {\sc c} {\sc a}$, or
${\sc a} {\sc b} \times {\sc b} {\sc c} \times {\sc c} {\sc a}$,
{\it of three successive sides of any triangle
${\sc a} {\sc b} {\sc c}$, is a pure vector, in the direction of
the tangent to the circumscribed circle}, at the point~${\sc a}$,
where the sides which are assumed as first and third factors of
the product meet each other. If ${\sc a}_\prime$ be the point
upon this circumscribed circle which is diametrically opposite to
${\sc a}$, we find for the length and direction of the diameter
${\sc a} {\sc a}_\prime$ in this notation, that is, for the
straight line {\it to ${\sc a}$ from ${\sc a}_\prime$}, the
expression:\footnote*{With respect to the {\it notation of
division}, in this theory, the author proposes to distinguish
between the two symbols
$${\sc q}^{-1} {\sc q}'
\quad\hbox{and}\quad
{{\sc q} \over {\sc q}'},$$
which he inadvertently used as interchangeable in his first
communication to the Academy: and to make them satisfy the two
separate equations,
$${\sc q} \times {\sc q}^{-1} {\sc q}' = {\sc q}';$$
$${{\sc q}' \over {\sc q}} \times {\sc q} = {\sc q}'.$$
He proposes to confine the symbol ${\sc q}' \div {\sc q}$ to the
signification thus assigned for the latter of the two symbols
which have been thus defined, and which, on account of the
non-commutative property of multiplication of quaternions, ought
not to be confounded with each other.}
$${\sc a} {\sc a}_\prime
= {{\sc a} {\sc b} {\sc c} {\sc a}
\over {\sc v} \mathbin{.} {\sc a} {\sc b} {\sc c}};
\eqno (16)$$
the denominator denoting a line which is in direction
perpendicular to the plane of the triangle, and in magnitude
represents the double of its area; while the numerator is, as we
have just seen, in direction tangential to the circle at
${\sc a}$, and its length represents the product of the lengths
of the three sides, or the volume of the solid constructed with
those sides as rectangular edges. We may add, that this
tangential line ${\sc a} {\sc b} {\sc c} {\sc a}$ is
distinguished from the equally long but {\it opposite\/} tangent
${\sc a} {\sc c} {\sc b} {\sc a}$ to the same circle
${\sc a} {\sc b} {\sc c}$ at the same point~${\sc a}$, by the
condition that the former is intermediate in direction, between
${\sc a} {\sc b}$ (prolonged through ${\sc a}$) and
${\sc c} {\sc a}$, while the latter in like manner lies between
${\sc a} {\sc c}$ (prolonged) and ${\sc b} {\sc a}$: or we may
say that the line ${\sc a} {\sc b} {\sc c} {\sc a}$ touches, at
${\sc a}$, the segment {\it alternate\/} to that segment of the
circle ${\sc a} {\sc b} {\sc c}$ which has for ${\sc a} {\sc c}$
the base, and contains the point~${\sc b}$; while the opposite
line ${\sc a} {\sc b} {\sc c} {\sc a}$ touches, at the same
point, the last mentioned segment itself. The condition for the
diameter~${\sc a} {\sc a}_\prime$ becoming infinite, or for the
three points ${\sc a} {\sc b} {\sc c}$ being situated on one
common straight line, is
$${\sc v} \mathbin{.} {\sc a} {\sc b} {\sc c} = 0.
\eqno (17)$$
This formula (17) is therefore, in this notation, the {\it
general equation of a straight line\/} in space; (15) is the
general {\it equation of a circle\/}; (14) of a plane, and (6) of
a sphere.\footnote*{The simpler equation of scalar form,
${\sc s} \mathbin{.} {\sc a} {\sc b} {\sc c} = 0$,
also represents a spheric surface, if ${\sc b}$ be regarded as
the variable point; but a plane, if ${\sc b}$ be fixed, and
either ${\sc a}$ or ${\sc c}$ alone variable.}
It may seem strange that the line and circle should here be
represented each by only {\it one\/} equation; but these
equations are of {\it vector forms}, and decompose themselves
each into three equations, equivalent, however, only to two
distinct ones, when we pass to rectangular coordinates, for the
sake of comparison with known results.
In the same notation of capitals, whatever five distinct points
may be denoted by
${\sc a}$, ${\sc b}$, ${\sc c}$, ${\sc d}$, ${\sc e}$,
we have the general transformation,
$${\sc a} {\sc b} {\sc c} {\sc d} {\sc e} {\sc a}
= {\sc a} {\sc b} {\sc c} {\sc a}
\times
{\sc a} {\sc c} {\sc d} {\sc a}
\times
{\sc a} {\sc d} {\sc e} {\sc a}
\div
{\sc a} {\sc c} {\sc a} {\sc d} {\sc a},
\eqno (18)$$
in which the divisor ${\sc a} {\sc c} {\sc a} {\sc d} {\sc a}$,
or ${\sc a} {\sc c} {\sc a} \times {\sc a} {\sc d} {\sc a}$, is
the product of two positive scalars; if then we had otherwise
established the interpretation lately assigned to the
symbol~${\sc a} {\sc b} {\sc c} {\sc a}$, as denoting a line
which touches at ${\sc a}$ the circle ${\sc a} {\sc b} {\sc c}$,
we might have in that way deduced the equation~(6) of a sphere,
as the {\it condition of coplanarity of the three tangents\/} at
${\sc a}$, {\it to the three circles}, ${\sc a} {\sc b} {\sc c}$,
${\sc a} {\sc c} {\sc d}$, ${\sc a} {\sc d} {\sc e}$. And we see
that when this condition is satisfied, so that the points
${\sc a}$,~${\sc b}$,~${\sc c}$,~${\sc d}$,~${\sc e}$
are homosph{\ae}ric, and that, therefore, the symbol
${\sc a} {\sc b} {\sc c} {\sc d} {\sc e} {\sc a}$
represents a vector, we can {\it construct the direction of this
vector\/} by drawing in the plane which touches the sphere at
${\sc a}$, a line ${\sc a}_1 {\sc a}_2$ parallel to the line
${\sc a} {\sc c} {\sc d} {\sc a}$ which touches the circle
${\sc a} {\sc c} {\sc d}$ at ${\sc a}$, and cutting, in the
points ${\sc a}_1$ and ${\sc a}_2$, the two lines
${\sc a} {\sc b} {\sc c} {\sc a}$ and
${\sc a} {\sc d} {\sc e} {\sc a}$, which are drawn at ${\sc a}$
to touch the circles ${\sc a} {\sc b} {\sc c}$,
${\sc a} {\sc d} {\sc e}$; for then the vector
${\sc a} {\sc b} {\sc c} {\sc d} {\sc e} {\sc a}$,
which is thus seen to be a tangent to the sphere, will touch, at
the same point~${\sc a}$, the
circle~${\sc a} {\sc a}_1 {\sc a}_2$,
described on the tangent plane. In the more general case, when
the condition~(6) is {\it not\/} satisfied, and when, therefore,
the rectilinear pentagon
${\sc a} {\sc b} {\sc c} {\sc d} {\sc e}$,
which we shall suppose to be uneven, cannot be inscribed in a
sphere, the scalar symbol
${\sc s} \mathbin{.}
{\sc a} {\sc b} {\sc c} {\sc d} {\sc e} {\sc a}$
which has been seen to vanish, when the pentagon can be so
inscribed, represents the continued {\it product of the lengths
of the five sides
${\sc a} {\sc b}$, ${\sc b} {\sc c}$, ${\sc c} {\sc d}$,
${\sc d} {\sc e}$, ${\sc e} {\sc a}$,
multiplied by the sextuple volume of that triangular pyramid
which is constructed with three coterminous edges, each equal to
the unit of length, and touching at the vertex~${\sc a}$ the
three circles
${\sc a} {\sc b} {\sc c}$,
${\sc a} {\sc c} {\sc d}$,
${\sc a} {\sc d} {\sc e}$,
which have respectively for chords the three remote sides of the
pentagon}, and are not now homosph{\ae}ric circles. And because,
in general, in this notation, the equation
$${\sc s} \mathbin{.}
{\sc a} {\sc b} {\sc c} {\sc d} {\sc e} {\sc a}
= {\sc s} \mathbin{.}
{\sc b} {\sc c} {\sc d} {\sc e} {\sc a} {\sc b}
\eqno (19)$$
holds good, it follows that {\it for any rectilinear pentagon\/}
(in space) {\it the five triangular pyramids constructed on the
foregoing plan}, with the five corners of the pentagon for their
respective vertices, {\it have equal volumes}.
Besides the characteristics ${\sc s}$ and ${\sc v}$, which serve
to decompose a quaternion~${\sc q}$ into {\it two parts}, of
distinct and determined kinds, the author frequently finds it to
be convenient to use two other characteristics of operation,
${\sc t}$ and ${\sc u}$, which serve to decompose the same
quaternion into {\it two factors}, of kinds equally distinct and
equally determinate; in such a manner that we may write
generally, with these characteristics, for any
quaternion~${\sc q}$,
$${\sc q} = {\sc s} {\sc q} + {\sc v} {\sc q}
= {\sc t} {\sc q} \times {\sc u} {\sc q}.
\eqno (20)$$
The factor~${\sc t} {\sc q}$ is always a positive, or rather an
absolute (or {\it signless\/}) number; it is what was called by
the author, in his first communication on this subject to the
Academy, the {\it modulus}, but he has since come to prefer to
call it the {\sc tensor} of the quaternion~${\sc q}$: and he
calls the other factor~${\sc u} {\sc q}$ the {\sc versor} of the
same quaternion. As the {\it scalar of a sum is the sum of the
scalars}, and the {\it vector of a sum is the sum of the
vectors}, so the {\it tensor of a product is the product of the
tensors}, and the {\it versor of a product is the product of the
versors\/}; relations or properties which may be concisely
expressed by the formulae:
$${\sc s} \Sigma = \Sigma {\sc s};\quad
{\sc v} \Sigma = \Sigma {\sc v};
\eqno (21)$$
$${\sc t} \Pi = \Pi {\sc t};\quad
{\sc u} \Pi = \Pi {\sc u}.
\eqno (22)$$
When we operate by the characteristics ${\sc t}$ and ${\sc u}$ on
a straight line, regarded as a vector, we obtain as the
{\it tensor\/} of this line a {\it signless number\/} expressing
its {\it length\/}; and, as the {\it versor\/} of the same line,
an {\it imaginary unit}, determining its {\it direction}. When
we operate on the product
${\sc a} {\sc b} {\sc c}
= {\sc a} {\sc b} \times {\sc b} {\sc c}$
of two successive lines, regarded as a quaternion, we obtain for
the tensor,
${\sc t} \mathbin{.} {\sc a} {\sc b} {\sc c}$,
the product of the lengths of the two lines, or the area of the
rectangle under them; and for the versor of the same product of
two successive sides of a triangle (or polygon), we obtain an
expression of the form
$${\sc u} \mathbin{.} {\sc a} {\sc b} {\sc c}
= \cos {\sc b} + \sqrt{-1} \sin {\sc b};
\eqno (23)$$
the symbol~${\sc b}$ in the second member denoting the
{\it internal angle\/} of the figure at the point denoted by the
same letter, which angle is thus the {\it amplitude of the
versor}, and at the same time (in the sense of the author's first
communication) the {\it amplitude of the quaternion\/} itself,
which quaternion is here denoted by the symbol
${\sc a} {\sc b} {\sc c}$. In this theory (as was shown by the
author to the Academy in that first communication), there are
{\it infinitely many\/} different square roots of negative unity,
constructed by lines equal to each other, and to the unit of
length, but distinguishable by their {\it directional\/} (or
{\it polar\/}) coordinates: the {\it particular\/} $\sqrt{-1}$
which enters into the expression~(23) is perpendicular to the
plane of the triangle ${\sc a} {\sc b} {\sc c}$. It is the
{\it versor of the vector\/} of that quaternion which is denoted
by the same symbol ${\sc a} {\sc b} {\sc c}$; and it may,
therefore, be replaced by the symbol
${\sc u} {\sc v} \mathbin{.} {\sc a} {\sc b} {\sc c}$,
which we may agree to abridge to
${\sc w} \mathbin{.} {\sc a} {\sc b} {\sc c}$, so that we may
establish the symbolic equation:
$${\sc u} {\sc v} {\sc q} = {\sc w} {\sc q},
\quad\hbox{or simply,}\quad
{\sc u} {\sc v} = {\sc w};
\eqno (24)$$
we may also call ${\sc w} {\sc q}$ the {\it vector unit\/} of the
quaternion~${\sc q}$. The expression~(23) suggests also the
denoting the {\it amplitude\/} of {\it any\/} quaternion by the
geometrical mark for an {\it angle}, which notation will also
agree with the original conception of such an amplitude; and thus
we are led to write, generally, as a transformed expression for a
versor,
$${\sc u} {\sc q}
= \cos \angle {\sc q}
+ {\sc w} {\sc q} \mathbin{.} \sin \angle {\sc q}.
\eqno (25)$$
The amplitude of a vector is in this theory a quadrant; that of a
positive number being, as usual, zero, and that of a negative
number two right angles. Applying the same principles and
notation to the case of the continued product
${\sc a} {\sc b} {\sc c} {\sc d} {\sc a}$
of the four successive sides of an uneven quadrilateral
${\sc a} {\sc b} {\sc c} {\sc d}$,
we find that the amplitude
$\angle {\sc a} {\sc b} {\sc c} {\sc d} {\sc a}$
of this quaternion product is equal to the {\it angle of the
lunule\/} ${\sc a} {\sc b} {\sc c} {\sc d} {\sc a}$, if we employ
this term ``lunule'' to denote a portion of a spherical surface
bounded by two arcs (which may be greater than halves) of
{\it small\/} circles, namely, here, the portion of the surface
of the sphere circumscribed about the quadrilateral
${\sc a} {\sc b} {\sc c} {\sc d}$, which portion is bounded by
the two arcs that go from the corner~${\sc a}$ of that
quadrilateral to the opposite corner~${\sc c}$, and which pass
respectively through the two other corners ${\sc b}$ and
${\sc d}$. The tensor and scalar of the continued product of the
four sides of the quadrilateral do not change when the sides are
taken in the order, second, third, fourth, first; and generally,
$$\cos \angle {\sc q} = {\sc s} {\sc q} \div {\sc t} {\sc q};
\eqno (26)$$
so that we have the equation
$$\angle {\sc a} {\sc b} {\sc c} {\sc d} {\sc a}
= \angle {\sc b} {\sc c} {\sc d} {\sc a} {\sc b};
\eqno (27)$$
hence the {\it two lunules\/}
${\sc a} {\sc b} {\sc c} {\sc d} {\sc a}$ and
${\sc b} {\sc c} {\sc d} {\sc a} {\sc b}$, which have for their
{\it diagonals\/} ${\sc a} {\sc c}$ and ${\sc b} {\sc d}$ the two
diagonals of the quadrilateral, and with which the lunules
${\sc c} {\sc d} {\sc a} {\sc b} {\sc c}$ and
${\sc d} {\sc a} {\sc b} {\sc c} {\sc d}$ respectively coincide,
are {\it mutually equiangular\/} at ${\sc a}$ and ${\sc b}$.
Thus, generally, for {\it any four points},
${\sc a} \, {\sc b} \, {\sc c} \, {\sc d}$, the two circles
${\sc a} {\sc b} {\sc c}$, ${\sc a} {\sc d} {\sc c}$ cross each
other at ${\sc a}$ and ${\sc c}$ (in space, or on one plane),
under the same angles as the two other circles,
${\sc b} {\sc c} {\sc d}$, ${\sc b} {\sc a} {\sc d}$,
at ${\sc b}$ and ${\sc d}$.
Again, it may be remarked, that the condition for a fifth
point~${\sc e}$ being contained on the plane which touches, at
${\sc a}$, the sphere circumscribed about the tetrahedron
${\sc a} {\sc b} {\sc c} {\sc d}$, is expressed by the equation
$${\sc s} \mathbin{.}
{\sc a} {\sc b} {\sc c} {\sc d} {\sc a} {\sc e}
= 0;
\eqno (28)$$
this equation, therefore, ought not to be compatible with the
equation~(6), which expressed that the point~${\sc e}$ was on the
sphere itself, except by supposing that the point~${\sc e}$
coincides with the point of contact~${\sc a}$; and accordingly
the principles and rules of this notation give, generally,
$${\sc s} \mathbin{.}
{\sc a} {\sc b} {\sc c} {\sc d} {\sc e} {\sc a}
+ {\sc s} \mathbin{.}
{\sc a} {\sc b} {\sc c} {\sc d} {\sc a} {\sc e}
= {\sc s} \mathbin{.}
{\sc a} {\sc b} {\sc c} {\sc d} \mathbin{.}
{\sc a} {\sc e} {\sc a},
\eqno (29)$$
in which, by (14) the first factor
${\sc s} \mathbin{.} {\sc a} {\sc b} {\sc c} {\sc d}$
of the second member does not vanish if the sphere be finite,
that is, if the volume of the tetrahedron does not vanish, while
the second factor may be thus transformed,
$${\sc a} {\sc e} {\sc a} = - ({\sc e} {\sc a})^2,
\eqno (30)$$
so that the coexistence of the two equations (6) and (28) of the
sphere and its tangent plane, is thus seen to require that we
shall have
$${\sc e} {\sc a} = 0;
\eqno (31)$$
which is, relatively to the sought position of ${\sc e}$, the
{\it equation of the point of contact}. These examples, though
not the most important that might be selected, may suffice to
show that there already exists a {\it calculus}, which may
deserve to be further developed, for combining and transforming
geometrical expressions of this sort. Several of the elements of
such a calculus, especially as regards geometrical {\it
addition\/} and {\it subtraction}, have been contributed by
other, and (as the author willingly believes) by better
geometers; what Sir William Hamilton considers to be peculiarly
his own contribution to this department of mathematical and
symbolical science consists of the introduction and development
of those conceptions of {\sc geometrical multiplication} (and
division), which were embodied by him (in 1843) in his
fundamental formulae for the {\it symbolic squares and products
of the three coordinate characteristics\/} (or algebraically
imaginary units) $i$,~$j$,~$k$, which entered into his original
expression of a {\sc quaternion} ($w + i x + j y + k z$), and by
which he succeeded in representing, symmetrically, that is,
{\it without any selection of one direction as eminent}, the
three dimensions of space.
It is, however, convenient, in many researches, to retain the
notation in which Greek letters denote {\it vectors}, instead of
employing that other notation, in which capital letters (a few
{\it characteristics\/} excepted), denote {\it points}. In the
former notation it was shown to the Academy in last December (see
formula (21) of the abstract of the author's communication of
that date), that the equation of an ellipsoid, with three unequal
axes, referred to its centre as the origin of vectors, may be put
under the form:\footnote*{Appendix, No.~V., page lviii.
[{\it Proceedings of the Royal Irish Academy}, vol.~3.]}
$$(\alpha \rho + \rho \alpha)^2 - (\beta \rho - \rho \beta)^2
= 1;$$
$\rho$ being the variable vector of the ellipsoid, and $\beta$
and $\alpha$ being two constant vectors, in the directions
respectively of the axes of one of the two circumscribed
cylinders of revolution, and of a normal to the plane of the
corresponding ellipse of contact. Decomposing the first member
of that equation of an ellipsoid into two factors of the first
degree, or writing the equation as follows:
$$ (\alpha \rho + \rho \alpha + \beta \rho - \rho \beta)
(\alpha \rho + \rho \alpha - \beta \rho + \rho \beta)
= 1,
\eqno (32)$$
we may observe that these two factors, which are thus
{\it separately linear\/} with respect to the variable
vector~$\rho$, are at the same time {\it conjugate
quaternions\/}; if we call two quaternions, ${\sc q}$ and
${\sc k} {\sc q}$, {\sc conjugate}, when they have {\it equal
scalars\/} but have {\it opposite vectors}, so that generally,
$${\sc k} {\sc q} = {\sc s} {\sc q} - {\sc v} {\sc q},
\quad\hbox{or, more concisely,}\quad
{\sc k} = {\sc s} - {\sc v}.
\eqno (33)$$
And if we further observe, that in general the {\it product of
two conjugate quaternions\/} is equal to {\it the square of their
common tensor},
$${\sc q} \times {\sc k} {\sc q}
= ({\sc s} {\sc q})^2 - ({\sc v} {\sc q})^2
= ({\sc t} {\sc q})^2,
\eqno (34)$$
we shall perceive that the equation~(32) of an ellipsoid may be
put, by extraction of a square root, under this simpler, but not
less general form:
$${\sc t} (\alpha \rho + \rho \alpha + \beta \rho - \rho \beta)
= 1.
\eqno (35)$$
Again, by employing the principle, that
${\sc t} \Pi = \Pi {\sc t}$, we may again decompose the first
member of (35) into two factors, and may write the {\it equation
of an ellipsoid\/} thus:
$${\sc t} (\alpha + \beta + \sigma) \mathbin{.} {\sc t} \rho
= 1,
\eqno (36)$$
if we introduce an auxiliary vector,~$\sigma$, connected with the
vector~$\rho$ by the relation
$$\sigma = \rho (\alpha - \beta) \rho^{-1},
\eqno (37)$$
which gives, by the same principle respecting the tensor of a
product,
$${\sc t} \sigma = {\sc t} (\alpha - \beta);
\eqno (38)$$
so that the auxiliary vector~$\sigma$ has a {\it constant
length}, although it has by (37) a {\it variable direction},
depending on, and in its turn assisting to determine or construct
the direction of the vector~$\rho$ of the ellipsoid; for the same
equation~(37) gives for the {\it versor\/} of that vector the
expression
$${\sc u} \rho = \pm {\sc u} (\alpha - \beta + \sigma).
\eqno (39)$$
Hence, by the second general decomposition~(20), and by the
equation~(36), the last-mentioned vector~$\rho$ may be expressed
as follows:
$$\rho = {{\sc u} (\alpha - \beta + \sigma)
\over {\sc t} (\alpha + \beta + \sigma)};
\eqno (40)$$
making then, in the notation of capital letters for points,
$$\alpha + \beta = {\sc c} {\sc b},\quad
\alpha - \beta = {\sc c} {\sc a},\quad
\sigma = {\sc d} {\sc c},\quad
\rho = {\sc e} {\sc a},
\eqno (41)$$
so that ${\sc a}$ is the centre of the ellipsoid, ${\sc e}$ a
variable point on its surface, ${\sc c}$ the fixed centre of an
auxiliary sphere, of which the surface passes through the fixed
point~${\sc a}$, and also through the auxiliary and variable
point~${\sc d}$, while ${\sc b}$ is another fixed point, we
obtain the equation:
$${\sc e} {\sc a}
= \pm {\sc u} \mathbin{.} {\sc d} {\sc a}
\div {\sc t} \mathbin{.} {\sc d} {\sc b};
\eqno (42)$$
which gives
$$({\sc e} {\sc a})^{-1}
= \mp {\sc u} \mathbin{.} {\sc d} {\sc a} \mathbin{.}
{\sc t} \mathbin{.} {\sc d} {\sc b},
\eqno (43)$$
and shows, therefore, that the {\it proximity
$({\sc e} {\sc a})^{-1}$ of a variable point~${\sc e}$, on the
surface of an ellipsoid, to the centre~${\sc a}$ of that
ellipsoid, is represented in direction by a variable
chord~${\sc d} {\sc a}$ of a fixed sphere}, of which one
extremity~${\sc a}$ is fixed, while the {\it magnitude\/} of the
same proximity, or the {\it degree of nearness\/} (increasing as
${\sc e}$ approaches to the centre~${\sc a}$, and diminishing as
it recedes), is {\it represented by the
distance~${\sc d} {\sc b}$ of the other extremity~${\sc d}$ of
the same chord~${\sc d} {\sc a}$ from another fixed
point\/}~${\sc b}$ which may be supposed to be external to the
sphere. This use of the {\it word\/} ``proximity,'' which
appears to be a very convenient one, is borrowed from Sir John
Herschel: the {\it construction for the ellipsoid\/} is perhaps
new, and may be also thus enunciated:---From a fixed
point~${\sc a}$ on the surface of a sphere, draw a variable
chord~${\sc d} {\sc a}$; let ${\sc d}'$ be the second point of
intersection of the spheric surface with the
secant~${\sc d} {\sc b}$, drawn to the variable
extremity~${\sc d}$ of this chord from a fixed external
point~${\sc b}$; take the radius vector~${\sc e} {\sc a}$ equal
in length to ${\sc d}' {\sc b}$, and in direction either
coincident with, or opposite to, the chord~${\sc d} {\sc a}$;
{\it the locus of the point~${\sc e}$, thus constructed, will be
an ellipsoid}, which will pass through the point~${\sc b}$. This
fixed point~${\sc b}$ (one of four known points upon the
principal ellipse) may, perhaps, be fitly called a
{\sc pole}, and the line~${\sc b} {\sc e}$ a {\it polar chord},
of the ellipsoid; and in the construction just stated, the two
variable points ${\sc d}$,~${\sc d}'$ may be said to be
{\it conjugate guide-points}, at the extremities of coinitial and
{\it conjugate guide-chords\/}
${\sc d} {\sc a}$,~${\sc d}' {\sc a}$ of a fixed
{\it guide-sphere}, which passes through the centre~${\sc a}$ of
the ellipsoid.
We may also say, that {\it if of a quadrilateral\/}
(${\sc a} {\sc b} {\sc e} {\sc d}'$) {\it of which one side\/}
(${\sc a} {\sc b}$) {\it is given in length and in position, the
two diagonals\/} (${\sc a} {\sc e}$, ${\sc b} {\sc d}'$) {\it be
equal to each other in length, and intersect\/} (in ${\sc d}$)
{\it on the surface of a given sphere\/} (with centre~${\sc c}$),
{\it of which a chord\/} (${\sc a} {\sc d}'$) {\it is a side of
the quadrilateral adjacent to the given side\/}
(${\sc a} {\sc b}$), {\it then the other side\/}
(${\sc b} {\sc e}$), {\it adjacent to the same given side, is a
(polar) chord of a given ellipsoid\/}: of which last surface, the
form, position, and magnitude, are thus seen to depend on the
form, position, and magnitude, of what may, therefore, be called
the {\it generating triangle\/} ${\sc a} {\sc b} {\sc c}$. Two
sides of this triangle, namely, ${\sc b} {\sc c}$ and
${\sc c} {\sc a}$, are perpendicular to the two planes of
{\it circular section\/}; and the third side~${\sc a} {\sc b}$ is
perpendicular to one of the two planes of {\it circular
projection\/} of the ellipsoid, being the axis of revolution of a
circumscribed circular cylinder. Many fundamental properties of
the ellipsoid may be deduced with extreme facility, as
geometrical\footnote*{For the following geometrical corollary,
from the construction assigned above, the author is indebted to
the Rev.\ J.~W. Stubbs, Fellow of Trinity College. If the
auxiliary point~${\sc d}$ describe, on the sphere, a circle of
which the plane is perpendicular to ${\sc b} {\sc c}$, the
point~${\sc e}$ on the ellipsoid will describe a spherical
conic.}
consequences of this mode of generation; for example, the
well-known proportionality of the difference of the squares of
the reciprocals of the semi-axes of a diametral section to the
product of the sines of the inclinations of its plane to the two
planes of circular section, presents itself under the form of a
proportionality of the same difference of squares to the
rectangle under the projections of the two sides
${\sc b} {\sc c}$ and ${\sc c} {\sc a}$ of the generating
triangle on the plane of the elliptic section.
If we put the equation~(35) of an ellipsoid under the form
$${\sc t} (\iota \rho + \rho \kappa) = \kappa^2 - \iota^2,
\eqno (44)$$
the constant vectors $\iota$ and $\kappa$ will be in the
directions of the normals to the planes of circular section, and
may represent the two sides ${\sc b} {\sc c}$ and
${\sc a} {\sc c}$ of the triangle, while $\iota - \kappa$ will be
one value of the variable vector~$\rho$ or ${\sc e} {\sc a}$,
namely, the remaining side of the same triangle, or the
semi-diameter ${\sc b} {\sc a}$ in the last-mentioned
construction of the surface; and by applying to this
equation~(44) the general methods which the author has
established for investigating by quaternions the tangent planes
and curvatures of surfaces, it is found that the {\it vector of
proximity\/}~$\nu$ of the tangent plane to the centre of the
ellipsoid (that is, the reciprocal of the perpendicular let fall
on this plane from this centre), is determined in length and in
direction by the equation,
$$(\kappa^2 - \iota^2)^2 \nu
= (\kappa^2 + \iota^2) \rho
+ \iota \rho \kappa + \kappa \rho \iota;
\eqno (45)$$
while the two rectangular directions of a vector~$\tau$,
tangential to a line of curvature, at the extremity of the
vector~$\rho$, are determined by the system of equations:
$$\nu \tau + \tau \nu = 0;\quad
\nu \tau \iota \tau \kappa - \kappa \tau \iota \tau \nu = 0;
\eqno (46)$$
which may be thus written:
$${\sc s} \mathbin{.} \nu \tau = 0;\quad
{\sc s} \mathbin{.} \nu \tau \iota \tau \kappa = 0.
\eqno (47)$$
Of these two equations (46) or (47), the former expresses merely
that the tangential vector~$\tau$ is perpendicular to the normal
vector~$\nu$; while the latter is found to express that the
tangent to either line of curvature of an ellipsoid is equally
inclined to the two traces of the planes of circular section on the
tangent plane, and therefore bisects one pair of the angles
formed by the two circular sections themselves, which pass
through the given point of contact. Indeed, it is easy to prove
this relation of bisection otherwise, not only for the ellipsoid,
but for the hyperboloids, by considering the common sphere which
contains the circular sections last mentioned; the author
believes that the result has been given in one of the excellent
geometrical works of M.~Chasles; it may also be derived without
difficulty from principles stated in the masterly Memoir on
Surfaces of the Second Order, which has been published by
Professor Mac Cullagh in the Proceedings of this Academy. (See
Part VIII, page 484.)
The length to which the present abstract has already extended,
prevents Sir William Hamilton from offering on the present
occasion any details respecting the processes (analogous in some
respects to the calculi of variations and partial differentials)
by which he applies the principles of his own method to
investigations respecting surfaces and curves in space, or to
physical problems connected therewith; he desires, however, to
mention here that, in investigations respecting normals to
surfaces, he finds it convenient to employ a new characteristic
of operation of the form
$$({\sc s} \mathbin{.} {\rm d} \rho)^{-1} \mathbin{.} {\rm d}
= \reversedD,
\eqno (48)$$
in order to obtain from a scalar function of a variable
vector~$\nu$ which shall be normal to the locus for which that
scalar function is constant; and that the following more general
characteristic of operation,
$$ i {{\rm d} \over {\rm d} x}
+ j {{\rm d} \over {\rm d} y}
+ k {{\rm d} \over {\rm d} z}
= \vecd,
\eqno (49)$$
in which $x$,~$y$,~$z$ are ordinary rectangular coordinates,
while $i$,~$j$,~$k$ are his own coordinate imaginary units,
appears to him to be one of great importance in many researches.
This will be felt (he thinks) as soon as it is perceived that
with this meaning of $\vecd$ the equation
$$ \left( {{\rm d} \over {\rm d} x} \right)^2
+ \left( {{\rm d} \over {\rm d} y} \right)^2
+ \left( {{\rm d} \over {\rm d} z} \right)^2
= - \vecd^2,
\eqno (50)$$
is satisfied in virtue of the fundamental relations between the
symbols $i$,~$j$,~$k$; which relations give also, as another
result of operating with the same characteristic, this other
important symbolic expression, which presents itself under the
form of a quaternion:
$$\vecd (it + ju + kv)
= - \left(
{{\rm d} t \over {\rm d} x}
+ {{\rm d} u \over {\rm d} y}
+ {{\rm d} v \over {\rm d} z}
\right)
+ i \left(
{{\rm d} v \over {\rm d} y}
- {{\rm d} u \over {\rm d} z}
\right)
+ j \left(
{{\rm d} t \over {\rm d} z}
- {{\rm d} v \over {\rm d} x}
\right)
+ k \left(
{{\rm d} u \over {\rm d} x}
- {{\rm d} t \over {\rm d} y}
\right).
\eqno (51)$$
\bye