% This paper has been transcribed in Plain TeX by
% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
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% Trinity College, 1st June 1999.
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\centerline{\Largebf ON GEOMETRICAL NETS IN SPACE}
\vskip24pt
\centerline{\Largebf By}
\vskip24pt
\centerline{\Largebf William Rowan Hamilton}
\vskip24pt
\centerline{\largerm (Proceedings of the Royal Irish Academy,
7 (1862), pp.~532--582.)}
\vskip36pt
\vfill
\centerline{\largerm Edited by David R. Wilkins}
\vskip 12pt
\centerline{\largerm 1999}
\vskip36pt\eject
\pageno=-1
\null\vskip36pt
\centerline{\Largebf NOTE ON THE TEXT}
\bigskip
The paper {\it On Geometrical Nets in Space\/} by Sir William
Rowan Hamilton was originally published in the seventh volume of
the {\it Proceedings of the Royal Irish Academy}. The following
typographical errors in that text have been corrected in this
edition:---
\smallskip
\item{}
in article [13.], equation (47), ${\sc f}_1$ has been corrected
to ${\sc f}'$;
\smallskip
\item{}
in article [21.], equation (100), a missing overline has been
placed over the 3rd occurrence of $\sigma$ on the line.
\smallskip
\item{}
in article [33.], `$($' has been corrected to `$[$' in the
ternary symbol for the plane
${\sc a} {\sc a}' {\sc d}_1 {\sc c}_1 {\sc b}_2$;
\smallskip
\item{}
in article [56.], the final point in the list of points on the
third typical line ${\sc b}' {\sc c}'$ was printed as
${\sc a}^{\rm IX}$, but this has been corrected to
${\sc a}_1^{\rm IX}$;
\smallskip
\item{}
in article [58.], the ternary symbol for the point ${\sc b}_0$ on
the line $[0 \, 1 \, 1]$ was printed as $(\overline{1} \, 1 \,1)$,
but has been corrected to $(1 \, \overline{1} \, 1)$;
\smallskip
\item{}
in article [74.], the ternary symbol for the line
${\sc a} {\sc a}'$ was printed as $[0 \, 1 \, 1]$, but this
has been corrected to $[0 \, 1 \, \overline{1}]$;
\smallskip
\item{}
in article [100.], `remaked' has been corrected to `remarked';
\smallskip
\item{}
in article [123.]: the third point on the line of intersection
of the planes ${\sc a} {\sc b} {\sc d}$ and
${\sc a}_1 {\sc b}_1 {\sc d}_1$ was printed as ${\sc a}''$; but
this has been corrected to ${\sc c}''$.
\smallskip
Also, from article [97.] onwards, the roman superscripts on the
points
${\sc a}^{\rm IV}$, ${\sc b}^{\rm IV}$, ${\sc c}^{\rm IV}$, etc.
were printed in lower case, as
${\sc a}^{\rm iv}$, ${\sc b}^{\rm iv}$, ${\sc c}^{\rm iv}$, etc.,
but these superscripts have been changed to uppercase, in
conformity with the notation established in the earlier articles.
Similarly ${\sc p}_0$, ${\sc p}_1$, and ${\sc p}_2$ in articles
[1.] and [2.] were originally printed in normal size uppercase
roman, but have been changed in this edition to `small capitals',
in conformity with the notation in the remainder of the paper.
\bigbreak\bigskip
\line{\hfil David R. Wilkins}
\vskip3pt
\line{\hfil Dublin, June 1999}
\vfill\eject
\pageno=1
\null\vskip36pt
\centerline{ON GEOMETRICAL NETS IN SPACE.}
\vskip 12pt
\centerline{Sir William Rowan Hamilton.}
\vskip12pt
\centerline{Read June 24th, 1861.}
\vskip12pt
\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~vii (1862), pp.~532--582.]}
\bigskip
[1.]
When any five points of space,
${\sc a} \, {\sc b} \, {\sc c} \, {\sc d} \, {\sc e}$,
are given, whereof no four are supposed to be complanar, we can
connect any two of them by a right line, and the three others by
a plane, and determine the point in which these last intersect
each other: {\it deriving\/} thus a system of {\it ten lines\/}
$\Lambda_1$, {\it ten planes\/} $\Pi_1$, and {\it ten points\/}
${\sc p}_1$, from the {\it given\/} system of {\it five points\/}
${\sc p}_0$, by what may be called a {\it First Construction}.
We may next propose to determine all the new and distinct lines
$\Lambda_2$, and planes $\Pi_2$, which connect the ten derived
points ${\sc p}_1$, with the five given points ${\sc p}_0$, and
with each other, and may then inquire what new and distinct
points ${\sc p}_2$ arise, as
intersections\footnote*{Intersections
$\Lambda \cdot \Lambda$ of {\it line with line\/} (when
complanar) are {\it included\/} in this class $\Lambda \cdot
\Pi$; and intersections $\Pi \cdot \Pi \cdot \Pi$ of {\it three
distinct planes}, when {\it not\/} included at this stage, may be
reserved for a {\it subsequent construction}, in which they
naturally offer themselves, as of the standard form $\Lambda
\cdot \Pi$.}
$\Lambda \cdot \Pi$ of lines and planes already obtained: all
{\it such\/} new lines, planes, and points being said to belong
to a {\it Second Construction}. And then we might proceed, on
the same plan, to a {\it Third\/} Construction, and to
indefinitely many others following: building up thus what
Professor {\it M\"{o}bius}, in his {\it Barycentric
Calculus},\footnote\dag{Der Calcul Barycentrische, Leipzig, 1827,
p.~291. Some first results connected with the subject were
given, according to the writer's recollection, in a Memoir by
{\it Carnot\/} on {\it Transversals}, to which he cannot at
present refer.}
has proposed to call a {\it Geometric Net in Space}.
\bigbreak
[2.]
In general, if $n$ denote five or any greater number of
{\it independent\/} points of space, the number of the derived
points of the {\it form\/} $\Lambda \cdot \Pi$, or
${\sc a} {\sc b} \cdot {\sc c} {\sc d} {\sc e}$,
which can be obtained by what is relatively to {\it them\/} a
First Construction, of the kind just now described, is easily
seen to be the function,
$$f(n) = {n (n-1) \over 2} \cdot
{(n-2) (n-3) (n-4) \over 2 \mathbin{.} 3};$$
so that $f(5) = 10$, as above, but $f(15) = 30030$. If then the
{\it fifteen points\/} ${\sc p}_0$,~${\sc p}_1$ were thus
{\it independent}, or {\it unconnected\/} with each other, we
might expect to find that the number of points ${\sc p}_2$
{\it derived\/} from them, at the next stage, should {\it exceed
thirty thousand}. And although it was obvious that many
{\it reductions\/} of this number must occur, on account of the
{\it dependence\/} of the ten points ${\sc p}_1$ on the five
points ${\sc p}_0$, yet when I happened to feel a curiosity, some
time ago, to determine the precise {\it number\/} of those which
have been above called {\it Points of Second Construction}, and
to assign their chief geometrical relations to each other, and to
the fifteen former points, it must be confessed that I thought
myself about to undertake the solution of a rather formidable
Problem. But the motive which led me to attack that problem,
namely the desire to try the efficiency of a certain system of
{\it Quinary Symbols}, for points, lines, and planes in space,
which the {\it Method of Vectors\/} had led me to invent,
inspired me with a hope, which I trust that the result of the
attempt has not altogether failed to justify. And, in the
present communication, I wish first to present some outline of
what may be called perhaps a {\it Quinary Calculus}, before
proceeding to give, in the second place, some sketch of the
results of its application to the geometrical {\it Net in Space}.
\bigbreak
\centerline{{\sc Part~I.}---{\it On a Quinary Calculus for Space}.}
\nobreak\bigskip
[3.]
Let
${\sc a} \, {\sc b} \, {\sc c} \, {\sc d} \, {\sc e}$
be (as in [1.]) any five given points of space, whereof no four
are situated in any common plane; then, by decomposing
${\sc e} {\sc d}$ in the directions of
${\sc e} {\sc a}$, ${\sc e} {\sc b}$, ${\sc e} {\sc c}$,
we can always obtain an equation of the form,
$$ a \mathbin{.} {\sc e} {\sc a}
+ b \mathbin{.} {\sc e} {\sc b}
+ c \mathbin{.} {\sc e} {\sc c}
+ d \mathbin{.} {\sc e} {\sc d}
= 0,
\eqno (1)$$
in which the coefficients $a \, b \, c \, d$ have determined
ratios. And if we next introduce a fifth coefficient~$e$, such
that
$$a + b + c + d + e = 0,
\eqno (2)$$
and add to (1) the identity
$$(a + b + c + d + e) \, {\sc o} {\sc e} = 0,
\eqno (3)$$
in which ${\sc o}$ is any arbitrary point (or origin of vectors),
we arrive at the following equivalent but more symmetric form,
$$ a \mathbin{.} {\sc o} {\sc a}
+ b \mathbin{.} {\sc o} {\sc b}
+ c \mathbin{.} {\sc o} {\sc c}
+ d \mathbin{.} {\sc o} {\sc d}
+ e \mathbin{.} {\sc o} {\sc e}
= 0,
\eqno (4)$$
in which $a \, b \, c \, d \, e$ may be called the {\it five
(numerical) constants\/} of the given system of {\it five
points}, ${\sc a} \, \ldots \, {\sc e}$, although only their
{\it ratios\/} are important, and (as above) their {\it sum\/} is
{\it zero}.
\bigbreak
[4.]
Let ${\sc p}$ be any other point of space, and let
$x \, y \, z \, w \, v$ be coefficients satisfying the equation,
$$ (x - v) a \mathbin{.} {\sc p} {\sc a}
+ (y - v) b \mathbin{.} {\sc p} {\sc b}
+ (z - v) c \mathbin{.} {\sc p} {\sc c}
+ (w - v) d \mathbin{.} {\sc p} {\sc d}
= 0;
\eqno (5)$$
then, adding the identity,
$$v ( a \mathbin{.} {\sc p} {\sc a}
+ b \mathbin{.} {\sc p} {\sc b}
+ c \mathbin{.} {\sc p} {\sc c}
+ d \mathbin{.} {\sc p} {\sc d}
+ e \mathbin{.} {\sc p} {\sc e} )
= 0,
\eqno (6)$$
which results from (4), we obtain this other symmetric formula,
$$ xa \mathbin{.} {\sc p} {\sc a}
+ yb \mathbin{.} {\sc p} {\sc b}
+ zc \mathbin{.} {\sc p} {\sc c}
+ wd \mathbin{.} {\sc p} {\sc d}
+ ve \mathbin{.} {\sc p} {\sc e}
= 0,
\eqno (7)$$
which may also be thus written,
$${\sc o} {\sc p}
= { xa \mathbin{.} {\sc o} {\sc a}
+ yb \mathbin{.} {\sc o} {\sc b}
+ zc \mathbin{.} {\sc o} {\sc c}
+ wd \mathbin{.} {\sc o} {\sc d}
+ ve \mathbin{.} {\sc o} {\sc e}
\over xa + yb + zc + wd + ve },
\eqno (8)$$
${\sc o}$ being again an arbitrary origin; and the {\it five new
and variable coefficients}, $x \, y \, z \, w \, v$, whereof the
{\it ratios of the differences\/} determine the {\it position of
the point\/}~${\sc p}$, when the five points
${\sc a} \, \ldots \, {\sc e}$ are given, may be called the
{\it Quinary Coordinates of that Point\/}~${\sc p}$, with respect
to the given system of five points.
[5.]
Under these conditions, we may agree to write, briefly,
$${\sc p} = (x, y, z, w, v),
\quad\hbox{or even}\quad
{\sc p} = (x \, y \, z \, w \, v),
\eqno (9)$$
whenever it seems that the omission of the commas will not give
rise to any confusion; and may call this form a {\it Quinary
Symbol of the Point\/}~${\sc p}$. But because (as above) only
the ratios of the differences of the coefficients or coordinates
are important, we may establish the following {\it Formula of
Quinary Congruence}, between two {\it equivalent Symbols\/} of
one {\it common point},
$$(x' \, y' \, z' \, w' \, v') \equiv (x \, y \, z \, w \, v),
\eqno (10)$$
$$\hbox{if}\quad
x' - v' : y' - v' : z' - v' : w' - v'
= x - v : y - v : z - v : w - v;
\eqno (11)$$
reserving the
{\it Quinary Equation},
$$(x' \, y' \, z' \, w' \, v') = (x \, y \, z \, w \, v),
\eqno (12)$$
to imply the coexistence of the {\it five\/} separate and
ordinary equations,
$$x' = x,\quad y' = y,\quad z' = z,\quad w' = w,\quad v' = v.
\eqno (13)$$
We shall also adopt, as abridgments of notation, the formul{\ae},
$$t (x, y, z, w, v) = (tx, ty, tz, tw, tv);
\eqno (14)$$
$$(x' \, \ldots v') \pm (x \, \ldots \, v)
= (x' \pm x, \, \ldots \, v' \pm v);
\eqno (15)$$
and shall find it convenient to employ occasionally what may be
called the {\it Quinary Unit Symbol},
$${\si U} = (1 \, 1 \, 1 \, 1 \, 1);
\eqno (16)$$
although {\it this\/} symbol represents {\it no determined
point}, because both the denominator and numerator of the
expression~(8) vanish, by (2) and (4), when the five coefficients
$x \, y \, z \, w \, v$ become each equal to unity.
\bigbreak
[6.]
With these notations, if ${\si Q}$ and
${\si Q}'$ be any {\it other\/} quinary symbols, and $t$
and $u$ any two coefficients, we shall have the congruence,
$${\si Q}' \equiv {\si Q},
\quad\hbox{if}\quad {\si Q}' = t {\si Q} + u {\si U};
\eqno (17)$$
the {\it two points\/} ${\sc p}$ and ${\sc p}'$, which are
denoted by these {\it two symbols}, in this case
{\it coinciding}. Again the equation
$${\si Q}'' = t {\si Q} + t' {\si Q}' + u {\si U},
\eqno (18)$$
is found to express that ${\si Q}$, ${\si Q}'$ ${\si Q}''$ are
symbols of {\it three collinear points\/}; and the
{\it complanarity of four points}, of which the symbols are
${\si Q}$, ${\si Q}'$, ${\si Q}''$, ${\si Q}'''$, is expressed by
this other equation of the same form,
$${\si Q}''' = t {\si Q} + t' {\si Q}' + t'' {\si Q}'' + u {\si U}.
\eqno (19)$$
\bigbreak
[7.]
If then a {\it variable point\/}~${\sc p}$ be thus
{\it complanar\/} with {\it three given points},
${\sc p}_0$,~${\sc p}_1$,~${\sc p}_2$, its coordinates [4.] must
be connected with theirs, by five equations of the form,
$$x = t_0 x_0 + t_1 x_1 + t_2 x_2 + u;\quad\ldots\quad
v = t_0 v_0 + t_1 v_1 + t_2 v_2 + u;
\eqno (20)$$
whence, by elimination of the four arbitrary coefficients
$t_0 \, t_1 \, t_2 \, u$, a {\it linear equation\/} is obtained,
of the form
$$lx + my + nz + rw + sv = 0,
\eqno (21)$$
with the general relation
$$l + m + n + r + s = 0
\eqno (22)$$
between its coefficients; and this equation (21) may be said to
be the {\it Quinary Equation of the
Plane\/}~${\sc p}_0 \, {\sc p}_1 \, {\sc p}_2$. The five new
coefficients $l \, m \, n \, r \, s$ may be called the
{\it Quinary Coordinates of that Plane\/}; and the plane itself
may be denoted by the {\it Quinary Symbol},
$$\Pi = [l, m, n, r, s],
\quad\hbox{or briefly,}\quad
\Pi = [l \, m \, n \, r \, s],
\eqno (23)$$
when the commas can be omitted without confusion.
If ${\si R}, {\si R'},\ldots$ be symbols of this form, for planes
$\Pi, \Pi',\ldots$, then the equation
$${\si R}' = t {\si R},
\eqno (24)$$
in which $t$ is an arbitrary coefficient, expresses that the
{\it two planes $\Pi$,~$\Pi'$ coincide\/}; the equation
$${\si R}'' = t {\si R} + t' {\si R}'
\eqno (25)$$
expresses that the {\it three planes\/} $\Pi$,~$\Pi'$,~$\Pi''$ are
{\it collinear}, or that the {\it third\/} passes {\it through
the line of intersection\/} of the {\it other two\/}; and the
equation
$${\si R}''' = t {\si R} + t' {\si R}' + t'' {\si R}''
\eqno (26)$$
expresses that the {\it four planes\/}
$\Pi$,~$\Pi'$,~$\Pi''$,~$\Pi'''$ are {\it compunctual\/} (or
{\it concurrent\/}), or that the {\it fourth\/} passes
{\it through the point of intersection\/} of the {\it other
three}.
\bigbreak
[8.]
It is easy to conceive how problems respecting {\it intersections
of lines and planes\/} can be resolved, on the foregoing
principles. And if we define that a point~${\sc p}$, or
plane~$\Pi$, is a {\it Rational Point}, or a {\it Rational
Plane\/} of the {\it System} determined by the {\it five given
Points\/} ${\sc a} \, \ldots \, {\sc e}$, or that it is
{\it rationally related\/} to those five points, when its
{\it coordinates\/} are equal (or proportional) to {\it whole
numbers}, it is obvious, from the nature of the
{\it eliminations\/} employed, that a {\it plane\/} which is
determined as containing {\it three rational points}, or a
{\it point\/} which is determined as the intersection of
{\it three rational planes}, is itself, in the above sense,
{\it rational}. We may also say that a {\it right
line\/}~$\Lambda$ is a {\it Rational Line}, when it is the line
${\sc p} \overline{\phantom{m}} {\sc p}$ which {\it connects\/}
two rational points, or the {\it intersection\/} $\Pi \cdot \Pi$
of two rational planes: and then the intersection of a rational
line with a rational plane, or of two complanar and rational lines
with each other, will be a rational point.
\bigbreak
[9.]
When any two points, ${\sc p}$,~${\sc p}'$, or any two planes
$\Pi$,~$\Pi'$, have symbols which differ only by the
{\it arrangement\/} or ({\it order\/}) of the five coefficients
or coordinates in each, those points, or those planes, may then
be said to have one {\it common type\/}; or briefly, to be
{\it syntypical}. For example, the five {\it given\/} points are
thus syntypical, because (omitting commas, as in [5.]) their
symbols are,
$${\sc a} = (1 \, 0 \, 0 \, 0 \, 0),\quad
{\sc b} = (0 \, 1 \, 0 \, 0 \, 0),\quad
{\sc c} = (0 \, 0 \, 1 \, 0 \, 0),\quad
{\sc d} = (0 \, 0 \, 0 \, 1 \, 0),\quad
{\sc e} = (0 \, 0 \, 0 \, 0 \, 1).
\eqno (27)$$
In general, any two syntypical points, or planes, admit of being
{\it derived\/} from the five given points, by precisely
{\it similar processes of construction}, the {\it order\/} only
of the {\it data\/} being {\it varied\/}; and in the {\it most
general case}, a {\it single type\/} includes 120 {\it distinct
points}, or {\it distinct planes}, although this {\it number\/}
may happen to be diminished, even when the coordinates are all
unequal: for example, the type $(1 \, 2 \, 3 \, 4 \, 5)$ includes
only {\it sixty\/} distinct points, because, by (17), we have in
this case the congruence,
$$(1 \, 2 \, 3 \, 4 \, 5) \equiv (5 \, 4 \, 3 \, 2 \, 1).
\eqno (28)$$
\bigbreak
[10.]
The {\it anharmonic function\/} of any group of four collinear
points ${\sc a} \, {\sc b} \, {\sc c} \, {\sc d}$ being denoted
by the symbol $( {\sc a} \, {\sc b} \, {\sc c} \, {\sc d} )$, and
defined by the equation,
$$( {\sc a} \, {\sc b} \, {\sc c} \, {\sc d} )
= {{\sc a} {\sc b} \over {\sc b} {\sc c}}
\cdot {{\sc c} {\sc d} \over {\sc d} {\sc a}}
= {{\sc a} {\sc b} \over {\sc c} {\sc b}}
: {{\sc a} {\sc d} \over {\sc c} {\sc d}},
\eqno (29)$$
it will be found that if ${\sc p}_0 \, \ldots \, {\sc p}_3$ be
thus {\it any four collinear points}, of which therefore, by
(18), the quinary symbols ${\si Q}_0 \, \ldots \, {\si Q}_3$ are
connected by two linear relations, of the forms,
$${\si Q}_1 = t_0 {\si Q}_0 + t_2 {\si Q}_2 + u {\si U},\quad
{\si Q}_3 = t'_0 {\si Q}_0 + t'_2 {\si Q}_2 + u' {\si U},
\eqno (30)$$
then the {\it anharmonic of this group of points\/} is given by
the formula,
$$( {\sc p}_0 \, {\sc p}_1 \, {\sc p}_2 \, {\sc p}_3 )
= {t_2 t'_0 \over t_0 t'_2},
\eqno (31)$$
of which the applications are numerous and important.
And in like manner, if $\Pi_0 \, \ldots \, \Pi_3$ be {\it any
four collinear planes}, of which consequently, by (25), the
symbols ${\si R}_0 \, \ldots \, {\si R}_3$ are connected by two
other linear relations, such as
$${\si R}_1 = t_0 {\si R}_0 + t_2 {\si R}_2,\quad
{\si R}_3 = t'_0 {\si R}_0 + t'_2 {\si R}_2,
\eqno (32)$$
we have then this other very useful formula of the same kind, of
the {\it anharmonic of this pencil of planes},
$$( \Pi_0 \, \Pi_1 \, \Pi_2 \, \Pi_3 )
= {t_2 t'_0 \over t_0 t'_2};
\eqno (33)$$
it being understood that the anharmonic function of such a
{\it pencil\/} is the same as that of the {\it group\/} of
{\it points}, in which its {\it planes\/} are {\it cut\/} by any
rectilinear {\it transversal\/}: so that we may write generally,
for {\it any six points\/} ${\sc a} \, \ldots \, {\sc f}$, the
formula,
$$( {\sc e} \, {\sc f} \mathbin{.} {\sc a} \, {\sc b} \, {\sc c} \, {\sc d} )
= ({\sc a}' \, {\sc b}' \, {\sc c}' \, {\sc d}'),
\eqno (34)$$
if any transversal ${\sc g} {\sc h}$ cut the four planes
${\sc e} {\sc f} {\sc a},\ldots \, {\sc e} {\sc f} {\sc d}$
in the four points ${\sc a}',\ldots \, {\sc d}'$; or in symbols,
if
$${\sc a}' = {\sc g} {\sc h} \cdot {\sc e} {\sc f} {\sc a},
\quad\ldots\quad
{\sc d}' = {\sc g} {\sc h} \cdot {\sc e} {\sc f} {\sc d}.
\eqno (35)$$
\bigbreak
[11.]
The expression of fractional form,
$$\varphi(x \, y \, z \, w \, v)
= {l' x + m' y + n' z + r' w + s' v
\over l x + m y + n z + r w + s v}
= {f' \over f},
\eqno (36)$$
in which the ten coefficients $l \, \ldots \, s$ and
$l' \, \ldots \, s'$, are supposed to be given, and to be such
(comp.~(22)) that
$$l + \cdots + s = 0,
\quad\hbox{and}\quad
l' + \cdots + s' = 0,
\eqno (37)$$
may represent the quotient of any two linear and homogeneous
functions, $f$ and $f'$, of the coordinates $x \, \ldots \, v$ of
a variable point~${\sc p}$, or rather of the {\it differences\/}
of those coordinates (comp.~[5.]); and if we assign any
{\it particular} or {\it constant value}, such as $k$, to this
{\it quotient}, or {\it fractional function}, $\varphi$, the
equation so obtained will represent (comp.~(21)) a {\it plane
locus\/} for that point, which {\it plane\/}~$\Pi$ will always
pass {\it through a given line\/}~$\Lambda$, determined by
equating separately the denominator and numerator of $\varphi$ to
zero. Hence the {\it four equations},
$$f = 0,\quad f'=f,\quad f' = 0,\quad f' = kf,
\eqno (38)$$
which answer to the four values,
$$\varphi = \infty,\quad
\varphi = 1,\quad
\varphi = 0,\quad
\varphi = k,
\eqno (39)$$
represent a {\it pencil of four planes\/}
$\Pi_0 \, \ldots \, \Pi_3$, of which the quinary symbols (23) may
be thus written:---
$${\si R}_0 = [l \, m \, n \, r \, s];\quad
{\si R}_2 = [l' \, m' \, n' \, r' \, s'];\quad
{\si R}_1 = {\si R}_2 - {\si R}_0;\quad
{\si R}_3 = {\si R}_2 - k {\si R}_0;
\eqno (40)$$
and of which the {\it anharmonic\/} is consequently, by (33), the
same {\it quotient},
$$( \Pi_0 \, \Pi_1 \, \Pi_2 \, \Pi_3)
= (k = \varphi = ) {f' \over f},
\eqno (41)$$
as before. We have therefore this {\it Theorem}:---
\medbreak
``{\it The Quotient of any two given homogeneous and linear
Functions, of the Differences of the Quinary Coordinates of a
variable Point in Space, can always be expressed as the
Anharmonic of a Pencil of Planes, whereof three are given, while
the fourth passes through the variable Point, and through a given
Right Line, which is common to the three former Planes.}''
\bigbreak
[12.]
For example, we find thus that
$${x - v \over w - v}
= ( {\sc b} \, {\sc c} \mathbin{.}
{\sc a} \, {\sc e} \, {\sc d} \, {\sc p});\quad
{y - v \over w - v}
= ( {\sc c} \, {\sc a} \mathbin{.}
{\sc b} \, {\sc e} \, {\sc d} \, {\sc p});\quad
{z - v \over w - v}
= ( {\sc a} \, {\sc b} \mathbin{.}
{\sc c} \, {\sc e} \, {\sc d} \, {\sc p});
\eqno (42)$$
and that
$${x - v \over y - v}
= ( {\sc c} \, {\sc d} \mathbin{.}
{\sc a} \, {\sc e} \, {\sc b} \, {\sc p});\quad
{y - v \over z - v}
= ( {\sc a} \, {\sc d} \mathbin{.}
{\sc b} \, {\sc e} \, {\sc c} \, {\sc p});\quad
{z - v \over x - v}
= ( {\sc b} \, {\sc d} \mathbin{.}
{\sc c} \, {\sc e} \, {\sc a} \, {\sc p});
\eqno (43)$$
the product of these three last anharmonics of pencils being
therefore equal to positive unity, so that we have, for {\it any
six points of space},
${\sc a} \, {\sc b} \, {\sc c} \, {\sc d} \, {\sc e} \, {\sc f}$,
the general equation,
$$ ( {\sc a} \, {\sc d} \mathbin{.}
{\sc b} \, {\sc e} \, {\sc c} \, {\sc f})
\mathbin{.}
( {\sc b} \, {\sc d} \mathbin{.}
{\sc c} \, {\sc e} \, {\sc a} \, {\sc f})
\mathbin{.}
( {\sc c} \, {\sc d} \mathbin{.}
{\sc a} \, {\sc e} \, {\sc b} \, {\sc f})
= 1.
\eqno (44)$$
If then we {\it suppress the fifth coefficient, $v$, in the
quinary symbol\/} (9) {\it of a point\/}~${\sc p}$, which comes
to first substituting, as the congruence (10) permits, the
differences
$x - v$, $y - v$, $z - v$, $w - v$, and $v - v$ or $0$,
for $x$, $y$, $z$, $w$, and $v$, and then writing simply
$x, \ldots \, w$ instead of $x - v,\ldots \, w - v$, and omitting
the final zero, whereby the quinary symbol
$(0 \, 0 \, 0 \, 0 \, 1)$ for the fifth given point~${\sc e}$
(27) becomes first $(-1, -1, -1, -1, 0)$, or
$(1 \, 1 \, 1 \, 1\, 0)$, and then is reduced to the
{\it quaternary unit symbol\/} $(1 \, 1 \, 1 \, 1)$, we shall
{\it fall back on that system of anharmonic coordinates in
space}, of which some account was given in a former
communication\footnote*{See the Proceedings for the Session of
1859--60.}
to this Academy: the {\it anharmonic\/} (or {\it quaternary\/})
{\it symbol of a plane\/}~$\Pi$ being, in like manner,
{\it derived from the quinary symbol\/} (23), by simply
{\it suppressing the fifth coefficient}, or coordinate,~$s$.
{\it Anharmonic coordinates}, whether for {\it point\/} or for
{\it plane}, are therefore {\it included in quinary ones\/}; but
although they have some advantages of {\it simplicity}, it
appears that their {\it less perfect symmetry}, of reference to
the {\it five given points\/} ${\sc a} \, \ldots \, {\sc e}$,
renders them less adapted to investigations respecting the
{\it Geometrical Net in Space}, which is constructed with those
{\it five\/} points as data: and that therefore they are less fit
than {\it quinary\/} coordinates for the purposes of the present
paper.
\bigbreak
[13.]
Retaining then the {\it quinary form}, we may next observe that
although, {\it when the five coefficients $l \, \ldots \, s$ are
given}, as in [7.], and the {\it coordinates\/}
$x \, \ldots \, v$ of a {\it point\/}~${\sc p}$ are
{\it variable}, the {\it linear equation\/}
$lx + \cdots + sv = 0$ (21)
may be said to be the {\it Local Equation of a Plane}, namely of
the {\it plane\/} $[l \, \ldots \, s]$, considered as the
{\it locus of the point\/} $(x \, \ldots \, v)$; yet if, on the
contrary, we {\it now\/} regard $x \, \ldots \, v$ as
{\it given}, and $l \, \ldots \, s$ as {\it variable}, the
{\it same linear equation\/} (21) expresses the {\it condition\/}
necessary, in order that a {\it variable plane\/}
$[l \, \ldots \, s]$ may pass {\it through a given point\/}
$(x \, \ldots \, v)$; and {\it in this view}, the formula (21)
may be considered to be the {\it Tangential Equation of that
given Point}. Thus the very simple equation,
$$l = 0,
\eqno (45)$$
expresses the condition requisite for the plane
$[l \, \ldots \, s]$ passing through the given point
$(1 \, 0 \, 0 \, 0 \, 0)$, or ${\sc a}$ (27); and it is, in that
sense, the tangential {\it equation of that point\/}: while
$m = 0$ is, in like manner, the equation of ${\sc b}$, \&c.
This being understood, if we suppose that ${\sc f}$ and
${\sc f}'$ denote two given, linear, and homogeneous functions of
the coordinates $l \, \ldots \, s$ of a variable plane~$\Pi$, we
may consider the four equations,
$${\sc f} = 0,\quad
{\sc f}' = {\sc f},\quad
{\sc f}' = 0,\quad
{\sc f}' = k {\sc f},
\eqno (46)$$
as the tangential equations of {\it four collinear points},
${\sc p}_0$,~${\sc p}_1$,~${\sc p}_2$,~${\sc p}_3$, whereof the
three first are entirely given, but the fourth varies with the
value of the coefficient~$k$, although always remaining on the
line~$\Lambda$ of the other three; and it is easy to deduce, from
the formula (31), by reasonings analogous to those employed in
[11.], the following {\it anharmonic of the group\/}:
$$( {\sc p}_0 \, {\sc p}_1 \, {\sc p}_2 \, {\sc p}_3 )
= k = {{\sc f}' \over {\sc f}}.
\eqno (47)$$
We have therefore this new {\it Theorem}, analogous to one lately
stated:---
\medbreak
``{\it The Quotient of any two given, homogeneous, and linear
Functions, of the Quinary Coordinates of a variable Plane, may
always be expressed as the Anharmonic of a Group of Points;
whereof three are given and collinear, while the fourth is the
Intersection of the variable Plane with the given Line on which
the other three are situated.}''
\bigbreak
[14.]
For example, if we wish in this way to {\it interpret the
quotient\/} $m : n$, of these two coordinates of a
{\it variable plane\/}~$\Pi$, or $[l \, m \, n \, r \, s]$ (23),
as denoting the {\it anharmonic of a group of points}, the first
three points ${\sc p}_0$,~${\sc p}_1$,~${\sc p}_2$ of that group
(47) have here for their tangential equations,
$$n = 0,\quad m - n = 0,\quad m = 0,
\eqno (48)$$
whereof the {\it third\/} has recently been seen [13.] to
represent the given point~${\sc b}$, and the {\it first\/}
represents in like manner another given point, namely ${\sc c}$,
of the initial system: while the {\it second\/} represents the
point $(0, 1, -1, 0, 0)$, or briefly
$(0 \, 1 \, \overline{1} \, 0 \, 0)$, if, to save commas, we
write $\overline{1}$ for $-1$. To {\it construct\/} this last
point, let us write
$${\sc a}' = (0 \, 1 \, 1 \, 0 \, 0)
\equiv (1 \, 0 \, 0 \, 1 \, 1),
\quad{and}\quad
{\sc a}'' = (0 \, 1 \, \overline{1} \, 0 \, 0);
\eqno (49)$$
then, by (18), these two new points ${\sc a}'$ and ${\sc a}''$
are each {\it collinear\/} with ${\sc b}$,~${\sc c}$, or are on
the line ${\sc b} {\sc c}$; and they are, with respect to that
line (or to its extreme points) {\it harmonically conjugate\/} to
each other, because the formula (31) gives easily, by the
{\it first\/} symbol for ${\sc a}'$, the {\it harmonic equation},
$$({\sc b} \, {\sc a}' \, {\sc c} \, {\sc a}'') = -1;
\eqno (50)$$
but also the {\it second\/} (or congruent) symbol for ${\sc a}'$
shows, by (19), that ${\sc a}'$ is in the {\it plane\/}
${\sc a} {\sc d} {\sc e}$; we may therefore write the
{\it formula of intersection},
$${\sc a}' = {\sc b} {\sc c} \cdot {\sc a} {\sc d} {\sc e},
\eqno (51)$$
whereby the point~${\sc a}'$ is entirely determined; and then the
point~${\sc a}''$, as being its harmonic conjugate with respect
to ${\sc b}$ and ${\sc c}$, or as satisfying the equation~(50),
is to be considered as being itself a known point. We have thus
assigned the three first points
${\sc p}_0$,~${\sc p}_1$,~${\sc p}_2$, of the {\it group\/} (47),
namely the points ${\sc c}$,~${\sc a}''$,~${\sc b}$; and if we
denote by ${\sc l}$ the point ${\sc b} {\sc c} \cdot \Pi$ in
which the variable plane~$\Pi$, or $[l \, \ldots \, s]$,
intersects the given line ${\sc b} {\sc c}$, so that
$${\sc l} = (0, n, -m, 0, 0),
\quad\hbox{or briefly,}\quad
{\sc l} = (0 \, n \, \overline{m} \, 0 \, 0),
\eqno (52)$$
writing $\overline{m}$ for $- m$, then the fourth
point~${\sc p}_3$ is ${\sc l}$; and the required {\it formula of
interpretation\/} for the quotient $m : n$ becomes,
$${m \over n} = ({\sc c} \, {\sc a}'' \, {\sc b} \, {\sc l}).
\eqno (53)$$
In like manner, if we write
$${\sc b}' = (1 \, 0 \, 1 \, 0 \, 0),\quad
{\sc c}' = (1 \, 1 \, 0 \, 0 \, 0),\quad
{\sc b}'' = (\overline{1} \, 0 \, 1 \, 0 \, 0),\quad
{\sc c}'' = (1 \, \overline{1} \, 0 \, 0 \, 0),
\eqno (54)$$
and
$${\sc m} = ( \overline{n} \, 0 \, l \, 0 \, 0),\quad
{\sc n} = ( m \, \overline{l} \, 0 \, 0 \, 0),
\eqno (55)$$
in which $\overline{n} = -n$ and $\overline{l} = -l$, so that
${\sc m} = {\sc c} {\sc a} \cdot \Pi$,
${\sc n} = {\sc a} {\sc b} \cdot \Pi$,
and
$${\sc b}' = {\sc c} {\sc a} \cdot {\sc b} {\sc d} {\sc e},\quad
{\sc c}' = {\sc a} {\sc b} \cdot {\sc c} {\sc d} {\sc e},\quad
({\sc c} \, {\sc b}' \, {\sc a} \, {\sc b}'')
= ({\sc a} \, {\sc c}' \, {\sc b} \, {\sc c}'')
= -1,
\eqno (56)$$
we shall have these two other formul{\ae} of interpretation,
analogous to (53),
$${n \over l} = ({\sc a} \, {\sc b}'' \, {\sc c} \, {\sc m}),\quad
{l \over m} = ({\sc b} \, {\sc c}'' \, {\sc a} \, {\sc n});
\eqno (57)$$
and therefore,
$$ ({\sc a} \, {\sc b}'' \, {\sc c} \, {\sc m})
\cdot ({\sc b} \, {\sc c}'' \, {\sc a} \, {\sc n})
\cdot ({\sc c} \, {\sc a}'' \, {\sc b} \, {\sc l})
= 1.
\eqno (58)$$
\bigbreak
[15.]
Again, if we denote by ${\sc q}$,~${\sc r}$,~${\sc s}$ the
intersections
${\sc d} {\sc a} \cdot \Pi$,
${\sc d} {\sc b} \cdot \Pi$,
${\sc d} {\sc c} \cdot \Pi$,
so that
$${\sc q} = (\overline{r} \, 0 \, 0 \, l \, 0),\quad
{\sc r} = (0 \, \overline{r} \, 0 \, m \, 0),\quad
{\sc s} = (0 \, 0 \, \overline{r} \, n \, 0),
\eqno (59)$$
where $\overline{r} = - r$; if also we introduce seven new points
syntypical [9.] with the three points
${\sc a}' \, {\sc b}' \, {\sc c}'$, and seven others syntypical
with
${\sc a}'' \, {\sc b}'' \, {\sc c}''$, as follows:
$${\sc a}_1 = (1 \, 0 \, 0 \, 0 \, 1),\quad
{\sc b}_1 = (0 \, 1 \, 0 \, 0 \, 1),\quad
{\sc c}_1 = (0 \, 0 \, 1 \, 0 \, 1),\quad
{\sc d}_1 = (0 \, 0 \, 0 \, 1 \, 1);
\eqno (60)$$
$${\sc a}_2 = (1 \, 0 \, 0 \, 1 \, 0),\quad
{\sc b}_2 = (0 \, 1 \, 0 \, 1 \, 0),\quad
{\sc c}_2 = (0 \, 0 \, 1 \, 1 \, 0);
\eqno (61)$$
$${\sc a}'_1 = (1 \, 0 \, 0 \, 0 \, \overline{1}),\quad
{\sc b}'_1 = (0 \, 1 \, 0 \, 0 \, \overline{1}),\quad
{\sc c}'_1 = (0 \, 0 \, 1 \, 0 \, \overline{1}),\quad
{\sc d}'_1 = (0 \, 0 \, 0 \, 1 \, \overline{1});
\eqno (62)$$
$${\sc a}'_2 = (1 \, 0 \, 0 \, \overline{1} \, 0),\quad
{\sc b}'_2 = (0 \, 1 \, 0 \, \overline{1} \, 0),\quad
{\sc c}'_2 = (0 \, 0 \, 1 \, \overline{1} \, 0);
\eqno (63)$$
so that, by principles already established, we shall have the
seven relations of intersection,
$${\sc a}_1 = {\sc e} {\sc a} \cdot {\sc b} {\sc c} {\sc d},\quad
{\sc b}_1 = {\sc e} {\sc b} \cdot {\sc c} {\sc a} {\sc d},\quad
{\sc c}_1 = {\sc e} {\sc c} \cdot {\sc a} {\sc b} {\sc d},\quad
{\sc d}_1 = {\sc e} {\sc d} \cdot {\sc a} {\sc b} {\sc c},
\eqno (64)$$
$${\sc a}_2 = {\sc d} {\sc a} \cdot {\sc b} {\sc c} {\sc e},\quad
{\sc b}_2 = {\sc d} {\sc b} \cdot {\sc c} {\sc a} {\sc e},\quad
{\sc c}_2 = {\sc d} {\sc c} \cdot {\sc a} {\sc b} {\sc e},
\eqno (65)$$
and the seven harmonic relations,
$$ ( {\sc e} \, {\sc a}_1 \, {\sc a} \, {\sc a}'_1 )
= ( {\sc e} \, {\sc b}_1 \, {\sc b} \, {\sc b}'_1 )
= ( {\sc e} \, {\sc c}_1 \, {\sc c} \, {\sc c}'_1 )
= ( {\sc e} \, {\sc d}_1 \, {\sc d} \, {\sc d}'_1 )
= -1,
\eqno (66)$$
$$ ( {\sc d} \, {\sc a}_2 \, {\sc a} \, {\sc a}'_2 )
= ( {\sc d} \, {\sc b}_2 \, {\sc b} \, {\sc b}'_2 )
= ( {\sc d} \, {\sc c}_2 \, {\sc c} \, {\sc c}'_2 )
= -1,
\eqno (67)$$
by means of which 14 last relations these 14 new points can all
be geometrically constructed; we shall then be able to interpret,
on the recent plan [13.], the three new quotients,
$l : r$, $m : r$, $n : r$, as anharmonics of groups, as follows:
$${l \over r} = ( {\sc d} \, {\sc a}'_2 \, {\sc a} \, {\sc q});\quad
{m \over r} = ( {\sc d} \, {\sc b}'_2 \, {\sc b} \, {\sc r});\quad
{n \over r} = ( {\sc d} \, {\sc c}'_2 \, {\sc c} \, {\sc s});
\eqno (68)$$
with the analogous interpretations,
$${l \over s} = ( {\sc e} \, {\sc a}'_1 \, {\sc a} \, {\sc x});\quad
{m \over s} = ( {\sc e} \, {\sc b}'_1 \, {\sc b} \, {\sc y});\quad
{n \over s} = ( {\sc e} \, {\sc c}'_1 \, {\sc c} \, {\sc z});\quad
{r \over s} = ( {\sc e} \, {\sc d}'_1 \, {\sc d} \, {\sc w}),
\eqno (69)$$
if ${\sc x}$,~${\sc y}$,~${\sc z}$,~${\sc w}$ denote the
intersections
${\sc e} {\sc a} \cdot \Pi$,
${\sc e} {\sc b} \cdot \Pi$,
${\sc e} {\sc c} \cdot \Pi$,
${\sc e} {\sc d} \cdot \Pi$,
so that
$${\sc x} = (\overline{s} \, 0 \, 0 \, 0 \, l),\quad
{\sc y} = (0 \, \overline{s} \, 0 \, 0 \, m),\quad
{\sc z} = (0 \, 0 \, \overline{s} \, 0 \, n),\quad
{\sc w} = (0 \, 0 \, 0 \, \overline{s} \, r),
\eqno (70)$$
where $\overline{s} = -s$.
\bigbreak
[16.]
As regards the {\it notations\/} employed, it may be observed that
although we have often, as in (9) or (27), \&c., {\it equated a
point}, or rather its {\it literal symbol}, ${\sc a}$ or
${\sc p}$, \&c., to the {\it corresponding quinary symbol\/}
$(1 \, 0 \, 0 \, 0 \, 0)$ or $(x \, y \, z \, w \, v)$, \&c., of
that point, yet in some formul{\ae}, such as (17) (18) (19), in
which we had occasion to treat of {\it linear combinations\/} of
such quinary symbols, we substituted {\it new letters}, such as
${\si Q}$,~${\si Q}'$, for ${\sc p}$,~${\sc p}'$, \&c., in order
to avoid the apparent strangeness of writing such
expressions\footnote*{Expressions of this {\it form\/} occur
continually in the {\it Barycentric Calculus\/} of {\it Moebius},
but with significations entirely different from those here
proposed.}
as $t {\sc p} + t' {\sc p}'$, \&c. To {\it economise symbols},
however, we may agree to {\it retain the literal symbols first
used}, for any system of given or derived points, but to
{\it enclose them in parentheses}, when we wish to employ them as
{\it denoting quinary symbols in combination with each other\/};
writing, at the same time, for the sake of uniformity $({\sc u})$
instead of ${\si U}$, as the {\it quinary unit symbol\/} (16).
And thus, if we agree also that an {\it equation\/} between
{\it two unenclosed\/} and {\it literal symbols of points},
${\sc p}$ and ${\sc p}'$, shall be understood as expressing that
the two points so denoted {\it coincide}, we may write anew those
formul{ae} (17) (18) (19) as follows:
$${\sc p}' = {\sc p},
\hbox{ if }
({\sc p}') = t ({\sc p}) + u ({\sc u});
\eqno (71)$$
$${\sc p}'' \hbox{\it on line } {\sc p} {\sc p}',
\hbox{ if }
({\sc p}'') = t ({\sc p}) + t' ({\sc p}') + u ({\sc u});
\eqno (72)$$
$${\sc p}''' \hbox{\it in plane } {\sc p} {\sc p}' {\sc p}'',
\hbox{ if }
({\sc p}''') = t ({\sc p}) + t' ({\sc p}') + t'' ({\sc p}'') + u ({\sc u}).
\eqno (73)$$
\bigbreak
[17.]
We may also occasionally denote a point {\it in the given
plane\/} of ${\sc a}$,~${\sc b}$,~${\sc c}$ by the {\it ternary
symbol},
$$(x, y, z), \quad\hbox{or}\quad (x \, y \, z),
\eqno (74)$$
considered here as an {\it abridgment\/} of the {\it quinary\/}
symbol $(x \, y \, z \, 0 \, 0)$; and the {\it right line\/}
which is the {\it trace on that plane}, of any {\it other
plane}~$\Pi$, or $[l \, m \, n \, r \, s]$ (23) may be denoted by
this {\it other ternary symbol},
$$[l, m, n], \quad\hbox{or}\quad [l \, m \, n];
\eqno (75)$$
these two last ternary symbols being {\it connected\/} by the
relation,
$$lx + my + nz = 0,
\eqno (76)$$
if the {\it point\/} $(x \, y \, z)$ be {\it on the line\/}
$[l \, m \, n]$. And the {\it point\/}~${\sc p}$ in which any
{\it other\/} line~$\Lambda$, {\it not\/} situated in the plane
${\sc a} {\sc b} {\sc c}$, {\it intersects\/} that {\it plane},
may be said to be the {\it trace\/} of that {\it line}.
\bigbreak
[18.]
For example, the {\it point\/}~${\sc d}_1$ is, by (64), the
{\it trace of the line\/}~${\sc d} {\sc e}$; and if we write,
$${\sc a}_0 = (\overline{1} \, 1 \, 1),\quad
{\sc b}_0 = (1 \, \overline{1} \, 1),\quad
{\sc c}_0 = (1 \, 1 \, \overline{1}),
\eqno (77)$$
then these three points are the respective traces of the three
lines
${\sc a}_1 {\sc a}_2$, ${\sc b}_1 {\sc b}_2$, ${\sc c}_1 {\sc c}_2$;
because they are, by the notation (74), in the given plane, and
we have, by (60) and (61), the three following symbolical
equations of the form (72),
$$ ({\sc a}_0) + ({\sc a}_1) + ({\sc a}_2)
= ({\sc b}_0) + ({\sc b}_1) + ({\sc b}_2)
= ({\sc c}_0) + ({\sc c}_1) + ({\sc c}_2)
= ({\sc u}),
\eqno (78)$$
which expresses the three collineations,
${\sc a}_0 {\sc a}_1 {\sc a}_2$,
${\sc b}_0 {\sc b}_1 {\sc b}_2$,
${\sc c}_0 {\sc c}_1 {\sc c}_2$.
We have also the three other collineations,
${\sc a} {\sc d}_1 {\sc a}'$,
${\sc b} {\sc d}_1 {\sc b}'$,
${\sc c} {\sc d}_1 {\sc c}'$,
because the quinary symbols (27) (49) (54) (60) give the
equations,
$$ ({\sc a}) + ({\sc a}') + ({\sc d}_1)
= ({\sc b}) + ({\sc b}') + ({\sc d}_1)
= ({\sc c}) + ({\sc c}') + ({\sc d}_1)
= ({\sc u});
\eqno (79)$$
and these {\it three lines}, ${\sc a} {\sc a}' {\sc d}_1$, \&c.,
are the {\it traces of the three planes\/}
${\sc a} {\sc d} {\sc e}$,
${\sc b} {\sc d} {\sc e}$,
${\sc c} {\sc d} {\sc e}$,
of which {\it planes\/} the respective {\it equations\/} (21),
and {\it quinary symbols\/} (23), are
$$y - z = 0,\quad
z - x = 0,\quad
x - y = 0,
\eqno (80)$$
and
$$[0 \, 1 \, \overline{1} \, 0 \, 0],\quad
[\overline{1} \, 0 \, 1 \, 0 \, 0],\quad
[1 \, \overline{1} \, 0 \, 0 \, 0];
\eqno (81)$$
so that the {\it ternary symbols\/} of the three last
{\it lines}, regarded as their {\it traces}, are simply, by (75),
$$[0 \, 1 \, \overline{1}],\quad
[\overline{1} \, 0 \, 1],\quad
[1 \, \overline{1} \, 0].
\eqno (82)$$
Accordingly, whether we consider the point
${\sc a} = (1 \, 0 \, 0)$, or
${\sc a}' = (0 \, 1 \, 1)$, or
${\sc d}_1 = (1 \, 1 \, 1)$,
(this {\it ternary\/} symbol of ${\sc d}_1$ being
{\it congruent\/} to the former {\it quinary\/} symbol
$(0 \, 0 \, 0 \, 1 \, 1)$ for that point (60),) we have in each
case the relation $y - z = 0$ between its coordinates; and
similarly for the two other lines.
\bigbreak
[19.]
As other examples, the {\it four planes},
$${\sc a}_1 {\sc b}_1 {\sc c}_1,\quad
{\sc a}_2 {\sc b}_2 {\sc c}_2,\quad
{\sc a}'_1 {\sc b}'_1 {\sc c}'_1,\quad
{\sc a}'_2 {\sc b}'_2 {\sc c}'_2,
\eqno (83)$$
have for their quinary equations,
$$x + y + z = 2w + v,\quad
x + y + z = w + 2v,\quad
x + y + z + v = 4w,\quad
x + y + z + w = 4v,
\eqno (84)$$
and for their quinary symbols,
$$[1 \, 1 \, 1 \, \overline{2} \, \overline{1}],\quad
[1 \, 1 \, 1 \, \overline{1} \, \overline{2}],\quad
[1 \, 1 \, 1 \, \overline{4} \, 1],\quad
[1 \, 1 \, 1 \, 1 \, \overline{4}];
\eqno (85)$$
they have therefore a {\it common trace}, namely the line
$$[1 \, 1 \, 1], \hbox{ or } {\sc a}'' {\sc b}'' {\sc c}'',
\eqno (86)$$
because, by (49) and (54), we may now write
$${\sc a}'' = (0 \, 1 \, \overline{1}),\quad
{\sc b}'' = (\overline{1} \, 0 \, 1),\quad
{\sc c}'' = (1 \, \overline{1} \, 0),
\eqno (87)$$
and the coordinates of each of these three last points satisfy
the equation,
$$x + y + z = 0.
\eqno (88)$$
Accordingly we have, by (60) (61) (62) (63), the three following
sets of symbolical equations of the form (72),
$$\left. \eqalign{
({\sc a}'')
&= ({\sc b}_1) - ({\sc c}_1)
= ({\sc b}_2) - ({\sc c}_2)
= ({\sc b}'_1) - ({\sc c}'_1)
= ({\sc b}'_2) - ({\sc c}'_2),\cr
({\sc b}'')
&= ({\sc c}_1) - ({\sc a}_1)
= ({\sc c}_2) - ({\sc a}_2)
= ({\sc c}'_1) - ({\sc a}'_1)
= ({\sc c}'_2) - ({\sc a}'_2),\cr
({\sc c}'')
&= ({\sc a}_1) - ({\sc b}_1)
= ({\sc a}_2) - ({\sc b}_2)
= ({\sc a}'_1) - ({\sc b}'_1)
= ({\sc a}'_2) - ({\sc b}'_2),\cr}
\right\}
\eqno (89)$$
we see that the {\it point\/}~${\sc a}''$ is the {\it common
trace\/} of the {\it four lines\/}, ${\sc b}_1 {\sc c}_1$,
${\sc b}_2 {\sc c}_2$, ${\sc b}'_1 {\sc c}'_1$,
${\sc b}'_2 {\sc c}'_2$; ${\sc b}''$ of ${\sc c}_1 {\sc a}_1$,
${\sc c}_2 {\sc a}_2$, ${\sc c}'_1 {\sc a}'_1$,
${\sc c}'_2 {\sc a}'_2$; and ${\sc c}''$ of ${\sc a}_1 {\sc b}_1$,
${\sc a}_2 {\sc b}_2$, ${\sc a}'_1 {\sc b}'_1$,
${\sc a}'_2 {\sc b}'_2$.
\bigbreak
[20.]
In all such cases as these, in which we have to consider a
{\it set of three points\/}~${\sc p}$, or a {\it set of three
planes\/}~$\Pi$, of which the {\it first\/} is
{\it geometrically derived\/} from
${\sc a} \, {\sc b} \, {\sc c} \, {\sc d} \, {\sc e}$
according to the {\it same rule of construction}, as that
according to which the {\it second\/} is derived from
${\sc b} \, {\sc c} \, {\sc a} \, {\sc d} \, {\sc e}$,
and the {\it third\/} from
${\sc c} \, {\sc a} \, {\sc b} \, {\sc d} \, {\sc e}$,
we can {\it symbolically derive the second from the first}, and
in like manner the {\it third\/} from the {\it second}, (or again
the first from the third,) by writing, in each case, the
{\it third}, {\it first}, and {\it second coefficients}, or
coordinates, in the places of the {\it first}, {\it second}, and
{\it third}, respectively. In symbols, we may express the
{\it law of successive derivation}, of certain {\it syntypical\/}
points or planes [9.] from one another, by the formul{\ae},
$$\hbox{if } {\sc p} ({\sc a} \, {\sc b} \, {\sc c})
= (x \, y \, z \, w \, v),
\hbox{ then }
{\sc p} ({\sc b} \, {\sc c} \, {\sc a})
= (z \, x \, y \, w \, v),
\hbox{ and }
{\sc p} ({\sc c} \, {\sc a} \, {\sc b})
= (y \, z \, x \, w \, v);
\eqno (90)$$
and if
$$\Pi ({\sc a} \, {\sc b} \, {\sc c})
= [l \, m \, n \, r \, s],
\hbox{ then }
\Pi ({\sc b} \, {\sc c} \, {\sc a})
= [n \, l \, m \, r \, s],
\hbox{ and }
\Pi ({\sc c} \, {\sc a} \, {\sc b})
= [m \, n \, l \, r \, s];
\eqno (91)$$
as has been already exemplified in the systems (27), (60), (61),
(62), (63), (77), (81), (87), for {\it points\/} or {\it planes},
and in (82) for {\it lines}, considered as {\it traces\/} of
planes. In all these cases, therefore, we can, with perfect
clearness and {\it definiteness\/} of signification, {\it abridge
the notation}, by writing {\it only the first\/} (or indeed
{\it any one\/}) of the {\it three\/} equations (90) or (91) and
then appending an ``\&c.''; for the {\it law\/} which has been
just stated will always enable us to {\it recover\/} (or deduce)
{\it the other two}. We may therefore briefly but sufficiently
express several of the foregoing results, by writing,
$$\left. \eqalign{
{\sc a} = (1 \, 0 \, 0), \hbox{ \&c.};\quad
{\sc a}' = (0 \, 1 \, 1), \hbox{ \&c.};\quad
{\sc a}'' = (0 \, 1 \, \overline{1}), \hbox{ \&c.};\quad
{\sc a}_0 = (\overline{1} \, 1 \, 1), \hbox{ \&c.};\cr
{\sc a}_1 = (1 \, 0 \, 0 \, 0 \, 1), \hbox{ \&c.};\quad
{\sc a}_2 = (1 \, 0 \, 0 \, 1 \, 0), \hbox{ \&c.};\cr
{\sc a}'_1 = (1 \, 0 \, 0 \, 0 \, \overline{1}), \hbox{ \&c.};\quad
{\sc a}'_2 = (1 \, 0 \, 0 \, \overline{1} \, 0), \hbox{ \&c.};\cr}
\right\}
\eqno (92)$$
$$\hbox{\it Plane } {\sc a} {\sc d} {\sc e}
= [0 \, 1 \, \overline{1} \, 0 \, 0], \hbox{ \&c.};\quad
\hbox{\it Line } {\sc a} {\sc d}_1 {\sc a}'
= [0 \, 1 \, \overline{1}], \hbox{ \&c.};
\eqno (93)$$
to which we may add these other symbols of planes and lines, each
supposed to be followed by an ``\&c.'':
$$\hbox{plane } {\sc b} {\sc c} {\sc d}
= [1 \, 0 \, 0 \, 0 \, \overline{1}];\quad
{\sc b} {\sc c} {\sc e}
= [1 \, 0 \, 0 \, \overline{1} \, 0];\quad
\hbox{trace} = {\sc b} {\sc c} = [1 \, 0 \, 0];
\eqno (94)$$
$$\hbox{plane } {\sc d} {\sc b}' {\sc b}_1 {\sc c}' {\sc c}_1
= [\overline{1} \, 1 \, 1 \, 0 \, \overline{1}];\quad
{\sc e} {\sc b}' {\sc b}_2 {\sc c}' {\sc c}_2
= [\overline{1} \, 1 \, 1 \, \overline{1} \, 0];\quad
\hbox{trace} = {\sc b}' {\sc c}' {\sc a}''
= [\overline{1} \, 1 \, 1];
\eqno (95)$$
$$\hbox{plane } {\sc a} {\sc b}_1 {\sc c}_2 {\sc c}_1 {\sc b}_2
= [0 \, 1 \, 1 \, \overline{1} \, \overline{1}];\quad
\hbox{trace} = {\sc a} {\sc a}''
= [0 \, 1 \, 1];
\eqno (96)$$
this line ${\sc a} {\sc a}''$ passing also, by (77), through the
two points ${\sc b}_0$ and ${\sc c}_0$;
$$\hbox{plane } {\sc b}_1 {\sc c}_1 {\sc d}_1
= [\overline{2} \, 1 \, 1 \, 1 \, \overline{1}];\quad
{\sc b}_2 {\sc c}_2 {\sc d}_1
= [\overline{2} \, 1 \, 1 \, \overline{1} \, 1];\quad
\hbox{trace} = {\sc d}_1 {\sc a}''
= [\overline{2} \, 1 \, 1];
\eqno (97)$$
$$\left. \eqalign{
\hbox{plane } {\sc a}' {\sc b}_1 {\sc b}_2
= [\overline{2} \, \overline{1} \, 1 \, 1 \, 1];\quad
\hbox{trace} = {\sc a}' {\sc b}_0
= [\overline{2} \, \overline{1} \, 1];\cr
\hbox{plane } {\sc a}' {\sc c}_1 {\sc c}_2
= [\overline{2} \, 1 \, \overline{1} \, 1 \, 1];\quad
\hbox{trace} = {\sc a}' {\sc c}_0
= [\overline{2} \, 1 \, \overline{1}];\cr}
\right\}
\eqno (98)$$
where it may be noticed that the symbol for
${\sc a}' {\sc c}_1 {\sc c}_2$, or for ${\sc a}' {\sc c}_0$, may
be deduced from that for
${\sc a}' {\sc b}_1 {\sc b}_2$, or for ${\sc a}' {\sc b}_0$, by
simply interchanging the second and third coefficients, or
coordinates. It is easy to see that the quinary symbol for the
plane ${\sc a} {\sc b} {\sc c}$ itself is on the same plan
$[0 \, 0 \, 0 \, 1 \, \overline{1}]$, the equation of that plane
being $w = v$; and it will be remembered that, by [18.], the
ternary symbol for the point ${\sc d}_1$ in that plane is
$(1 \, 1 \, 1)$.
\bigbreak
[21.]
A {\it right Line~$\Lambda$ in Space\/} may be regarded in two
principal views, as follows. Ist, it may be considered as the
{\it locus of a variable point~${\sc p}$, collinear with two given
points\/} ${\sc p}_0$,~${\sc p}_1$; and in this view, the
{\it symbol}
$$t_0 ({\sc p}_0) + t_1 ({\sc p}_1),
\eqno \hbox{(comp. (72),)}$$
for the variable {\it point\/} upon the line, may be regarded as
a {\it Local Symbol\/} (or {\it Point-Symbol\/}) {\it of the
Line~$\Lambda$ itself}. Thus
$$(0 \, t \, t'), \hbox{ or } (0 \, y \, z),
\eqno (99)$$
may either represent {\it an arbitrary point on the line\/}
${\sc b} {\sc c}$; or, {\it as a local symbol}, that {\it line
itself}. Or IInd, we may consider a line~$\Lambda$ as a
{\it hinge, round which a plane~$\Pi$ turns}, so as to be always
{\it collinear\/} [7.] {\it with two given planes\/}
$\Pi_0$,~$\Pi_1$ through the line; and then a symbol of the form
$$t_0 [\Pi_0] + t_1 [\Pi_1],
\eqno \hbox{(comp. (25),)}$$
which represents immediately the {\it variable plane\/}~$\Pi$,
may be regarded as being {\it also a Tangential Symbol\/} (or
{\it Plane-Symbol\/}) {\it for the line\/}~$\Lambda$. For
example the line ${\sc b} {\sc c}$ may be thus represented, not
only by the {\it local\/} symbol (99), but also by the
{\it tangential\/} symbol,
$$[\overline{\sigma} \, 0 \, 0 \, t \, u],
\hbox{ if } \sigma = t + u, \hbox{ and }
\overline{\sigma} = - \sigma.
\eqno (100)$$
In fact, this last symbol can be derived, by linear combinations,
from the symbols (94) for the two planes
${\sc b} {\sc c} {\sc d}$, ${\sc b} {\sc c} {\sc e}$,
which intersect in the line ${\sc b} {\sc c}$; and if any
particular value be assigned to the ratio $t : u$, a particular
{\it plane through that line\/} results. But it is time to apply
these general principles to the {\it Geometrical Net in Space}.
\bigbreak
\centerline{{\sc Part~II.}---{\it Applications to the Net in
Space: Enumeration and Classification of the Lines,}}
\centerline{{\it Planes, and Points of that Net, to the
end of the Second Construction.}}
\nobreak\bigskip
[22.]
The {\it data\/} of the {\it Geometrical Net\/} are, by [1.] the
{\it five points\/}
${\sc a} \, {\sc b} \, {\sc c} \, {\sc d} \, {\sc e}$,
or ${\sc p}_0$; of which the {\it quinary symbols\/} (27) have
been assigned, and shown to be {\it syntypical\/} [9.]; and also
the {\it ternary\/} symbols (92) of the three first of them. Of
these the symbol
$${\sc a} = (1 \, 0 \, 0)$$
may be taken as the {\it type\/}; and the point~${\sc a}$ itself
my be said to be a {\it First Typical Point}.
\bigbreak
[23.]
The {\it derived lines\/}~$\Lambda_1$ of {\it First
Construction\/} [1.], are the {\it ten\/} following,
$${\sc b} {\sc c}, \hbox{ \&c.};\quad
{\sc d} {\sc a}, \hbox{ \&c.};\quad
{\sc e} {\sc a}, \hbox{ \&c.};\quad
\hbox{and}\quad {\sc d} {\sc e};$$
the ``\&c.'' being interpreted as in [20.]; and each
line~$\Lambda_1$ connecting, by this construction, {\it two\/}
points ${\sc p}_0$. Among these the line ${\sc b} {\sc c}$ may
be selected as a {\it First Typical Line\/}; and its
{\it symbols\/} [21.], namely,
$$(0 \, y \, z),
\hbox{ and }
[\overline{\sigma} \, 0 \, 0 \, t \, u],$$
whereof the former represents this line ${\sc b} {\sc c}$
considered as the {\it locus\/} of a {\it variable point}, while
the latter represents the same line considered as the
{\it hinge\/} of a {\it variable plane}, may be taken as
{\it types\/} (the {\it point-type\/} and the {\it plane-type\/})
of the {\it group\/} of the {\it ten lines\/}~$\Lambda_1$.
\bigbreak
[24.]
The {\it derived planes\/}~$\Pi_1$ of {\it first\/} construction
are in like manner {\it ten\/}; namely
$${\sc a} {\sc d} {\sc e}, \hbox{ \&c.};\quad
{\sc b} {\sc c} {\sc e}, \hbox{ \&c.};\quad
{\sc b} {\sc c} {\sc d}, \hbox{ \&c.};\quad
\hbox{and}\quad {\sc a} {\sc b} {\sc c},$$
each obtained by connecting {\it three\/} points ${\sc p}_0$. Of
these the last has, by [20.] the quinary {\it symbol},
$${\sc a} {\sc b} {\sc c} = [0 \, 0 \, 0 \, 1 \, \overline{1}],$$
which may be taken as a {\it type\/} of the
{\it group\/}~$\Pi_1$; and the plane ${\sc a} {\sc b} {\sc c}$
itself may be called a {\it First Typical Plane}. As a
verification, we see that when we make $\sigma = t + u = 0$, in
the second symbol [23.], and divide by $t$, we are led to the
recent symbol for ${\sc a} {\sc b} {\sc c}$, as one of the planes
which pass through the line ${\sc b} {\sc c}$.
\bigbreak
[25.]
The {\it derived points\/}~${\sc p}_1$, of the same {\it first
construction}, which are all, by [1.], of the form
$\Lambda_1 \cdot \Pi_1$, are in like manner {\it ten\/}; namely
the intersections,
$${\sc b} {\sc c} \cdot {\sc a} {\sc d} {\sc e}, \hbox{ \&c.};\quad
{\sc d} {\sc a} \cdot {\sc b} {\sc c} {\sc e}, \hbox{ \&c.};\quad
{\sc e} {\sc a} \cdot {\sc b} {\sc c} {\sc d}, \hbox{ \&c.};\quad
\hbox{and}\quad {\sc d} {\sc e} \cdot {\sc a} {\sc b} {\sc c},$$
which have been denoted in [14.] and [15.] by the letters, or
{\it literal symbols},
$${\sc a}', \hbox{ \&c.};\quad
{\sc a}_2, \hbox{ \&c.};\quad
{\sc a}_1, \hbox{ \&c.};\quad
\hbox{and}\quad {\sc d}_1,$$
and for which {\it quinary symbols\/} (49) (54) (60) (61) have
been assigned. Of these ten points {\it four\/}, namely
${\sc a}'$,~${\sc b}'$,~${\sc c}'$,~${\sc d}_1$, are situated
{\it in the plane\/}~${\sc a} {\sc b} {\sc c}$, and have
accordingly been represented [20.] by {\it ternary symbols\/}
also: and we may take the particular symbol of this sort,
$${\sc a}' = (0 \, 1 \, 1),$$
as a {\it type\/} of this {\it group\/}~${\sc p}_1$;
understanding, however, that the {\it full\/} or {\it quinary
type\/} is to be recovered from this {\it ternary type}, by
{\it restoring the two omitted zeros\/}; so that we have, more
fully,
$${\sc a}'
= (0 \, 1 \, 1 \, 0 \, 0)
\equiv (1 \, 0 \, 0 \, 1 \, 1).$$
And the {\it point\/}~${\sc a}'$ itself may be considered as a
{\it Second Typical Point}.
\bigbreak
[26.]
We have thus denoted, by {\it literal\/} and by {\it quinary
symbols}, whereof some have been {\it abridged\/} to
{\it ternary\/} ones [17.], and have been also represented by
{\it types\/} [9.], not only the {\it five given
points\/}~${\sc p}_0$, but all the {\it ten lines\/}~$\Lambda_1$,
{\it ten planes\/}~$\Pi_1$, and {\it ten points\/}~${\sc p}_1$,
of what has been called, in [1.], the {\it First Construction}.
And it is evident that we have, at this stage, {\it ten
triangles\/}~${\sc t}_1$, namely the ten,
$${\sc a} {\sc d} {\sc e}, \hbox{ \&c.};\quad
{\sc b} {\sc c} {\sc e}, \hbox{ \&c.};\quad
{\sc b} {\sc c} {\sc d}, \hbox{ \&c.};\quad
\hbox{and}\quad {\sc a} {\sc b} {\sc c},$$
whereof each is contained in a plane~$\Pi_1$; and also {\it five
pyramids\/}~${\sc r}_1$, each bounded by {\it four\/} of these
{\it triangles}, namely, the pyramids,
$${\sc b} {\sc c} {\sc d} {\sc e},\quad
{\sc c} {\sc a} {\sc d} {\sc e},\quad
{\sc a} {\sc b} {\sc d} {\sc e},\quad
{\sc a} {\sc b} {\sc c} {\sc d},$$
which may be called the pyramids
${\sc a}$,~${\sc b}$,~${\sc c}$,~${\sc d}$,~${\sc e}$; each
marked by the literal symbol of {\it that one\/} of the five
points~${\sc p}_0$, which is {\it not a corner\/} of the pyramid.
\bigbreak
[27.]
It may be remarked, that {\it ten arbitrary lines\/} in space
{\it intersect}, generally, {\it ten arbitrary planes}, in
{\it one hundred points\/}; but that this {\it number\/} of
intersections $\Lambda_1 \cdot \Pi_1$ is {\it here\/} reduced to
{\it fifteen}, whereof only {\it ten\/} are {\it new\/}; because
{\it each\/} of the {\it five points\/}~${\sc p}_0$
{\it counts\/} as {\it twelve\/}, since in each of those points
{\it four lines cut\/} (each) {\it three planes}, while
{\it each\/} of the {\it ten planes contains three lines\/}; so
that {\it thirty binary combinations\/} are {\it not cases of
intersection}, and {\it sixty\/} such cases conduct only to the
five {\it old\/} (or given) points. This sort of
{\it arithmetical verification\/} of the accuracy of an
{\it enumeration\/} of {\it derived points}, or lines, or planes,
will be found useful in more complex cases, although it was not
necessary here.
\bigbreak
[28.]
Proceeding to a {\it Second Construction\/} [1.], we may begin by
determining the {\it lines\/}~$\Lambda_2$, whereof each connects
some {\it two\/} (at least) of the {\it fifteen points\/}
${\sc p}_0$,~${\sc p}_1$, but {\it not\/} any two of the
{\it five\/} points~${\sc p}_0$, since otherwise it would be a
line~$\Lambda_1$. If the 15 points to be connected were
{\it independent}, they would give, generally, by their binary
combinations, 105 lines; but the {\it ten collineations of
construction},
$${\sc b} {\sc c} {\sc a}', \hbox{ \&c.};\quad
{\sc d} {\sc a} {\sc a}_2, \hbox{ \&c.};\quad
{\sc e} {\sc a} {\sc a}_1, \hbox{ \&c.};\quad
\hbox{and}\quad {\sc e} {\sc d} {\sc d}_1,$$
show that 30 of these {\it combinations\/} are to be rejected, as
giving only the ten old lines. The remaining number, 75, is
still farther reduced by the consideration that we have
(comp.~(79)) the {\it fifteen derived collineations},
$${\sc a} {\sc a}' {\sc d}_1, \hbox{ \&c.};\quad
{\sc a} {\sc b}_1 {\sc c}_2, \hbox{ \&c.};\quad
{\sc a} {\sc c}_1 {\sc b}_2, \hbox{ \&c.};\quad
{\sc d} {\sc a}' {\sc a}_1, \hbox{ \&c.};\quad
{\sc e} {\sc a}' {\sc a}_2, \hbox{ \&c.};$$
which represent only {\it fifteen new lines\/}, of a
{\it group\/} which we shall denote by $\Lambda_{2,1}$, but
{\it count\/} (comp.~[27.]) as 45 binary combinations of the 15
points. There remain only 30 such combinations to be
considered; and these give in fact a {\it second group},
$\Lambda_{2,2}$, consisting of {\it thirty lines of second
construction\/}: namely the {\it thirty edges\/} of the {\it five
new pyramids\/}~${\sc r}_2$,
$${\sc c}' {\sc b}' {\sc a}_2 {\sc a}_1,\quad
{\sc a}' {\sc c}' {\sc b}_2 {\sc b}_1,\quad
{\sc b}' {\sc a}' {\sc c}_2 {\sc c}_1,\quad
{\sc a}_2 {\sc b}_2 {\sc c}_2 {\sc d}_1,\quad
{\sc a}_1 {\sc b}_1 {\sc c}_1 {\sc d}_1,$$
which are respectively {\it inscribed\/} in the five former
pyramids~${\sc r}_1$ [26.], and are {\it homologous\/} to them,
the five given points ${\sc a} \, \dots \, {\sc e}$ being the
respective {\it centres of homology\/}; for example,
${\sc c}' = {\sc a} {\sc b} \cdot {\sc c} {\sc d} {\sc e}$,
\&c. The corresponding {\it planes of homology\/} will present
themselves somewhat later, in connexion with the
points~${\sc p}_2$.
\bigbreak
[29.]
On the whole, then, there are only {\it forty-five distinct lines
of second construction\/} $\Lambda_2$; and these naturally divide
themselves into {\it two groups}, of 15 lines $\Lambda_{2,1}$,
and 30 lines $\Lambda_{2,2}$, as above. {\it Each\/} line of the
{\it first group\/}~$\Lambda_{2,1}$ connects {\it one\/}
point~${\sc p}_0$ with {\it two\/} points ${\sc p}_1$; as each
line~$\Lambda_1$ had connected {\it one\/} point~${\sc p}_1$ with
{\it two\/} points~${\sc p}_0$, but {\it no\/} line of the
{\it second group\/}~$\Lambda_{2,2}$ connects, at this stage of
the construction, more than {\it two\/} points, which are
{\it both\/} points~${\sc p}_1$. Through {\it no\/}
point~${\sc p}_0$, therefore, can we draw {\it any
line\/}~$\Lambda_{2,2}$; but through {\it each\/}
point~${\sc p}_0$ we can draw {\it three
lines\/}~$\Lambda_{2,1}$; and each of these is determined as the
{\it intersection of two planes\/}~$\Pi_1$ through that point, or
as {\it crossing two opposite edges\/} of that
{\it pyramid\/}~${\sc r}_1$, which has {\it not\/} the
point~${\sc p}_0$ for a corner (comp.~[26.]): for example,
${\sc a} {\sc a}' {\sc d}_1$ is the intersection of
${\sc a} {\sc b} {\sc c}$, ${\sc a} {\sc d} {\sc e}$, and crosses
the lines ${\sc b} {\sc c}$, ${\sc d} {\sc e}$. And besides
being, as in [28.], the {\it edges\/} of certain {\it other\/}
and {\it inscribed\/} pyramids~${\sc r}_2$, the 30
lines~$\Lambda_{2,2}$ are also the {\it sides\/} of {\it ten new
triangles\/}~${\sc t}_2$, namely,
$${\sc d}_1 {\sc a}_1 {\sc a}_2, \hbox{ \&c.};\quad
{\sc c}_1 {\sc b}_1 {\sc a}', \hbox{ \&c.};\quad
{\sc c}_2 {\sc b}_2 {\sc a}', \hbox{ \&c.};\quad
\hbox{and}\quad {\sc a}' {\sc b}' {\sc c}',$$
situated {\it in the ten planes\/}~$\Pi_1$, and {\it inscribed\/}
in the {\it ten old triangles\/}~${\sc t}_1$, to which also they
are {\it homologous\/}; the corresponding {\it centres of
homology\/} being the ten points~${\sc p}_1$, in the same order,
$${\sc a}', \hbox{ \&c.};\quad
{\sc a}_2, \hbox{ \&c.};\quad
{\sc a}_1, \hbox{ \&c.};\quad
\hbox{and}\quad {\sc d}_1, \hbox{ as before}.$$
The {\it axes of homology\/} of these {\it ten pairs of
triangles\/} ${\sc t}_1$,~${\sc t}_2$, will offer themselves a
little later, in connexion with points~${\sc p}_2$.
\bigbreak
[30.]
All this may be considered as evident from {\it geometry\/}
alone, at least with the assistance of {\it literal symbols},
such as those used above. But to deduce the same things by
{\it calculation\/}, with {\it quinary symbols\/} and
{\it types}, on the plan of the present Paper, we may observe
that the symbolical equation,
$$(1 \, 0 \, 0 \, 0 \, 0)
+ (0 \, 1 \, 1 \, 0 \, 0)
+ (0 \, 0 \, 0 \, 1 \, 1)
= (1 \, 1 \, 1 \, 1 \, 1),$$
considered as a type of all equations of the same form, proves by
(18) or (72) that each point~${\sc p}_1$ can, in three different
ways, be combined with another point~${\sc p}_1$, so that their
joining line shall pass through a point~${\sc p}_0$; and that
thus the {\it group\/} of the 15 lines $\Lambda_{2,1}$ arises, of
which the line ${\sc a} {\sc a}' {\sc d}_1$ is a specimen, and may
be called a {\it Second Typical Line\/} (the {\it first\/} such
line having been ${\sc b} {\sc c}$, by [23.]). The
{\it complete\/} quinary symbol of a {\it point\/} on this line
is $(t \, u \, u \, v \, v)$, which is however congruent to
one of the form $(t \, u \, u \, 0 \, 0)$, and may therefore be
abridged to the ternary symbol $(t \, u \, u)$, or
$(x \, y \, y)$; and the quinary symbol of a {\it plane\/}
through the same line is of the form
$[0 \, m \, \overline{m} \, r \overline{r}]$, or
$[0 \, t \, \overline{t} \, u \overline{u}]$; we may therefore,
by [21.] (comp.~[23.]) consider the two expressions,
$$(x \, y \, y),
\hbox{ and }
[0 \, t \, \overline{t} \, u \overline{u}],$$
as being not only {\it local and tangential symbols\/} for the
{\it particular\/} (or {\it typical\/}) {\it line
${\sc a} {\sc a}' {\sc d}_1$ itself}, but also {\it local and
tangential types\/} for the {\it group\/}~$\Lambda_{2,1}$; or as
the {\it point-type}, and the {\it plane-type}, of that group.
\bigbreak
[31.]
The two points ${\sc p}_1$, of which the quinary symbols have
been thus combined in [30.], had {\it no common coordinate
different from zero\/}; but there remains to be considered the
case, in which two points of that group {\it have\/} such a
coordinate: for example, when the points have for their symbols,
$$(1 \, 0 \, 1 \, 0 \, 0) \hbox{ and } (1 \, 1 \, 0 \, 0 \, 0),
\hbox{ or } (1 \, 0 \, 1) \hbox{ and } (1 \, 1 \, 0).$$
The {\it point-symbol\/} and {\it plane-symbol\/} of the
{\it line\/}~$\Lambda_2$ connecting {\it these\/} two
points~${\sc p}_1$ are easily seen to be (with the same
significations of $\sigma$ and $\overline{\sigma}$ as before),
$$(\sigma \, t \, u \, 0 \, 0),
\hbox{ or }
(\sigma \, t \, u),
\hbox{ and }
[\overline{t} \, t \, t \, u \overline{\sigma}];$$
but no choice of the arbitrary {\it ratio}, $t : u$, with
$\sigma = t + u$, will reduce the symbol
$(\sigma \, t \, u)$ to denote {\it any one\/} of the 15 points
${\sc p}_0$,~${\sc p}_1$, except the {\it two\/} points
${\sc p}_1$ (in this case ${\sc b}'$ and ${\sc c}'$), by joining
which the line is obtained; considering therefore the last two
{\it symbols\/} as {\it types\/}, we see that they represent a
{\it second group}, consisting of {\it thirty
lines\/}~$\Lambda_{2,2}$; but that there can be {\it no third
group}, of {\it lines\/}~$\Lambda_2$ of {\it second
construction}. The {\it particular line\/} ${\sc b}' {\sc c}'$,
which the symbols in the present paragraph represent may be taken
as {\it typical\/} of this {\it second group\/}; and may be
called (comp.~[23.] and [30.]) a {\it Third Typical Line\/} of
the System, or {\it Net}, determined by the five given points
${\sc a} \, \ldots \, {\sc e}$. And the {\it pyramids\/}
${\sc r}_1$,~${\sc r}_2$, and {\it triangles\/}
${\sc t}_1$,~${\sc t}_2$, of first and second constructions, of
which the {\it literal symbols\/} have been assigned in [26.]
[28.] [29.], might also have easily been suggested and studied,
by {\it quinary\/} symbols and types alone.
\bigbreak
[32.]
As regards the {\it Planes\/}~$\Pi_2$ of {\it Second
Construction\/} [1.], it is easily seen that no such plane
contains any {\it two\/} points~${\sc p}_0$, or any {\it one\/}
line~$\Lambda_1$; for example, the {\it first typical line\/}
${\sc b} \, {\sc c}$ [23.] {\it contains\/} the point ${\sc a}'$;
and if we {\it connect\/} it with any one of the four points
${\sc a}$,~${\sc b}'$,~${\sc c}'$,~${\sc d}_1$, we only get a
plane~$\Pi_1$, namely ${\sc a} {\sc b} {\sc c}$; if with
${\sc d}$,~${\sc a}_1$,~${\sc b}_2$, or ${\sc c}_2$, we get
another plane~$\Pi_1$, namely ${\sc b} {\sc c} {\sc d}$; and if
with any one of the four remaining points ${\sc e}$, ${\sc a}_2$,
${\sc b}_1$, ${\sc c}_1$, the plane ${\sc b} {\sc c} {\sc e}$ is
obtained. Accordingly the general symbol
$[\overline{\sigma} \, 0 \, 0 \, t \, u]$, in [23.], for a plane
through the line ${\sc b} {\sc c}$, gives $\sigma = 0$, or
$t = 0$, or $u = 0$, when we seek to particularize it, by the
first, the second, or the third of these three sets of conditions
respectively.
\bigbreak
[33.]
But if we take the symbol
$[0 \, t \, \overline{t} \, u \, \overline{u}]$, in [30.], for a
plane through the {\it second typical line\/}
${\sc a} {\sc a}' {\sc d}_1$, and seek to particularize
{\it this\/} symbol by the condition of passing through some one
of the eight points~${\sc p}_1$ which are not situated upon it,
we are conducted to the following results. The points
${\sc b}'$,~${\sc c}'$ give $t = 0$, and the points
${\sc a}_1$,~${\sc a}_2$ give $u = 0$; these points therefore
give only two planes $\Pi_1$, namely the two planes
${\sc a} {\sc b} {\sc c}$ and ${\sc a} {\sc d} {\sc e}$,
of which the line~$\Lambda_2$ is the intersection. But the
points ${\sc b}_1$,~${\sc c}_2$ give $t = u$, and the points
${\sc c}_1$,~${\sc b}_2$ give $t = - u$; {\it these points\/}
therefore give {\it two planes\/} of a {\it new group\/}
$\Pi_{2,1}$, namely (comp.~[20.]) the two following:
$$\hbox{plane } {\sc a} {\sc a}' {\sc d}_1 {\sc b}_1 {\sc c}_2
= [0 \, 1 \, \overline{1} \, 1 \, \overline{1}];\quad
\hbox{plane } {\sc a} {\sc a}' {\sc d}_1 {\sc c}_1 {\sc b}_2
= [0 \, 1 \, \overline{1} \, \overline{1} \, 1];$$
which are of the same {\it type\/} as the plane (96), namely,
$$\hbox{plane } {\sc a} {\sc b}_1 {\sc c}_2 {\sc c}_1 {\sc b}_2
= [0 \, 1 \, 1 \, \overline{1} \, \overline{1}].$$
There are {\it fifteen\/} such {\it planes\/}~$\Pi_{2,1}$, as the
type sufficiently shows; each passes through {\it one
point\/}~${\sc p}_0$, and contains {\it two
lines\/}~$\Lambda_{2,1}$, containing also {\it four lines\/}
$\Lambda_{2,2}$; as, for instance, the last-mentioned plane
${\sc a} {\sc b}_1 {\sc c}_2 {\sc c}_1 {\sc b}_2$, which we shall
call (comp.~[24.]) the {\it Second Typical Plane}, contains the
{\it two\/} lines
${\sc a} {\sc b}_1 {\sc c}_2$,
${\sc a} {\sc c}_1 {\sc b}_2$
[28.], and the {\it four\/} lines
${\sc b}_1 {\sc c}_1$,
${\sc c}_1 {\sc c}_2$,
${\sc c}_2 {\sc b}_2$,
${\sc b}_2 {\sc b}_1$;
that is to say, the {\it two diagonals\/} and the {\it four
sides\/} of the {\it quadrilateral\/}
${\sc b}_1 \, {\sc c}_1 \, {\sc c}_2 \, {\sc b}_2$, of which the
{\it plane\/}~$\Pi_{2,1}$ passes through~${\sc a}$.
\bigbreak
[34.]
We have now exhausted all the planes~$\Pi_2$ which contain any
point~${\sc p}_0$; but there exists a {\it second group of
planes}, $\Pi_{2,2}$, each of which is determined as connecting
{\it three\/} points ${\sc p}_1$, although passing through
{\it no\/} point ${\sc p}_0$. Thus if we take the {\it third
typical line\/} ${\sc b}' {\sc c}'$ [31.], and the symbol
$[\overline{t} \, t \, t \, u \, \overline{\sigma}]$ for a plane
through it, we get indeed $t = 0$, or a plane~$\Pi_1$, namely,
${\sc a} {\sc b} {\sc c}$, if we oblige the plane through
${\sc b}' {\sc c}'$ to contain ${\sc a}$, or ${\sc b}$, or
${\sc c}$, or ${\sc a}'$, or ${\sc d}_1$; and we get $u = 0$, or
$[\overline{1} \, 1 \, 1 \, 0 \, \overline{1}]$, or a
plane~$\Pi_{2,1}$, namely
${\sc d} {\sc b}' {\sc b}_1 {\sc c}' {\sc c}_1$,
as in (95), if we oblige it to contain ${\sc d}$, or ${\sc b}_1$,
or ${\sc c}_1$; while we get $\sigma = 0$, or
$[\overline{1} \, 1 \, 1 \, \overline{1} \, 0]$,
or ${\sc e} {\sc b}' {\sc b}_2 {\sc c}' {\sc c}_2$,
again as in (95), if we oblige it to contain ${\sc e}$, or
${\sc b}_2$, or ${\sc c}_2$. But there remain the two points
${\sc a}_1$ and ${\sc a}_2$, determining the two new planes
${\sc b}' {\sc c}' {\sc a}_1$ and ${\sc b}' {\sc c}' {\sc a}_2$,
for the former of which we have $t + \sigma = 0$, or $u = -2t$,
$\sigma = -t$, and therefore have the symbol
$[\overline{1} \, 1 \, 1 \, \overline{2} \, 1]$;
while for the latter we have $u = t$, $\sigma = 2t$, and
therefore the syntypical symbol
$[\overline{1} \, 1 \, 1 \, 1 \, \overline{2}]$.
There are {\it twenty planes\/} of this {\it
group\/}~$\Pi_{2,2}$, as may be at once concluded from inspection
of the {\it type\/}; among which (comp.~[19.]) we shall select
the following,
$$\hbox{plane } {\sc a}_1 {\sc b}_1 {\sc c}_1
= [1 \, 1 \, 1 \, \overline{2} \, \overline{1}],$$
and call this a {\it Third Typical Plane}. And it is evident
that these 20 planes $\Pi_{2,2}$ are the {\it twenty faces\/} of
the {\it five inscribed pyramids\/} ${\sc r}_2$ [28.], of which
the {\it edges\/} have been seen to be the {\it thirty
lines\/}~$\Lambda_{2,2}$. On the whole, then, there are only
{\it thirty-five planes\/}~$\Pi_2$ of {\it second
construction\/}; which thus divide themselves into {\it two
groups}, of {\it fifteen\/} and {\it twenty}, respectively.
\bigbreak
[35.]
To {\it verify arithmetically\/} (comp. [27.] [28.]) the
{\it completeness\/} of the foregoing {\it enumeration\/} of the
{\it planes\/}~$\Pi_2$, we may proceed as follows. In general
{\it fifteen independent points\/} would determine 455 planes,
by their {\it ternary combinations\/}; but the 25
{\it collineations\/} [28.], which give only the {\it lines\/}
$\Lambda_1$,~$\Lambda_{2,1}$ {\it account\/} for 25 such
combinations, leaving only 430 to be accounted for, by so many
{\it triangles}. Now each plane $\Pi_1$ contains three
points~${\sc p}_0$, and four points ${\sc p}_1$, connected by six
collineations; it contains therefore 29 ($= 35 - 6$) triangles,
and thus the ten planes $\Pi_1$ account for 290 triangles,
leaving only 140, situated in planes $\Pi_2$. But each of the 15
planes $\Pi_{2,1}$ contains one point~${\sc p}_0$, and four
points ${\sc p}_1$, connected by two collineations; it contains
therefore 8 ($= 10 - 2$) triangles, and thus 120 are accounted for,
leaving only 20 ternary combinations to be represented, by
triangles in other planes $\Pi_2$. And these accordingly have
presented themselves, as the twenty faces $\Pi_{2,2}$ of the five
inscribed pyramids~${\sc r}_2$. It must be mentioned, that the
{\it enumeration\/} and {\it classification\/} of the foregoing
{\it lines\/} and {\it planes\/} had been completely performed by
{\sc M\"{o}bius}, although with an entirely different
{\it notation\/} and {\it analysis}.
\bigbreak
[36.]
It is much more difficult, however, or at least without the aid
of {\it types\/} it {\it would\/} be so, to {\it enumerate\/} and
{\it classify\/} what we have called in [1.] the
{\it Points\/}~${\sc p}_2$ of {\it Second Construction\/}; and to
assign their chief {\it geometrical relations\/}, to each other,
and to the {\it five given\/} and {\it ten\/} (formerly)
{\it derived\/} points, ${\sc p}_0$ and ${\sc p}_1$. In fact, it
is obvious that these {\it new points\/}~${\sc p}_2$, being (by
their definition) {\it all the intersections\/} of lines
$\Lambda_1$ or $\Lambda_2$ with planes $\Pi_1$ or $\Pi_2$, which
have {\it not already occurred}, as points ${\sc p}_0$ or
${\sc p}_1$, may be expected to be (comp.~[2.]) considerably
{\it more numerous}, than either the {\it lines\/} or the
{\it planes\/} themselves.
\bigbreak
[37.]
The {\it total number\/} of {\it derived lines and planes}, so
far, is exactly {\it one hundred\/}; namely 55 lines~$\Lambda$,
and 45 planes~$\Pi$, of first and second constructions. Their
{\it binary combinations}, of the form $\Lambda \Pi$, are
therefore 2475 in number; but as it is not difficult to prove
that there are 240 distinct cases of {\it coincidence\/} of line
with plane (or of a plane {\it containing\/} a line), we must
subtract this from the former number, and thus there remain only
2235 cases of {\it intersection}, of the kind which we have
proposed to consider. {\it Every one}, however, of these 2235
cases, must be accounted for, either as a {\it given
point\/}~${\sc p}_0$, or as a {\it derived point\/}~${\sc p}_1$
of {\it first\/} construction, or finally as one of those
{\it new points\/}~${\sc p}_2$, of which we have proposed to
accomplish the {\it enumeration}, and to determine the natural
{\it groups}, as represented by their respective {\it types}.
\bigbreak
[38.]
We saw, in [27.], that each point ${\sc p}_0$, as for instance
the point~${\sc a}$, represents {\it twelve intersections\/} of
the form $\Lambda_1 \cdot \Pi_1$: and it is easy to prove that
the same point~${\sc p}_0$ represents {\it twelve other\/}
intersections of the form $\Lambda_1 \cdot \Pi_{2,1}$;
{\it twelve\/} of the form $\Lambda_{2,1} \cdot \Pi_1$; and
{\it three}, of the form $\Lambda_{2,1} \cdot \Pi_{2,1}$; but
none of any other form. It represents therefore, in the whole, a
system of 39 {\it intersections}, included in the {\it general
form\/} $\Lambda \cdot \Pi$; and we must, for this reason,
subtract 195 ($= 5 \times 39$) from 2235, leaving 2040
{\it other\/} cases of intersection of line with plane, to be
accounted for by the old and new {\it derived points},
${\sc p}_1$ and ${\sc p}_2$.
\bigbreak
[39.]
An analysis of the same kind shows, that each of the {\it ten
points\/} of {\it first construction}, as for example the
{\it typical point\/}~${\sc a}'$ [25.], represents {\it one\/}
intersection of the form $\Lambda_1 \cdot \Pi_1$; {\it six}, of
the form, $\Lambda_1 \cdot \Pi_{2,1}$; {\it six}, of the form
$\Lambda_1 \cdot \Pi_{2,2}$; {\it six}, of the form
$\Lambda_{2,1} \cdot \Pi_1$; {\it twelve}, of the form
$\Lambda_{2,1} \cdot \Pi_{2,1}$; {\it eighteen}, of the form
$\Lambda_{2,1} \cdot \Pi_{2,2}$; {\it eighteen}, of the form
$\Lambda_{2,2} \cdot \Pi_1$; {\it twenty-four}, of the form
$\Lambda_{2,2} \cdot \Pi_{2,1}$; and {\it twenty-four\/} others,
of the remaining form $\Lambda_{2,2} \cdot \Pi_{2,2}$. It
represents, therefore, in all, 115 intersections
$\Lambda \cdot \Pi$; and there remain only 890
($= 2040 - 1150$) cases of intersection to be accounted for, or
represented, by the points~${\sc p}_2$ of which we are in search.
But all these 890 cases of intersection {\it must\/} be accounted
for by {\it such new points}, if the investigation is to be
considered as {\it complete}.
[40.]
A {\it first\/}, but important, and well-known {\it group\/} of
such points~${\sc p}_2$, consists of the {\it ten points\/}
(already considered in Part~I. of this Paper),
$${\sc a}'', \hbox{ \&c.};\quad
{\sc a}'_2, \hbox{ \&c.};\quad
{\sc a}'_1, \hbox{ \&c.};\quad
\hbox{and}\quad {\sc d}'_1,$$
namely, the {\it harmonic conjugates\/} of the {\it ten points\/}
${\sc p}_1$, with respect to the {\it ten lines\/}~$\Lambda_1$,
which we shall call collectively the points, or the group,
${\sc p}_{2,1}$; and among which we shall select the point
$${\sc a}'' = (0 \, 1 \, \overline{1}),$$
as a {\it Third Typical Point\/} of the {\it Net}. In fact, it
is what we have called a point~${\sc p}_2$, because, without
belonging to either of the two former groups
${\sc p}_0$~${\sc p}_1$, it is an {\it intersection\/}
$\Lambda_1 \cdot \Pi_{2,2}$; or rather, it represents {\it six\/}
such intersections, of the line ${\sc b} {\sc c}$ with planes of
second construction, and of the second group: namely, with two
such through ${\sc b}' {\sc c}'$, two through
${\sc b}_2 {\sc c}_2$, and two through ${\sc b}_1 {\sc c}_1$,
being pairs of faces [28.] of three pyramids ${\sc r}_2$,
inscribed in those three pyramids ${\sc r}_1$, which have been
distinguished, in [26.] by the letters
${\sc a}$,~${\sc d}$,~${\sc e}$. The same point~${\sc a}''$ is
also the intersection of the same line ${\sc b} {\sc c}$ with
{\it three planes\/} $\Pi_{2,1}$; namely, with the three which
connect, two by two, the three lines ${\sc b}' {\sc c}'$,
${\sc b}_2 {\sc c}_2$, ${\sc b}_1 {\sc c}_1$, and contain the
three points ${\sc a}$,~${\sc d}$,~${\sc e}$. It is also, in
{\it six\/} ways, the intersection of one or other of these three
last lines~$\Lambda_{2,2}$ with a plane~$\Pi_1$; in {\it three\/}
ways, with a plane~$\Pi_{2,1}$; and in {\it twelve\/} ways, with
a plane $\Pi_{2,2}$; so that a {\it single
point\/}~${\sc p}_{2,1}$ represents {\it thirty intersections\/}
of the form $\Lambda \cdot \Pi$; and the group of the {\it ten\/}
such points represents 300 such intersections. We have therefore
only to account for 590 ($= 890 - 300$) intersections
$\Lambda \cdot \Pi$, by {\it other groups\/}~${\sc p}_{2,2}$,
\&c., of points of {\it second construction}.
\bigbreak
[41.]
A {\it second group},~${\sc p}_{2,2}$ of such points~${\sc p}_2$
has already presented itself, in the case of the {\it traces\/}
${\sc a}_0$,~${\sc b}_0$,~${\sc c}_0$ [18.] of the {\it lines\/}
${\sc a}_1 {\sc a}_2$, ${\sc b}_1 {\sc b}_2$,
${\sc c}_1 {\sc c}_2$, on the plane ${\sc a} {\sc b} {\sc c}$.
The {\it ternary\/} symbol of the point~${\sc a}_0$ has been
found (77) (92) to be $(\overline{1} \, 1 \, 1)$, its
{\it quinary\/} symbol is therefore
$(\overline{1} \, 1 \, 1 \, 0 \, 0)$, which is {\it congruent\/}
(10) with $(2 \, 0 \, 0 \, 1 \, 1)$; hence in the {\it full}, or
{\it quinary sense\/} [9.], this point ${\sc a}_0$ is
{\it syntypical\/} with the following {\it other point, in the
same plane\/}~${\sc a} {\sc b} {\sc c}$,
$${\sc a}''' = (2 \, 1 \, 1),$$
which we shall call a {\it Fourth Typical Point}, and shall
consider as representing the {\it group\/}~${\sc p}_{2,2}$; this
group consisting of {\it thirty\/} such {\it points}, namely of
two on each of the 15 lines~$\Lambda_{2,1}$.
\bigbreak
[42.]
Each of these thirty points~${\sc p}_{2,2}$ represents {\it seven
intersections\/} of line with plane; namely two of each of the
three forms
$\Lambda_{2,1} \cdot \Pi_{2,1}$,
$\Lambda_{2,1} \cdot \Pi_{2,2}$,
$\Lambda_{2,2} \cdot \Pi_{2,1}$,
and one of the form
$\Lambda_{2,2} \cdot \Pi_1$.
For example, the typical point ${\sc a}'''$, which is the
intersection of the {\it two lines\/}
${\sc a} {\sc a}' {\sc d}_1$ and ${\sc b}' {\sc c}'$,
is at the same time the intersection of the former
line~$\Lambda_{2,1}$ with each of the four planes~$\Pi_2$ which
contain the latter line~$\Lambda_{2,2}$; being also the
intersection of this last line ${\sc b}' {\sc c}'$ with a
plane~$\Pi_{1}$, namely ${\sc a} {\sc d} {\sc e}$, and with two
planes $\Pi_{2,1}$ which contain the first line
${\sc a} {\sc a}' {\sc d}_1$. The {\it group\/}~${\sc p}_{2,2}$
represents therefore 210 intersections $\Lambda \cdot \Pi$; and
there remain only 380 ($= 590 - 210$) intersections of this
standard form, to be accounted for by {\it other groups\/} of
{\it second construction}, such as ${\sc p}_{2,3}$ \&c.
\bigbreak
[43.]
In investigating such {\it groups}, we need only seek for
{\it typical points\/}; and because every such {\it point\/} is
on a {\it line\/} of one of the {\it three forms\/} $\Lambda_1$,
$\Lambda_{2,1}$, $\Lambda_{2,2}$, we may confine ourselves to the
{\it three typical lines},
$${\sc b} {\sc c},\quad
{\sc a} {\sc a}' {\sc d}_1,\quad
{\sc b}' {\sc c}';
\quad \hbox{or}\quad
(0 \, t \, u),\quad
(t \, u \, u),\quad
(\sigma \, t \, u);$$
in which, as before, $\sigma = t + u$, and in which the ratio of
$t$ to $u$ is to be determined. And because a line in the
plane~${\sc a} {\sc b} {\sc c}$ intersects any {\it other
plane\/} in the point in which it intersects the {\it line\/}
which is the {\it trace\/} of the latter plane upon the former,
we need only, for the present purpose, consider these lines, or
traces: whereof there are, by what has been already seen,
{\it seven\/} distinct {\it ternary types}, namely the following:
$$[1 \, 0 \, 0],\quad
[0 \, 1 \, \overline{1}],\quad
[\overline{1} \, 1 \, 1],\quad
[1 \, 1 \, 1],\quad
[0 \, 1 \, 1],\quad
[\overline{2} \, 1 \, 1],\quad
[\overline{2} \, 1 \, \overline{1}];$$
which answer to the {\it seven typical traces\/} of planes,
$${\sc b} {\sc c},\quad
{\sc a} {\sc a}' {\sc d}_1,\quad
{\sc b}' {\sc c}',\quad
{\sc a}'' {\sc b}'' {\sc c}'',\quad
{\sc a} {\sc a}'',\quad
{\sc d}_1 {\sc a}'',\quad
{\sc a}' {\sc c}_0.$$
There are 22 ($= 3 + 3 + 3 + 1 + 3 + 3 + 6$) such {\it lines},
answering to 44
($= {3 \mathbin{.} 2} + {3 \mathbin{.} 3} + {3 \mathbin{.} 4}
+ {1 \mathbin{.} 2} + {3 \mathbin{.} 1} + {3 \mathbin{.} 2}
+ {6 \mathbin{.} 1}$)
{\it planes\/}; namely to {\it all\/} the 45 planes
$\Pi_1$,~$\Pi_2$, {\it except\/} the particular plane
${\sc a} {\sc b} {\sc c}$, on which the {\it traces\/} are thus
taken. And we have now to {\it combine\/} these {\it seven types
of lines}, with the {\it three symbols of points},
$(0 \, t \, u)$, $(t \, u \, u)$, $(\sigma \, t \, u)$,
according to the general law $lx + my + nz = 0$ (76).
\bigbreak
[44.]
The line ${\sc b} {\sc c}$ is itself one of the three traces of
the first type; and it intersects the twelve other traces, of the
five first types, only in points which have been already
considered. The line ${\sc a} {\sc a}' {\sc d}_1$ is, in like
manner, a trace of the second type; and it gives no new point, by
its intersections with the eight other traces, of the three first
types; but its intersection with the common trace
${\sc a}'' {\sc b}'' {\sc c}''$, of the two planes
${\sc a}_1 {\sc b}_1 {\sc c}_1$ and
${\sc a}_2 {\sc b}_2 {\sc c}_2$ [19.], which is the only line of
the fourth type, gives what we shall call a {\it Fifth Typical
Point}, namely the following:
$${\sc a}^{\rm IV} = (\overline{2} \, 1 \, 1);
\hbox{ or more fully, }
{\sc a}^{\rm IV} = (\overline{2} \, 1 \, 1 \, 0 \, 0)
\equiv (3 \, 0 \, 0 \, 1 \, 1).$$
This last quinary symbol shows that the point~${\sc a}^{\rm IV}$
is syntypical with this other point in the
plane~${\sc a} {\sc b} {\sc c}$,
$${\sc a}_1^{\rm IV} = (3 \, 1 \, 1 \, 0 \, 0) = (3 \, 1 \, 1);$$
so that this {\it plane\/} contains {\it six
points\/}~${\sc p}_{2,3}$, which (in the {\it quinary\/} sense)
belong to one {\it common group}, although their two {\it ternary
types\/} are {\it different}. In fact, the
point~${\sc a}_1^{\rm IV}$ is the common intersection of the line
${\sc a} {\sc a}' {\sc d}_1$ with the two planes
$[1 \, \overline{2} \, \overline{1} \, 1 \, 1]$ and
$[1 \, \overline{1} \, \overline{2} \, 1 \, 1]$, or
${\sc b}' {\sc c}_1 {\sc c}_2$ and
${\sc c}' {\sc b}_1 {\sc b}_2$,
as the point~${\sc a}^{\rm IV}$ is the common intersection of the
same line with the two planes
$[1 \, 1 \, 1 \, \overline{2} \, \overline{1}]$
$[1 \, 1 \, 1 \, \overline{1} \, \overline{2}]$,
or ${\sc a}_1 {\sc b}_1 {\sc c}_1$ and
${\sc a}_2 {\sc b}_2 {\sc c}_2$, as above.
\bigbreak
[45.]
There are {\it thirty\/} distinct points ${\sc p}_{2,3}$, of this
{\it third group\/} of {\it second construction\/}; and
{\it each\/} represents {\it two\/} (but only two) intersections,
which are both of the form $\Lambda_{2,1} \cdot \Pi_{2,2}$. The
{\it group\/} therefore represents a system of 60 intersections
$\Lambda \cdot \Pi$; and there remain only 320 ($= 380 - 60$)
such intersections to be accounted for by {\it other\/} points,
or groups, such as ${\sc p}_{2,4}$, \&c. It will be found that
we have now exhausted all the points, or groups, of
{\it second\/} construction, which are situated on
lines~$\Lambda_{2,1}$; but that two other groups of
points~${\sc p}_2$ may be determined on lines~$\Lambda_1$, by
combining the typical line ${\sc b} {\sc c}$ with the two last
sets of traces [43.] as follows.
\bigbreak
[46.]
Combining thus ${\sc b} {\sc c}$ with ${\sc d}_1 {\sc c}''$ and
${\sc d}_1 {\sc b}''$, or with the traces
$[1 \, 1 \, \overline{2}]$ and $[1 \, \overline{2} \, 1]$, we get
the two following points, of a {\it fourth group of second
construction},
$${\sc a}^{\rm V} = (0 \, 2 \, 1);\quad
{\sc a}_1^{\rm V} = (0 \, 1 \, 2);$$
whereof the former may be taken as a {\it Sixth Typical Point}.
There are {\it twenty points\/} of this group ${\sc p}_{2,4}$,
whereof each represents {\it three\/} intersections, of the form
$\Lambda_1 \cdot \Pi_{2,2}$; for example, the typical
point~${\sc a}^{\rm V}$ is the common intersection of the
line~${\sc b} {\sc c}$ with the three planes
${\sc c}' {\sc a}_1 {\sc a}_2$,
${\sc d}_1 {\sc a}_1 {\sc b}_1$,
${\sc d}_1 {\sc a}_2 {\sc b}_2$;
the group therefore represents {\it sixty\/} intersections
$\Lambda \cdot \Pi$, and there remain 260 ($= 320 - 60$) to be
accounted for.
\bigbreak
[47.]
Again, combining ${\sc b} {\sc c}$ with ${\sc c}' {\sc b}_0$, and
with ${\sc b}' {\sc c}_0$, or with
$[1 \, \overline{1} \, \overline{2}]$ and
$[1 \, \overline{2} \, \overline{1}]$, we get the two following
other points, belonging to a {\it fifth group\/} of {\it second
construction},
$${\sc a}^{\rm VI} = (0 \, 2 \, \overline{1});\quad
{\sc a}_1^{\rm VI} = (0 \, \overline{1} \, 2);$$
whereof the first may be said to be a {\it Seventh Typical
Point}. There are {\it twenty\/} points of this new
group~${\sc p}_{2,5}$, whereof each represents only {\it one\/}
intersection, of the form $\Lambda_1 \cdot \Pi_{2,2}$; for
example,
${\sc a}^{\rm VI} = {\sc b} {\sc c} \cdot {\sc c}' {\sc b}_1 {\sc b}_2$.
We are therefore to subtract 20 from the recent number 260; and
thus there remain still 240 intersections to be accounted for, by
new points ${\sc p}_2$ upon the lines $\Lambda_{2,2}$; since the
lines $\Lambda_1$ as well as $\Lambda_{2,1}$ have been exhausted,
as on examination will easily appear.
\bigbreak
[48.]
The line ${\sc b}' {\sc c}'$ intersects the traces
${\sc b} {\sc b}''$ and ${\sc c} {\sc c}''$ of the {\it fifth
type\/} [43.] in the two following points, of a {\it sixth
group\/} of {\it second construction},
$${\sc a}^{\rm VII} = (1 \, 2 \, \overline{1});\quad
{\sc a}_1^{\rm VII} = (1 \, \overline{1} \, 2);$$
whereof the former may be called an {\it Eighth Typical Point}.
There are {\it sixty\/} points of this new group,
${\sc p}_{2,6}$, whereof each represents {\it one\/}
intersection, of the form $\Lambda_{2,2} \cdot \Pi_{2,1}$; for
example ${\sc a}^{\rm VII}$ is the intersection of the
line~${\sc b}' {\sc c}'$ with the
plane~${\sc b} {\sc c}_1 {\sc a}_2 {\sc a}_1 {\sc c}_2$;
there remain therefore 180 ($= 240 - 60$) intersections
$\Lambda \cdot \Pi$ to be still accounted for, by other
points~${\sc p}_2$, on the same set of lines~$\Lambda_{2,2}$.
\bigbreak
[49.]
The traces ${\sc d}_1 {\sc b}''$, ${\sc d}_1 {\sc c}''$, which
belong to the {\it sixth type\/} [43.] intersect the line
${\sc b}' {\sc c}'$ in two new points, namely
$${\sc a}^{\rm VIII} = (3 \, 2 \, 1);\quad
{\sc a}_1^{\rm VIII} = (3 \, 1 \, 2);$$
which belong to a {\it seventh group\/}~${\sc p}_{2,7}$, of
{\it second construction}, and of which the former may be regarded
as a {\it Ninth Typical Point}. There are {\it sixty\/} points
of this group, namely two on each of the 30
lines~$\Lambda_{2,2}$; and each is the intersection of
{\it one\/} such line with {\it two\/} distinct
planes~$\Pi_{2,2}$; their {\it group\/} therefore represents a
system of 120 such intersections; and only 60 ($= 180 - 120$)
intersections {\it remain\/} to be accounted for, by
{\it other\/} points of this last {\it form},
$\Lambda_{2,2} \cdot \Pi_{2,2}$.
\bigbreak
[50.]
Accordingly, when we combine the line ${\sc b}' {\sc c}'$ with
the traces ${\sc a}' {\sc c}_0$, ${\sc a}' {\sc b}_0$, which are
of the {\it seventh type\/} [43.], we obtain, for the
intersections of that line~$\Lambda_{2,2}$ with two new
planes~$\Pi_{2,2}$, namely with ${\sc a}' {\sc c}_1 {\sc c}_2$
and ${\sc a}' {\sc b}_1 {\sc b}_2$ (98), two new points,
belonging to a new or {\it eighth group\/}~${\sc p}_{2,8}$ of
{\it second construction}, namely,
$${\sc a}^{\rm IX} = (2 \, 3 \, \overline{1});\quad
{\sc a}_1^{\rm IX} = (2 \, \overline{1} \, 3);$$
whereof the former may be selected, as a {\it Tenth\/} (and, for
our purpose, {\it last\/}) {\it Typical Point\/}: for the
{\it sixty\/} points of this last group represent each
{\it one\/} intersection, and thus account for {\it all\/} the
intersections which lately {\it remained\/} [49.], after all the
preceding groups had been exhausted.
\bigbreak
[51.]
We are now therefore enabled to assert that the proposed
{\it Enumeration of the Points~${\sc p}_2$ of Second
Construction}, and the proposed {\it Classification of such
Points in Groups}, have both been completely effected. For the
{\it number\/} of such {\it groups\/}
${\sc p}_{2,1},\ldots \, {\sc p}_{2,8}$
has been seen to be {\it eight\/}, represented by the 8
{\it typical points}, ${\sc a}'' \, \ldots \, {\sc a}^{\rm IX}$;
which, along with the {\it first given point\/}~${\sc a}$, and
the {\it first derived point\/}~${\sc a}'$, make up a system of
{\it ten types}, as follows:
$${\sc a} = (1 \, 0 \, 0);\quad
{\sc a}' = (0 \, 1 \, 1);\quad
{\sc a}'' = (0 \, 1 \, \overline{1});\quad
{\sc a}''' = (2 \, 1 \, 1);\quad
{\sc a}^{\rm IV} = (\overline{2} \, 1 \, 1);$$
$${\sc a}^{\rm V} = (0 \, 2 \, 1);\quad
{\sc a}^{\rm VI} = (0 \, 2 \, \overline{1});\quad
{\sc a}^{\rm VII} = (1 \, 2 \, \overline{1});\quad
{\sc a}^{\rm VIII} = (3 \, 2 \, 1);\quad
{\sc a}^{\rm IX} = (2 \, 3 \, \overline{1});$$
and the {\it number\/} of the {\it points\/}~${\sc p}_2$ is
($10 + 30 + 30 + 20 + 20 + 60 + 60 + 60 =$) 290; so that, when
combined with the points~${\sc p}_1$, they make up a system of
exactly {\it three hundred points, ${\sc p}_1$,~${\sc p}_2$,
derived from the five points\/}~${\sc p}_0$.
\bigbreak
[52.]
It is to be remembered that the three other {\it ternary types},
$${\sc d}_1 = (1 \, 1 \, 1),\quad
{\sc a}_0 = (\overline{1} \, 1 \, 1),\quad
{\sc a}_1^{\rm IV} = (3 \, 1 \, 1),$$
have been seen to represent points which are, in the
{\it quinary\/} sense, {\it syntypical\/} with
${\sc a}'$,~${\sc a}'''$,~${\sc a}^{\rm IV}$, and therefore
belong to the same three groups,
${\sc p}_1$,~${\sc p}_{2,2}$,~${\sc p}_{2,3}$; all these three
points being in the plane~${\sc a} {\sc b} {\sc c}$, and on the
line ${\sc a} {\sc a}' {\sc d}_1$. And it is evident that the
five other points,
$${\sc a}_1^{\rm V} = (0 \, 1 \, 2);\quad
{\sc a}_1^{\rm VI} = (0 \, \overline{1} \, 2);\quad
{\sc a}_1^{\rm VII} = (1 \, \overline{1} \, 2);\quad
{\sc a}_1^{\rm VIII} = (3 \, 1 \, 2);\quad
{\sc a}_1^{\rm IX} = (2 \, \overline{1} \, 3),$$
belong (as has been seen) to the same five last groups
${\sc p}_{2,4}, \ldots \, {\sc p}_{2,8}$, as the five points
above selected as typical thereof, namely the points
${\sc a}^{\rm V} \, \ldots \, {\sc a}^{\rm IX}$, and are situated
on the same two typical lines ${\sc b} {\sc c}$ and
${\sc b}' {\sc c}'$. The transition from ${\sc a}'$ to
${\sc b}'$,~${\sc c}'$, or from ${\sc a}''$ to
${\sc b}''$,~${\sc c}''$, \&c., is very easily made, by a rule
already stated [20.]; and therefore it is unnecessary to write
down here the symbols for {\it these\/} derived points,
${\sc b}'$,~${\sc b}''$, \&c., or ${\sc c}'$,~${\sc c}''$ \&c.
But we must now proceed, in the remainder of this Paper, to
investigate some of the chief {\it Geometrical Relations\/} which
connect the points, lines, and planes of the {\it Net}, so far as
they have been hitherto determined: namely to the end of the
{\it Second Construction}.
\bigbreak
\centerline{{\sc Part~III.}---{\it Applications to the Net,
continued: Enumeration and Classification of the}}
\centerline{{\it Collineations of the Fifty-Two Points in
a Plane of First Construction.}}
\nobreak\bigskip
[53.]
The plane~${\sc a} {\sc b} {\sc c}$ has been seen to contain,
besides the three points~${\sc p}_0$ which determine it, four
points~${\sc p}_1$, namely
${\sc a}'$,~${\sc b}'$,~${\sc c}'$, and ${\sc d}_1$;
and it contains forty-five points~${\sc p}_2$, namely the three
points ${\sc a}''$,~${\sc b}''$,~${\sc c}''$ of the
group~${\sc p}_{2,1}$, and six points of each of the seven
remaining groups of second construction. This
{\it plane\/}~$\Pi_1$ contains therefore {\it fifty-two points\/}
${\sc p}_0$,~${\sc p}_1$,~${\sc p}_2$; and we propose to
examine, in the first place, the various {\it relations of
collinearity\/} which connect these different points among
themselves: intending afterwards to investigate their principal
{\it harmonic and involutionary relations}.
\bigbreak
[54.]
The points on the {\it first typical line\/}~${\sc b} {\sc c}$
[23.] are, in number, {\it eight\/}; their literal symbols being,
by what precedes,
$${\sc b},\enspace
{\sc c},\enspace
{\sc a}',\enspace
{\sc a}'',\enspace
{\sc a}^{\rm V},\enspace
{\sc a}_1^{\rm V},\enspace
{\sc a}^{\rm VI},\enspace
{\sc a}_1^{\rm VI};$$
the ternary symbols corresponding to which have been shown to be,
$$(0 \, 1 \, 0),\quad
(0 \, 0 \, 1),\quad
(0 \, 1 \, 1),\quad
(0 \, 1 \, \overline{1}),\quad
(0 \, 2 \, 1),\quad
(0 \, 1 \, 2),\quad
(0 \, 2 \, \overline{1}),\quad
(0 \, \overline{1} \, 2).$$
In fact, that these eight points are all on the
line~${\sc b} {\sc c}$, is evident on mere inspection of their
{\it symbols}, which are of the common {\it form},
$$(0 \, y \, z) \quad \hbox{[23.]}.$$
\bigbreak
[55.]
The points on the {\it second typical line}, ${\sc a} {\sc a}'$
[30.], are in number {\it seven\/}: their literal symbols being,
$${\sc a},\enspace
{\sc a}',\enspace
{\sc d}_1,\enspace
{\sc a}''',\enspace
{\sc a}_0,\enspace
{\sc a}^{\rm IV},\enspace
{\sc a}_1^{\rm IV};$$
and their ternary symbols being,
$$(1 \, 0 \, 0),\quad
(0 \, 1 \, 1),\quad
(1 \, 1 \, 1),\quad
(2 \, 1 \, 1),\quad
(\overline{1} \, 1 \, 1),\quad
(\overline{2} \, 1 \, 1),\quad
(3 \, 1 \, 1).$$
In fact, each of these seven symbols is evidently of the form
$(t \, u \, u)$, or $(x \, y \, y)$ [30.].
\bigbreak
[56.]
The points on the {\it third typical line}, ${\sc b}' {\sc c}'$
[31.], are in number {\it ten\/}; namely the points,
$${\sc b}',\enspace
{\sc c}',\enspace
{\sc a}'',\enspace
{\sc a}''',\enspace
{\sc a}^{\rm VII},\enspace
{\sc a}_1^{\rm VII},\enspace
{\sc a}^{\rm VIII},\enspace
{\sc a}_1^{\rm VIII},\enspace
{\sc a}^{\rm IX},\enspace
{\sc a}_1^{\rm IX},$$
of which the ternary symbols are,
$$(1 \, 0 \, 1),\quad
(1 \, 1 \, 0),\quad
(0 \, 1 \, \overline{1}),\quad
(2 \, 1 \, 1),\quad
(1 \, 2 \, \overline{1}),\quad
(1 \, \overline{1} \, 2),\quad
(3 \, 2 \, 1),\quad
(3 \, 1 \, 2),\quad
(2 \, 3 \, \overline{1}),\quad
(2 \, \overline{1} \, 3);$$
each of these ten symbols being of the form $(\sigma \, t \, u)$
[31.], with $\sigma = t + u$, as before.
\bigbreak
[57.]
These {\it three\/} typical {\it lines}, in the
plane~${\sc a} {\sc b} {\sc c}$, which may be denoted by the
ternary symbols, $[1 \, 0 \, 0]$, $[0 \, 1 \, \overline{1}]$,
$[\overline{1} \, 1 \, 1]$, and represent a system of {\it nine
lines\/} $\Lambda_1$,~$\Lambda_2$ in that plane~$\Pi_1$, are also
three typical {\it traces\/} [43.] of {\it other\/} planes
thereon; and the remaining traces of such planes are in number
{\it thirteen}, represented by {\it four\/} other lines, as
{\it types\/}: of which lines, considered as such traces, the
ternary symbols have been found [43.] to be,
$$[1 \, 1 \, 1],\quad
[0 \, 1 \, 1],\quad
[\overline{2} \, 1 \, 1],\quad
[\overline{2} \, 1 \, \overline{1}];$$
answering to the literal symbols,
$${\sc a}'' {\sc b}'' {\sc c}'',\quad
{\sc a} {\sc a}'',\quad
{\sc d}_1 {\sc a}'',\quad
{\sc a}' {\sc c}_0,$$
and serving as abridged expressions for the four
{\it equations\/} of ternary form,
$$x + y + z = 0,\quad
y + z = 0,\quad
2x = y + z,\quad
2x = y - z.$$
\bigbreak
[58.]
Each of these four last lines passes through {\it six\/} points;
thus the trace $[1 \, 1 \, 1]$ passes through the points
$(0 \, 1 \, \overline{1})$
$(\overline{1} \, 0 \, 1)$
$(1 \, \overline{1} \, 0)$
$(\overline{2} \, 1 \, 1)$
$(1 \, \overline{2} \, 1)$
$(1 \, 1 \, \overline{2})$,
or through
${\sc a}'' \, {\sc b}'' \, {\sc c}''
\, {\sc a}^{\rm IV} \, {\sc b}^{\rm IV} \, {\sc c}^{\rm IV}$;
$[0 \, 1 \, 1]$ through
$(1 \, 0 \, 0)$
$(0 \, 1 \, \overline{1})$
$(1 \, \overline{1} \, 1)$
$(1 \, 1 \, \overline{1})$
$(2 \, \overline{1} \, 1)$
$(2 \, 1 \, \overline{1})$,
or
${\sc a} \, {\sc a}'' \, {\sc b}_0 \, {\sc c}_0
\, {\sc c}^{\rm VII}
\, {\sc b}_1^{\rm VII}$;
$[\overline{2} \, 1 \, 1]$ through
$(1 \, 1 \, 1)$
$(0 \, 1 \, \overline{1})$
$(1 \, 0 \, 2)$
$(1 \, 2 \, 0)$
$(2 \, 1 \, 3)$
$(2 \, 3 \, 1)$,
or
${\sc d}_1 \, {\sc a}'' \, {\sc b}^{\rm V} \, {\sc c}_1^{\rm V}
\, {\sc c}^{\rm VIII} \, {\sc b}_1^{\rm VIII}$;
and $[\overline{2} \, 1 \, \overline{1}]$ through
$(0 \, 1 \, 1)$
$(1 \, 1 \, \overline{1})$
$(1 \, 3 \, 1)$
$(1 \, 2 \, 0)$
$(\overline{1} \, 0 \, 2)$
$(2 \, 3 \, \overline{1})$,
or
${\sc a}' \, {\sc c}_0 \, {\sc b}_1^{\rm IV} \, {\sc c}_1^{\rm V}
\, {\sc b}^{\rm VI} \, {\sc a}^{\rm IX}$;
the correctness of the {\it ternary symbols\/} being evident on
inspection, if the law $lx + my + nz = 0$ (76) be remembered: and
the {\it literal symbols\/} being thence at once deduced, by
[51.] and [52.].
\bigbreak
[59.]
{\it So far}, then, that is when we attend only to the
{\it twenty-two traces\/} [43.] of planes $\Pi_1$,~$\Pi_2$ on the
plane~${\sc a} {\sc b} {\sc c}$, we find a system of three
collineations of eight points; three of seven points; three of
ten points; and thirteen of six points each. Each collineation
of the first of these four systems {\it counts\/} as 28 binary
combinations of the 52 points in the plane [53.]; each of the
second system counts as 21 such combinations; each of the third
system as 45; and each of the fourth as 15. We therefore
account, in this way, for $84 + 63 + 135 + 195 = 477$ binary
combinations; but the total number is
$26 \mathbin{.} 51 = 1326$; there remain then 849 to be accounted
for, by lines~$\Lambda_3$ which are {\it not traces}, of any one
of the foregoing groups.
\bigbreak
[60.]
In seeking for such new lines, it is natural to consider first
those which pass through one or other of the three given points
${\sc a}$,~${\sc b}$,~${\sc c}$; and the {\it types\/} of such
are found to be the five following, each representing a new group
of six lines~$\Lambda_3$:
$$[0 \, 2 \, 1];\quad
[0 \, 2 \, \overline{1}];\quad
[0 \, 3 \, \overline{1}];\quad
[0 \, 3 \, \overline{2}];\quad
[0 \, 3 \, 1].$$
As {\it symbols}, these answer respectively to the five
{\it new lines\/}:
$$\vbox{\halign{#\hfil &\quad#\hfil \cr
$(1 \, 0 \, 0)$
$(1 \, 1 \, \overline{2})$
$(0 \, \overline{1} \, 2)$
$(1 \, \overline{1} \, 2)$
$(3 \, \overline{1} \, 2)$,
&or
${\sc a} \, {\sc c}^{\rm IV} \, {\sc a}_1^{\rm VI}
\, {\sc a}_1^{\rm VII} \, {\sc c}^{\rm IX}$;\cr
$(1 \, 0 \, 0)$
$(1 \, 1 \, 2)$
$(0 \, 1 \, 2)$
$(\overline{1} \, 1 \, 2)$
$(3 \, 1 \, 2)$,
&or
${\sc a} \,{\sc c}''' \, {\sc a}_1^{\rm V}
\, {\sc b}^{\rm VII} \, {\sc a}_1^{\rm VIII}$;\cr
$(1 \, 0 \, 0)$
$(1 \, 1 \, 3)$
$(2 \, 1 \, 3)$,
&or
${\sc a} \, {\sc c}_1^{\rm IV} \, {\sc c}^{\rm VIII}$;\cr
$(1 \, 0 \, 0)$
$(1 \, 2 \, 3)$
$(\overline{1} \, 2 \, 3)$,
&or
${\sc a} \, {\sc c}_1^{\rm VIII} \, {\sc b}^{\rm IX}$;\cr
$(1 \, 0 \, 0)$
$(2 \, \overline{1} \, 3)$,
&or
${\sc a} \, {\sc a}_1^{\rm IX}$.\cr}}$$
We have thus {\it twelve\/} lines~$\Lambda_3$, each connecting a
point~${\sc p}_0$, with {\it four\/} points~${\sc p}_2$, and
counting as {\it ten\/} binary combinations; {\it twelve\/} other
lines, each connecting a point~${\sc p}_0$ with {\it two\/}
points~${\sc p}_2$, and counting as {\it three\/} such
combinations; and {\it six\/} lines, each of which connects a
point~${\sc p}_0$ with {\it one\/} point~${\sc p}_2$, and counts
as only {\it one\/} combination. In this manner, then, we
account for $120 + 36 + 6 = 162$, out of the 849 which had
remained in [59.]; but there still remain 687 combinations to be
accounted for, by new lines of third construction, which pass
through no given point.
\bigbreak
[61.]
Considering next the new lines which connect a point of the
{\it first\/} construction, with one or more points of the
{\it second}, we find these five new types,
$$[3 \, 1 \, \overline{1}];\quad
[1 \, 2 \, \overline{2}];\quad
[1 \, 2 \, \overline{3}];\quad
[1 \, 3 \, \overline{3}];\quad
[1 \, 3 \, \overline{4}];$$
which as symbols denote the five lines,
$$\left. \eqalign{
(0 \, 1 \, 1) \enspace
(1 \, \overline{2} \, 1) \enspace
(1 \, \overline{1} \, 2);\quad
(0 \, 1 \, 1) \enspace
(2 \, 0 \, 1) \enspace
(2 \, \overline{1} \, 0);\quad
(1 \, 1 \, 1) \enspace
(2 \, \overline{1} \, 0) \enspace
(\overline{1} \, 2 \, 1);\cr
(0 \, 1 \, 1) \enspace
(3 \, 1 \, 2);\quad
(1 \, 1 \, 1) \enspace
(\overline{1} \, 3 \, 2);\cr}
\right\}$$
$$\hbox{ or
${\sc a}' \, {\sc b}^{\rm IV} \, {\sc a}_1^{\rm VII}$;
${\sc a}' \, {\sc b}_1^{\rm V} \, {\sc c}^{\rm VI}$;
${\sc d}_1 \, {\sc c}^{\rm VI} \, {\sc c}_1^{\rm VII}$;
${\sc a}' \, {\sc a}_1^{\rm VIII}$;
and ${\sc d}_1 \, {\sc c}_1^{\rm IX}$;}$$
but as {\it types\/} represent each a {\it group of six lines}.
We thus get 18 new lines, each passing through 1
point~${\sc p}_1$, and 2 points~${\sc p}_2$; and 12 other lines,
each connecting a point~${\sc p}_1$ with only {\it one\/}
point~${\sc p}_2$. And these thirty lines~$\Lambda_3$ account
for $54 + 12 = 66$ binary combinations of points; leaving however
621 such combinations to be accounted for, by new
lines~$\Lambda_3$, of which each must connect at least two
points~${\sc p}_2$, without passing through any point ${\sc p}_0$
or ${\sc p}_1$, and without being any one of the traces already
considered.
\bigbreak
[62.]
The {\it symbol\/} $[\overline{2} \, 3 \, 3]$, which denotes a
line passing through {\it two\/} points~${\sc p}_2$, namely,
$(0 \, 1 \, \overline{1})$ and $(3 \, 1 \, 1)$, or
${\sc a}''$ and ${\sc a}_1^{\rm IV}$, but through {\it no
other\/} point, represents, when considered as a {\it type}, a
group of {\it three\/} such lines; and 40 {\it other types}, as
for example $[1 \, \overline{3} \, 4]$, which as a symbol denotes
the line $(\overline{1} \, 1 \, 1) \enspace (1 \, 3 \, 2)$, or
${\sc a}_0 \, {\sc b}^{\rm VIII}$, are found to exist,
representing each a group of {\it six\/} lines, whereof each
connects in like manner {\it two\/} points ${\sc p}_2$, but
{\it only\/} those two points. We have thus a system of 243 new
lines, which represent only so many binary combinations: and
there remain 378 such combinations to be accounted for, by new
lines~$\Lambda_3$, whereof each must connect
{\it at least three points\/}~${\sc p}_2$.
\bigbreak
[63.]
For lines connecting {\it three\/} such points, and {\it no
more}, it is found that there are {\it twenty types\/}; whereof
{\it eight}, as for instance the type $[\overline{3} \, 1 \, 1]$,
which as a symbol denotes the line
$(0 \, 1 \, \overline{1}) \enspace (1 \, 2 \, 1)
\enspace (1 \, 1 \, 2)$,
or ${\sc a}'' \, {\sc b}''' \, {\sc c}'''$,
represent each a group of {\it three\/} such lines; while each of
the {\it twelve others}, like $[1 \, \overline{2} \, 3]$, which
as a symbol denotes the line
$(\overline{1} \, 1 \, 1) \enspace (1 \, 2 \, 1)
\enspace (2 \, 1 \, 0)$,
or ${\sc a}_0 \, {\sc b}''' \, {\sc c}^{\rm V}$,
represents a group of {\it six\/} lines. We have thus 96 new
lines, counting as 288 binary combinations: but we must still
account for 90 {\it other\/} combinations, by new
lines~$\Lambda_3$, connecting each {\it more than three
points\/}~${\sc p}_2$.
\bigbreak
[64.]
Accordingly, we find {\it three new types of lines}, which
{\it alone remain}, when all those which have been above
exhibited, or alluded\footnote*{It has been thought that it could
not be interesting to set down {\it all\/} the {\it types of
lines}, above referred to; especially as those which relate to
lines {\it not\/} passing through {\it at least four points\/}
give rise, at the present stage of the construction, to no
{\it theorems\/} of {\it harmonic\/} (or {\it anharmonic\/})
{\it ratio}.}
to are set aside: namely
$$[1 \, \overline{2} \, 4];\quad
[\overline{1} \, 2 \, 4];\quad
[1 \, 1 \, 2].$$
And those represent, respectively, groups of {\it six\/}, of
{\it six}, and of {\it three\/} new lines, and therefore on the
whole a system of {\it fifteen\/} new lines, each passing
{\it through four points\/}~${\sc p}_2$, and consequently
counting as {\it six\/} combinations; for example, as
{\it symbols}, they denote the three following lines:
$$\vbox{\halign{#\hfil &\quad#\hfil \cr
$(2 \, 1 \, 0)$
$(\overline{2} \, 1 \, 1)$
$(0 \, 2 \, 1)$
$(2 \, 3 \, 1)$,
&or
${\sc c}^{\rm V} \, {\sc a}^{\rm IV} \, {\sc a}^{\rm V}
\, {\sc b}_1^{\rm VIII}$;\cr
$(2 \, 1 \, 0)$
$(2 \, \overline{1} \, 1)$
$(0 \, 2 \, \overline{1})$
$(2 \, 3 \, \overline{1})$,
&or
${\sc c}^{\rm V} \, {\sc c}^{\rm VII} \, {\sc a}^{\rm VI}
\, {\sc a}^{\rm IX}$;\cr
$(2 \, 0 \, \overline{1})$
$(1 \, \overline{1} \, 0)$
$(0 \, 2 \, \overline{1})$
$(1 \, 1 \, \overline{1})$,
&or
${\sc b}_1^{\rm VI} \, {\sc c}'' \, {\sc a}^{\rm VI}
\, {\sc c}_0$.\cr}}$$
But $6 \mathbin{.} 15 = 90$; we are therefore entitled to say,
that {\it all the\/} 1326 {\it binary combinations\/} [59.],
{\it of the\/} 52 {\it points\/}
${\sc p}_0$,~${\sc p}_1$,~${\sc p}_2$ [53.] {\it in the plane
${\sc a} {\sc b} {\sc c}$, have now been fully accounted for}.
\bigbreak
[65.]
Collecting the results, respecting the {\it collineations in the
plane\/}~${\sc a} {\sc b} {\sc c}$, it has been found that there
are 261 {\it lines\/}~$\Lambda_3$, whereof each {\it connects
two}, but {\it only two}, of the the 52 {\it points\/} in that
plane; and that {\it these lines}, which at the present stage of
the construction are not properly {\it cases of collinearity\/}
at all, are represented by a system of 44 {\it ternary types}.
\bigbreak
[66.]
There are 126 {\it other lines\/}~$\Lambda_3$, each connecting
{\it three\/} (but {\it only three\/}) {\it points\/}; they are
represented by a system of 25 {\it types\/}; and account for 378
binary {\it combinations}.
\bigbreak
[67.]
There are 15 lines~$\Lambda_3$, each connecting {\it four
points\/}~${\sc p}_2$; they are represented by a system of 3
types, and account for 90 combinations.
\bigbreak
[68.]
There are 12 lines~$\Lambda_3$, each connecting {\it one\/}
point~${\sc p}_0$ with {\it four\/} points ${\sc p}_2$; they are
represented by 2 types, and represent 120 combinations.
\bigbreak
[69.]
There are 13 other lines~$\Lambda_3$, namely the {\it traces of
planes\/}~$\Pi_1$ or $\Pi_2$, whereof each connects {\it six
points}, namely a point ${\sc p}_0$ or ${\sc p}_1$ with five
points ${\sc p}_2$, or else six points~${\sc p}_2$ with each
other; they are represented by 4 types, and account for 195
combinations.
\bigbreak
[70.]
There are 3 lines~$\Lambda_{2,2}$, each connecting {\it two\/}
points ${\sc p}_1$ with {\it eight\/} points~${\sc p}_2$; they
have one common type, and represent 135 combinations.
\bigbreak
[71.]
There are, in like manner, 3 lines~$\Lambda_{2,1}$, each
connecting {\it one point\/}~${\sc p}_0$ with {\it two
points\/}~${\sc p}_1$, and with {\it four points\/}~${\sc p}_2$,
but having only {\it one\/} common type; and they represent 63
combinations.
\bigbreak
[72.]
Finally, there are (in the same plane) 3
{\it lines\/}~$\Lambda_1$, each connecting {\it two
points\/}~${\sc p}_0$ with {\it one point\/}~${\sc p}_1$, and
with {\it five points\/}~${\sc p}_2$; these lines also have all
{\it one type\/}; and they account for 84 {\it combinations\/}:
with the {\it arithmetical verification}, that
$$261 + 378 + 90 + 120 + 195 + 135 +63 + 84 = 1326
= 26 \mathbin{.} 51;$$
which proves that the {\it enumeration\/} is {\it complete}.
\bigbreak
[73.]
The {\it total number of distinct lines}, above obtained, is
$261 + 126 + 15 + 12 + 13 + 3 + 3 + 3 = 436$; and the total
number of their {\it ternary types\/} is 81. But {\it if we set
aside\/} (as conducting to {\it no general metric relations\/})
{\it all those lines which contain fewer than four points}, there
{\it then remain only forty-nine lines}, and {\it only twelve
types}, to be discussed, with reference to {\it harmonic\/} (or
{\it anharmonic\/}) {\it relations}, of the points upon those
lines.
\bigbreak
[74.]
For the purpose of studying completely all {\it such\/}
relations, it will therefore be permitted to confine ourselves to
the {\it three first typical lines},
${\sc b} {\sc c}$, ${\sc a} {\sc a}'$, ${\sc b}' {\sc c}'$,
or $[1 \, 0 \, 0]$, $[0 \, 1 \, \overline{1}]$,
$[\overline{1} \, 1 \,1]$;
the {\it four other typical traces},
${\sc a}'' {\sc b}'' {\sc c}''$, ${\sc a} {\sc a}''$,
${\sc d}_1 {\sc a}''$, ${\sc a}' {\sc c}_0$, or
$[1 \, 1 \, 1]$, $[0 \, 1 \, 1]$, $[\overline{2} \, 1 \, 1]$,
$[\overline{2} \, 1 \, \overline{1}]$;
and {\it five new typical lines\/}~$\Lambda_3$, connecting each
{\it at least four points\/}: namely the {\it two lines},
$[0 \, 2 \, 1]$ and $[0 \, 2 \, \overline{1}]$, of [60.], whereof
each connects the given point~${\sc a}$ with {\it four\/}
points~${\sc p}_2$; and the {\it three lines\/}
$[1 \, \overline{2} \, 4]$, $[\overline{1} \, 2 \, 4]$, $[1 \, 1 \, 2]$,
of [64.], of which each connects {\it four\/} other
points~${\sc p}_2$ among themselves, but does not pass through
any point~${\sc p}_0$ or ${\sc p}_1$.
\bigbreak
\centerline{{\sc Part~IV.}---{\it Applications to the Net,
continued: Harmonic and Involutionary Relations,}}
\centerline{{\it of the Points situated on the Twelve
Typical Lines, in a Plane of First Construction.}}
\nobreak\bigskip
[75.]
Commencing here with the examination of the last typical lines,
because they contain only {\it four\/} points each, let us adopt,
as temporary symbols, of the {\it literal\/} kind, the ten
following:
$$\vbox{\halign{$\displaystyle #$\hfil &&\quad $\displaystyle #$\hfil \cr
a = (2 \, 1 \, 0), &
b = (\overline{2} \, 1 \, 1), &
c = (0 \, 2 \, 1), &
d = (2 \, 3 \, 1);\cr
&
b' = (2 \, \overline{1} \, 1), &
c' = (0 \, 2 \, \overline{1}), &
d' = (2 \, 3 \, \overline{1});\cr
a'' = (2 \, 0 \, \overline{1}), &
b'' = (1 \, \overline{1} \, 0), &
&
d'' = (1 \, 1 \, \overline{1});\cr}}$$
instead of the more systematic but less simple symbols,
${\sc c}^{\rm V}$ ${\sc a}^{\rm IV}$ ${\sc a}^{\rm V}$
${\sc b}_1^{\rm VIII}$ ${\sc c}^{\rm VII}$ ${\sc a}^{\rm VI}$
${\sc a}^{\rm IX}$ ${\sc b}_1^{\rm VI}$ ${\sc c}''$ ${\sc c}_0$.
\bigbreak
[76.]
The three lines referred to [64.], are then the three following:
$$a b c d;\quad a b' c' d';\quad a'' b'' c' d''.$$
And because we have (comp.~[16.]) the six symbolical relations,
$$(c) - (a) = (b);\quad (c) + (a) = (d);$$
$$(a) - (c') = (b');\quad (a) + (c') = (d');$$
$$(a'') - (c') = 2(b'');\quad (a'') + (c') = 2 (d''),$$
it results (31) that the three {\it harmonic equations\/} exist:
$$(a \, b \, c \, d)
= (a \, b' \, c' \, d')
= (a'' \, b'' \, c' \, d'') = -1.$$
We have therefore this {\it Theorem\/}:---
\medbreak
``{\it Each of the\/} 150 {\it lines~$\Lambda_3$, which connect
four points~${\sc p}_2$, in any one of the ten planes~$\Pi_1$,
and pass through no other of the\/} 305 {\it points
${\sc p}_0$,~${\sc p}_1$,~${\sc p}_2$, is harmonically
divided.}''
\bigbreak
[77.]
As verifications, the three right lines $b b'$, $c c'$, $d d'$
concur in the point ${\sc c}$; $b d'$, $c c'$, $d b'$, in
${\sc b}$; $a a''$, $b' b''$, $d' d''$, in ${\sc a}'$; and
$a a''$, $b' d''$, $d' b''$, in a point~${\sc p}_3$, namely in
$(4 \, 1 \, \overline{1})$: the existence of which four
{\it concurrences\/} of lines was to be expected, from a known
principle of {\it homography}, as consequences of the harmonic
relations [76.]. It is worth noticing, however, how simply these
concurrences are here {\it expressed}, by the {\it ternary
symbols\/} of the {\it points}, according to the {\it law} (18);
or, if we choose, by the corresponding symbols of the
{\it lines}, with the analogous law (25): for example, the three
last concurrent lines, $a a''$, \&c., have for their respective
symbols, $[1 \, \overline{2} \, 2]$, $[0 \, 1 \, 1]$, and
$[1 \, 1 \, 5] = [1 \, \overline{2} \, 2] + [0 \, 3 \, 3]$.
\bigbreak
[78.]
To examine, in like manner, the analogous relations of
arrangement, on the two new typical lines [60.], namely
$[0 \, 2 \, 1]$ and $[0 \, 2 \, \overline{1}]$, whereof each
connects the given point~${\sc a}$ with four points of second
construction, let us write as eight new temporary symbls of the
literal kind, more convenient than the former symbols,
${\sc c}^{\rm IV}$ ${\sc a}_1^{\rm VI}$ ${\sc a}_1^{\rm VII}$
${\sc c}^{\rm IX}$ ${\sc b}^{\rm VII}$ ${\sc a}_1^{\rm V}$
${\sc c}'''$ ${\sc a}_1^{\rm VIII}$,
the following:
$$\vbox{\halign{$\displaystyle #$\hfil &&\quad $\displaystyle #$\hfil \cr
b = (1 \, 1 \, \overline{2}), &
c = (0 \, \overline{1} \, 2), &
d = (1 \, \overline{1} \, 2), &
e = (3 \, \overline{1} \, 2);\cr
\beta = (\overline{1} \, 1 \, 2), &
\gamma = (0 \, 1 \, 2), &
\delta = (1 \, 1 \, 2), &
\epsilon = (3 \, 1 \, 2), \cr}}$$
so that the two lines in question are,
$${\sc a} b c d e,
\quad\hbox{and}\quad
{\sc a} \beta \gamma \delta \epsilon.$$
We have thus the eight following new symbolical relations,
${\sc a}$ being still $= (1 \, 0 \, 0)$:
$$({\sc a}) - (c) = (b),\quad
({\sc a}) + (c) = (d);\qquad
(e) - (b) = 2(d),\quad
(e) + (b) = 4({\sc a});$$
$$(\gamma) - ({\sc a}) = (\beta),\quad
(\gamma) + ({\sc a}) = (\delta);\qquad
(\epsilon) + (\beta) = 2(\delta),\quad
(\epsilon) - (\beta) = 4({\sc a});$$
whence result at once the {\it four harmonic relations},
$$({\sc a} \, b \, c \, d)
= ({\sc a} \, b \, d \, e)
= ({\sc a} \, \beta \, \gamma \, \delta)
= ({\sc a} \, \beta \, \delta \, \epsilon)
= -1.$$
These {\it two\/} lines from ${\sc a}$ are therefore
{\it homographically divided}, the point~${\sc a}$ corresponding
to {\it itself}, and $b$ to $\beta$, \&c.; and accordingly the
{\it four right lines\/},
$b \beta$, $c \gamma$, $d \delta$, $e \epsilon$,
{\it which connect corresponding points}, concur in one common
point, which is easily found to be ${\sc b}$. And other
{\it verifications}, by such {\it concurrences}, can be assigned
with little trouble.
\bigbreak
[79.]
It may assist the conception of the {\it common law of
arrangement}, of the {\it five points\/} on each of the {\it two
typical lines\/} last considered, to suppose that the joining
line~$b \beta$ is {\it thrown off}, by projection, {\it to
infinity\/}; or, what comes to the same thing, that the {\it two
points\/} $b$ and $\beta$, themselves, are thus made infinitely
distant. For thus the harmonic equations [78.] will simply
express that, {\it in this projected state of the figure}, the
{\it four points}, $d$, $e$, $\delta$, $\epsilon$, {\it bisect\/}
respectively the {\it four intervals},
${\sc a} c$, ${\sc a} d$, ${\sc a} \gamma$, ${\sc a} \delta$;
whence it is easy to construct a diagram, not necessary here to
be exhibited. The consideration of the {\it two other lines\/}
through the same given point~${\sc a}$, which have
$[0 \, 1 \, 2]$ $[0 \, \overline{1} \, 2]$ for their symbols, and
belong to the same two types as the two last, would offer to our
notice a {\it pencil of four rays}, which has some interesting
properties, especially as regards its {\it intersections\/} with
{\it other\/} pencils, but which we cannot here delay to
describe.
\bigbreak
[80.]
It may, however, be worth while to state here, as a consequence
from the preceding discussion, this other {\it Theorem\/}:---
\medbreak
``{\it The\/} 120 {\it lines~$\Lambda_3$ in the ten
planes~$\Pi_1$, whereof each connects a point~${\sc p}_0$ with
four points~${\sc p}_2$, and with no other of the\/} 305 {\it
points, although not all syntypical, are all homographically
divided.}''
\bigbreak
[81.]
Proceeding to consider the arrangements of those six typical
lines [58.] which contain each {\it six points}, we find that
whether we write, as new temporary and literal symbols,
$$a = (0 \, 1 \, \overline{1}),\quad
b = (\overline{1} \, 0 \, 1),\quad
c = (1 \, \overline{1} \, 0),\quad
a' = (\overline{2} \, 1 \, 1),\quad
b' = (1 \, \overline{2} \, 1),\quad
c' = (1 \, 1 \, \overline{2}),$$
or
$$a = (0 \, 1 \, 1),\quad
b = (1 \, 1 \, \overline{1}),\quad
c = (1 \, 2 \, 0),\quad
a' = (2 \, 3 \, \overline{1}),\quad
b' = (1 \, 3 \, 1),\quad
c' = (\overline{1} \, 0 \, 2),$$
the six points $a \, b \, c \, a' \, b' \, c'$ being in the one
case on the line $[1 \, 1 \, 1]$, and in the other case on the
line $[\overline{2} \, 1 \, \overline{1}]$, we have in each case
the three harmonic equations:
$$(c \, a \, b \, a')
= (a \, b \, c \, b')
= (b \, c \, a \, c')
= -1.$$
We may then at once infer this {\it Theorem\/}:
\medbreak
``{\it The\/} 70 {\it lines~$\Lambda_3$, in the ten
planes~$\Pi_1$, which are represented by the fourth and seventh
typical traces of planes on the plane~${\sc a} {\sc b} {\sc c}$,
although not all syntypical\/} (or generated by similar processes
of construction), {\it are all homographically divided.}''
\bigbreak
[82.]
This {\it common mode\/} of their {\it division\/} may deserve,
however, a somewhat closer examination, its consequences being
not without interest. When any six collinear points,
$a \, \ldots \, c'$, are connected by the three equations [81.],
we are permitted to suppose that their symbols are so
{\it prepared\/} (if necessary), by
{\it coefficients},\footnote*{For example, in the second case
[81.], we should change the symbols for $c$ and $b'$ to their
negatives, before employing the formul{\ae} of [82.].}
as to give,
$$(a) + (b) + (c) = 0;$$
$$(a') = (b) - (c),\quad
(b') = (c) - (a),\quad
(c') = (a) - (b);$$
and therefore,
$$(a') + (b') + (c') = 0,$$
$$3(a) = (c') - (b'),\quad
3(b) = (a') - (c'),\quad
3(c) = (b') - (a').$$
{\it Whenever, then, the three harmonic equations\/} [81.]
{\it exist, for a system of six collinear points},
$a \, \ldots \, c'$, {\it the three other harmonic equations},
formed by interchanging accented and unaccented letters,
$$(c' \, a' \, b' \, a)
= (a' \, b' \, c' \, b)
= (b' \, c' \, a' \, c)
= -1,$$
{\it are also satisfied\/}; and {\it the three pairs\/} (or
{\it segments\/}),
$$a a',\quad b b',\quad c c',$$
{\it which connect corresponding points, compose an
involution}.\footnote*{Compare p.~127 of the
{\it G\'{e}om\'{e}trie Sup\'{e}rieure\/} (Paris, 1852). In
general, the reader is supposed to be acquainted with the chapter
(chap.~ix.) of that excellent work of M.~{\it Chasles}, which
treats of {\it Involution}.}
\bigbreak
[83.]
Under the same conditions, the two points $a$ and $a'$ are
harmonically conjugate to each other, not only with respect to
$b$ and $c$, but also with respect to $b'$ and $c'$; they are
therefore the {\it double points\/} (or {\it foci\/}) of that
{\it other involution\/} which is determined by the {\it two
pairs\/} of points, $b c$,~$b' c'$. In like manner, $b$,~$b'$
are the double points of the involution, determined by the two
pairs, or segments $c a$, $c' a'$; and $c$,~$c'$ are the double
points of the involution determined by $a b$, $a' b'$.
\bigbreak
[84.]
From any one of the three last involutions [83.], we could
{\it return}, by known principles, to the involution [82.]; we
can also infer from them that the {\it three new pairs of
points\/} (or {\it segments of the common line\/}), $a a'$,
$b c'$, $c b'$; the three pairs, or segments, $b b'$, $c a'$,
$a c'$; and the three others, $c c'$, $a b'$, $b a'$, form
{\it three other involutions}, making {\it seven distinct
involutions of the six points}, so far: in {\it three\/} of
which, as we have seen in [83.] {\it two\/} of those {\it six
points\/} are {\it their own conjugates}.
\bigbreak
[85.]
For these and other reasons we propose to say, that {\it when any
three collinear points\/} (as $a$,~$b$,~$c$) {\it are assumed\/}
(or {\it given\/}), {\it and three other points on the same line
are derived from them, by the condition that each shall be the
harmonic conjugate of one, with respect to the other two, then
these two sets of points are two Triads of Points in Involution}.
And it is easy to extend this definition so as to include cases
of two {\it triads\/} of complanar and co-initial {\it lines}, or
of collinear {\it planes}, which shall be, in the same general
but (as it is supposed) {\it new sense}, in {\it involution\/}
with each other: every such {\it involution of triads\/}
including, by what precedes, {\it a system of seven
involutions\/} of the {\it old\/} or {\it usual\/} kind.
\bigbreak
[86.]
For example, because the two {\it triads of points},
${\sc a}'' {\sc b}'' {\sc c}''$ and
${\sc a}^{\rm IV} {\sc b}^{\rm IV} {\sc c}^{\rm IV}$,
are thus in involution, by the equations [81.] applied to the
fourth typical trace [43.], it follows that the {\it two
pencils}, each of {\it three rays},
$${\sc d}_1 \mathbin{.} {\sc a}'' {\sc b}'' {\sc c}'',
\quad\hbox{and}\quad
{\sc d}_1 \mathbin{.} {\sc a} {\sc b} {\sc c},$$
are {\it triads of lines, in involution\/} with each other; and
that, for a similar reason, the {\it two triads of planes}, all
passing through the line ${\sc d} {\sc e}$,
$${\sc d} {\sc e} {\sc a},\quad
{\sc d} {\sc e} {\sc b},\quad
{\sc d} {\sc e} {\sc c},
\quad\hbox{and}\quad
{\sc d} {\sc e} {\sc a}'',\quad
{\sc d} {\sc e} {\sc b}'',\quad
{\sc d} {\sc e} {\sc c}'',$$
are, in the sense above explained, in {\it involution}. In fact,
when the point~${\sc d}_1$ is thus taken as a {\it vertex of the
pencils\/} in the plane ${\sc a} {\sc b} {\sc c}$, the three
harmonic equations of the first case [81.], namely,
$$ ({\sc c}'' \, {\sc a}'' \, {\sc b}'' \, {\sc a}^{\rm IV})
= ({\sc a}'' \, {\sc b}'' \, {\sc c}'' \, {\sc b}^{\rm IV})
= ({\sc b}'' \, {\sc c}'' \, {\sc a}'' \, {\sc c}^{\rm IV})
= -1,$$
or rather the three reciprocal equations (comp.~[82.]),
$$ ({\sc c}^{\rm IV} \, {\sc a}^{\rm IV} \, {\sc b}^{\rm IV} \, {\sc a}'')
= ({\sc a}^{\rm IV} \, {\sc b}^{\rm IV} \, {\sc c}^{\rm IV} \, {\sc b}'')
= ({\sc b}^{\rm IV} \, {\sc c}^{\rm IV} \, {\sc a}^{\rm IV} \, {\sc c}'')
= -1,$$
correspond simply to the elementary equations, (50), (56),
$$ ({\sc c} \, {\sc a}' \, {\sc b} \, {\sc a}'')
= ({\sc a} \, {\sc b}' \, {\sc c} \, {\sc b}'')
= ({\sc b} \, {\sc c}' \, {\sc a} \, {\sc c}'')
= -1,$$
which may be employed to {\it define\/} the three important
points ${\sc a}''$,~${\sc b}''$,~${\sc c}''$, (87), of the
{\it first group\/} of {\it second construction\/} [40.], as
being the (well known) {\it harmonic conjugates\/} of the points
${\sc a}'$,~${\sc b}'$,~${\sc c}'$ of {\it first\/} construction,
with respect to the three {\it lines\/} of the same first
construction, ${\sc b} {\sc c}$, ${\sc c} {\sc a}$,
${\sc a} {\sc b}$, on which those points are situated.
\bigbreak
[87.]
The equations [82.], which connect the {\it symbols\/}
$(a) \, \ldots \, (c')$ of the {\it six points}, give, by easy
eliminations, these other equations of the same kind:
$$(b') = (b) + 2 (c);\quad
- (c') = 2(b) + (c);$$
we have therefore, by (31), the following {\it anharmonic of the
group\/} $b$,~$b'$,~$c$,~$c'$:
$$(b \, b' \, c \, c') = + 4;$$
and other easy calculations of the same sort given, in like
manner, the equal anharmonics,
$$(c \, c' \, a \, a') = + 4;\quad
(a \, a' \, b \, b') = + 4.$$
But in general, for any four collinear points,
$a$,~$b$,~$c$,~$d$, the {\it definition\/} (29) of the
{\it symbol\/} $(a \, b \, c \, d)$ gives easily the relation,
$$(a \, b \, c \, d) + (a \, c \, b \, d) = 1;$$
and hence, or immediately by calculations such as those recently
used, we have this {\it other\/} set of anharmonics, with a
{\it new\/} common value:
$$(b \, c \, b' \, c')
= (c \, a \, c' \, a')
= (a \, b \, a' \, b')
= -3;$$
the {\it negative\/} character of which shows, by the same
definition (29), that the segment (or interval) $a a'$, for
example, is cut {\it internally\/} by {\it one\/} of the two
points $b$,~$b'$, or by one of the two points $c$,~$c'$, and
{\it externally\/} by the {\it other\/}; with similar results for
each of the two other segments, $b b'$, $c c'$.
\bigbreak
[88.]
We may then say that {\it each of the three segements}, $a a'$,
$b b'$, $c c'$, {\it overlaps each of the two others}, in the
sense that {\it any two\/} of them have a {\it common part}, and
also {\it parts not common\/}: whence it immediately follows that
the {\it involution\/} [82.], {\it to which these three segments
belong, has its double points imaginary\/}: whereas it may be
proved, on the same plan, that each of the three involutions of
segments mentioned in [84.], namely $a a'$, $b c'$, $c b'$;
$b b'$, $c a'$, $a c'$; $c c'$, $a b'$, $b a'$, has
{\it real\/}\footnote*{The determination of these double points
gives rise naturally to some new theorems, which cannot
conveniently be stated here.} {\it double points\/}; and the
double points of the three other involutions, determined by the
three {\it pairs\/} of segments, $b c$, $b' c'$; $c a$, $c' a'$;
$a b$, $a' b'$, are likewise {\it real}, and have been assigned
[83.]; namely, in each of these three last cases, the two
remaining points of the system.
\bigbreak
[89.]
Now, in general, when the {\it foci\/} (or double points) of an
involution of collinear segments, $a a'$, $b b',\ldots$ are
{\it imaginary}, so that {\it conjugate points}, $a$,~$a'$, or
$b$,~$b'$, \&c., fall at {\it opposite sides\/} of the
{\it central point\/}~${\sc o}$, it is known, and may indeed by
considered as evident, that if an
{\it ordinate\/}~${\sc o} {\sc p}$ be erected, equal to the
constant {\it geometrical mean\/} between the two distances
${\sc o} a$,~${\sc o} a'$, or ${\sc o} b$,~${\sc o} b'$, \&c.,
then, {\it all the segments\/} $a a'$, $b b'$, \&c., {\it subtend
right angles, at the extremity~${\sc p}$ of this ordinate}. It
follows, then, by what has been proved in [82.] and [88.], and by
the {\it first case\/} of [81.], that {\it each of the three
segments\/}
${\sc a}'' {\sc a}^{\rm IV}$,
${\sc b}'' {\sc b}^{\rm IV}$,
${\sc c}'' {\sc c}^{\rm IV}$,
{\it of the fourth typical trace\/} [43.] {\it subtends a right
angle at some one point}, ${\sc p}$, {\it in the
plane\/}~${\sc a} {\sc b} {\sc c}$, or rather generally at
{\it each\/} of {\it two\/} such points: and in like manner, by
the {\it second case\/} [81.], that each of the {\it three other
segments},
${\sc a}' {\sc a}^{\rm IX}$,
${\sc c}_0 {\sc b}_1^{\rm IV}$,
${\sc c}_1^{\rm V} {\sc b}^{\rm VI}$,
of the {\it seventh typical trace}, subtends a {\it right angle},
at each of two {\it other\/} points, ${\sc p}$,~${\sc p}'$, in
the same plane.
\bigbreak
[90.]
{\it These\/} results, by their nature, like {\it all the
foregoing results\/} of the present Paper, are quite
{\it independent of the assumed arrangement of the five given\/}
(or {\it initial\/}) {\it points of space\/}
${\sc a} \, \ldots {\sc e}$, and are {\it unaffected by
projection}, or {\it perspective}. In saying this, it is not
meant, of course, that one {\it right angle\/} will generally be
{\it projected\/} into {\it another\/}; or that the {\it new
point\/}~${\sc p}$, at which the {\it three new segments\/}
${\sc a}'' {\sc a}^{\rm IV}$,
${\sc b}'' {\sc b}^{\rm IV}$,
${\sc c}'' {\sc c}^{\rm IV}$,
or
${\sc a}' {\sc a}^{\rm IX}$,
${\sc c}_0 {\sc b}_1^{\rm IV}$,
${\sc c}_1^{\rm V} {\sc b}^{\rm VI}$,
{\it subtend\/} right angles, will be itself (what may be called)
the {\it projection\/} of the {\it old point\/}~${\sc p}$ [89.],
which was so related to the three {\it old segments}, denoted by
the same literal symbols, when the {\it arrangement\/} (or
{\it configuration\/}) of the five {\it initial\/} points is
{\it varied}, by a process {\it analogous\/} to projection. We
can only assert that there will {\it always}, in {\it every
state\/} of the Figure, or of the {\it Net}, be {\it some
point\/}~${\sc p}$, possessing the above-mentioned property: or
rather that there will be a {\it circle\/} of such points {\it in
space}, having for its {\it axis\/} the {\it line\/} to which the
three segments belong.
\bigbreak
[91.]
To fix a little more definitely the conceptions, let
${\sc a}$,~${\sc b}$,~${\sc c}$,~${\sc d}$ be supposed, for the
moment, to be the {\it corners\/} of a {\it regular pyramid},
with ${\sc e}$ for its {\it mean point}, or {\it centre of
gravity}. With this {\it arrangement\/} of the five
{\it given\/} points~${\sc p}_0$, {\it six\/} of the
{\it derived\/} points~${\sc p}_1$, namely
${\sc a}'$,~${\sc b}'$,~${\sc c}'$,
${\sc a}_2$,~${\sc b}_2$,~${\sc c}_2$,
{\it bisect\/} the {\it six edges\/}
${\sc b} {\sc c}$, ${\sc c} {\sc a}$, ${\sc a} {\sc b}$,
${\sc d} {\sc a}$, ${\sc d} {\sc b}$, ${\sc d} {\sc c}$,
of the given pyramid; and the {\it four\/} other
points~${\sc p}_1$, namely
${\sc a}_1$,~${\sc b}_1$,~${\sc c}_1$,~${\sc d}_1$,
are the {\it mean points\/} of the {\it four faces}, opposite to
${\sc a}$,~${\sc b}$,~${\sc c}$,~${\sc d}$. {\it Six\/} of the
ten points~${\sc p}_{2,1}$, namely
${\sc a}''$,~${\sc b}''$,~${\sc c}''$,
${\sc a}_2'$,~${\sc b}_2'$,~${\sc c}_2'$,
are now {\it infinitely distant\/}; and the {\it line\/}
${\sc a}'' {\sc b}'' {\sc c}''
{\sc a}^{\rm IV} {\sc b}^{\rm IV} {\sc c}^{\rm IV}$
to which three of the lately mentioned {\it segments\/} belong,
becomes the {\it line at infinity\/} in the plane
${\sc a} {\sc b} {\sc c}$: which might seem, at first sight, to
render difficult, with respect at least to {\it them}, the
verification of a recent theorem [89.]. That theorem is,
however, verified in a very simple manner, by observing that,
with the arrangement here conceived, {\it the three angles\/}
${\sc a}'' {\sc d}_1 {\sc a}^{\rm IV}$,
${\sc b}'' {\sc d}_1 {\sc b}^{\rm IV}$,
${\sc c}'' {\sc d}_1 {\sc c}^{\rm IV}$,
which those {\it infinite\/} and {\it infinitely distant
segments\/} may be imagined to {\it subtend\/} at the
point~${\sc d}_1$, {\it are all right angles\/};
${\sc d}_1 {\sc a}''$, for example, being {\it parallel\/} to the
{\it side\/}~${\sc b} {\sc c}$ of the
{\it triangle\/}~${\sc a} {\sc b} {\sc c}$, which is now an
{\it equilateral\/} one; while ${\sc d}_1 {\sc a}^{\rm IV}$ is
{\it perpendicular\/} to the same side, because it is drawn from
the {\it mean point\/}~${\sc d}_1$, and passes through the
{\it opposite corner},~${\sc a}$. As another verification of the
theorem [89.], it will be found that, with the arrangement here
supposed, the {\it segments\/}
${\sc a}' {\sc a}^{\rm IX}$,
${\sc c}_0 {\sc b}_1^{\rm IV}$,
${\sc c}_1^{\rm V} {\sc b}^{\rm VI}$,
{\it of the seventh trace\/} [43.], {\it subtend right angles at
the given point\/}~${\sc b}$.
\bigbreak
[92.]
The {\it involution of the three segments\/} [82.] is only
{\it one\/} of the consequences of the {\it three harmonic
equations\/} [81.], or of what we have called in [85.] the
{\it Involution of the two Triads}, $a b c$ and $a' b' c'$. We
can therefore {\it infer more}, respecting the {\it geometrical
relations\/} of the {\it six points}, even in the {\it general
state\/} of the whole {\it Figure}, or {\sc Net}, than merely
that those three segments subtend {\it right angles}, as above,
{\it at every point of one real circle}, which has its
{\it centre on the common line}, and its {\it plane perpendicular
thereto}. The {\it order of succession\/} of the six points
being supposed to be the following,
$a c' b a' c b'$,
from which it can only differ, if at all, by changes not
important to the argument, let ${\sc p}$ be, as in [90.], a point
such that the angles
$a {\sc p} a'$, $b {\sc p} b'$, $c {\sc p} c'$
are {\it right}. Then, because the {\it three pencils},
$${\sc p} \mathbin{.} a c' b c,\quad
{\sc p} \mathbin{.} c' b a' b',
\quad\hbox{and}\quad
{\sc p} \mathbin{.} b a' c a,$$
are all {\it harmonic\/} pencils by [81.], it follows that (with
the supposed {\it order\/} of the points) the lines
${\sc p} c'$ and ${\sc p} c$ are respectively the
{\it internal\/} and {\it external bisectors\/} of the
{\it angle\/}~$a {\sc p} b$;
${\sc p} b$ and ${\sc p} b'$, of the angle~$c' {\sc p} a'$;
and ${\sc p} a'$, ${\sc p} a$, of ~$b {\sc p} c$:
the line~${\sc p} c$ bisecting also the angle
$a' {\sc p} b'$ internally. Hence it is easy to infer the
following {\it continued equation between angles\/} (which is
supposed to be new):
$$a {\sc p} c' = c' {\sc p} b = b {\sc p} a' = a' {\sc p} c = c {\sc p} b'
= {\pi \over 6};$$
and therefore we may enunciate this
{\it Theorem\/}:---``{\it When six collinear points form a system
of two triads in involution, their five successive intervals
subtend angles each equal to the third part of a right angle, at
every point of a certain circle, of which the axis is their
common line.}''
For example, with the particular arrangement [91.] of the five
initial points ${\sc a} \, \ldots \, {\sc e}$, it is found that
the five successive portions,
${\sc c}_0 {\sc a}^{\rm IX}$,
${\sc a}^{\rm IX} {\sc c}_1^{\rm V}$,
${\sc c}_1^{\rm V} {\sc b}_1^{\rm IV}$,
${\sc b}_1^{\rm IV} {\sc a}'$,
${\sc a}' {\sc b}^{\rm VI}$,
of the seventh trace, subtend each an angle of {\it thirty
degrees}, at the given point~${\sc b}$; and the six lines
${\sc d}_1 {\sc a}''$,
${\sc d}_1 {\sc c}^{\rm IV}$,
${\sc d}_1 {\sc b}''$,
${\sc d}_1 {\sc a}^{\rm IV}$,
${\sc d}_1 {\sc c}''$,
${\sc d}_1 {\sc b}^{\rm IV}$,
if suitably distingished from their own opposites, succeed each
other at angular intervals, of the same common amount.
\bigbreak
[93.]
In general, if {\it three equally inclined diameters\/} of a
circle, forming a regular and {\it six-rayed star}, be taken as
a {\it given triad of lines\/} [85.], the {\it triad in
involution\/} therewith is represented by that {\it other star\/}
of the same kind, of which the diameters {\it bisect the
angles\/} between those of the former star: so that if we consider
any {\it six successive rays\/} of the {\it compound\/} or
{\it twelve-rayed star}, which results from the combination of
these {\it two}, their {\it successive angles\/} are evidently
each equal to thirty degrees. But now we see further, that if a
{\it star\/} of this last kind be {\it cut in six points\/} by an
{\it arbitrary transversal\/} in its plane; and if these six
points of section be in any manner put into perspective, by any
{\it new\/} pencil and transversal: the {\it six new points},
thus obtained, as forming still {\it two triads in involution},
must admit of having their {\it five successive intervals seen},
from {\it every point\/} of some {\it new circle}, under
{\it angles still equal each\/} to the same {\it third part of a
right angle}.
\bigbreak
[94.]
We have not yet considered the arrangement of the six points on
either the {\it fifth\/} or the {\it sixth\/} typical
{\it trace\/} [43.]; but it is easy to do this as follows. Let
$a \, b \, c \, \alpha \, \beta \, \gamma$
denote, as new temporary symbols, either the six points of the
fifth trace (comp.~[58.]),
$$\hbox{I.}\quad
a = (1 \, 0 \, 0),\quad
b = (1 \, \overline{1} \, 1),\quad
c = (1 \, 1 \, \overline{1}),\quad
\alpha = (0 \, 1 \, \overline{1}),\quad
\beta = (2 \, 1 \, \overline{1}),\quad
\gamma = (2 \, \overline{1} \, 1);$$
or these six other points, belonging to the sixth trace,
$$\hbox{II.}\quad
a = (1 \, 1 \, 1),\quad
b = (1 \, 0 \, 2),\quad
c = (1 \, 2 \, 0),\quad
\alpha = (0 \, 1 \, \overline{1}),\quad
\beta = (2 \, 3 \, 1),\quad
\gamma = (2 \, 1 \, 3);$$
we shall then have, in each case, the three harmonic equations,
$$(b \, a \, c \, \alpha)
= (c \, \beta \, a \, \alpha)
= (a \, \gamma \, b \, \alpha)
= -1.$$
In {\it each\/} case, therefore, we may consider ourselves as
first deriving from three points a fourth, as the harmonic
conjugate of the first with respect to the other two; and then
deriving a fifth point, and a sixth, as the harmonic conjugates
of that fourth point, with respect, on the one hand, to the third
and first points; and on the other hand, to the first and second
points of the system.
\bigbreak
[95.]
Having regard merely to this {\it common law}, we may enunciate
(comp.\ [80.] [81.]) this theorem:---
\medbreak
``{\it The sixty lines, in the ten planes of first construction,
represented by the fourth and fifth typical traces of the planes
on the plane~${\sc a} {\sc b} {\sc c}$, although not all
syntypical, are all homographically divided.}''
\medbreak
And this {\it common mode\/} of their {\it division\/} is such,
that if the fourth point be thrown off to infinity, the first
point bisects the interval between the second and third; the
fifth point bisects the interval between third and first; and the
sixth point bisects the interval between first and second: so
that, on the whole, we have a {\it finite line},~$b c$,
{\it quadrisected\/} in the points $\gamma$,~$a$,~$\beta$, and
cut at infinity in $\alpha$; whereas if, on either the
{\it fourth\/} or the {\it seventh\/} trace, {\it one\/} of the
six points, but only one, had been thus made {\it infinitely
distant}, the {\it five others\/} would have presented the figure
of a {\it finite right line, bisected\/} and {\it trisected}.
With the equations [94.], if $a$, instead of $\alpha$, be
projected to infinity, it is then the line $\beta \gamma$ which
is quadrisected, namely, in the points $c$, $\alpha$, $b$. In
general, with these last equations, the {\it first set\/} of
three points, $a \, b \, c$, can be {\it derived\/} from the
{\it second set}, $\alpha \, \beta \, \gamma$, by the {\it same
rule\/} [94.], as that by which the second set has been derived
from the first: so that there is a sense in which {\it these two
sets\/} may be said to be {\it reciprocal triads}, although they
are {\it not triads in involution}, according to the definition
[85.].
\bigbreak
[96.]
It may be added that, on either the {\it fifth\/} or the
{\it sixth\/} trace, the two points which we have called
{\it first\/} and {\it fourth}, are the {\it double points\/} of
a new {\it involution}, determined by the {\it two pairs},
{\it second\/} and {\it third}, {\it fifth\/} and {\it sixth};
or, with the recent notations [94.], that $a \, \alpha$ are the
{\it foci\/} of the involution $b c$,~$\beta \gamma$; because the
three last harmonic equations conduct to this fourth equation,
$$(\beta \, a \gamma \alpha) = -1.$$
\bigbreak
[97.]
And, as regards the {\it homography\/} of the divisions on the
same two traces, if we denote, for the sake of distinction, the
six points on the sixth trace by $a' \, \ldots \, \gamma'$, then
(because $\alpha' = \alpha$) the {\it five lines\/}
$a a'$, $b b'$, $c c'$, $\beta \beta'$ $\gamma \gamma'$,
or (comp.~[58.]) the five lines
$${\sc a} {\sc d}_1,\quad
{\sc b}_0 {\sc b}^{\rm V},\quad
{\sc c}_0 {\sc c}_1^{\rm V},\quad
{\sc b}_1^{\rm VII} {\sc b}_1^{\rm VIII},\quad
{\sc c}^{\rm VII} {\sc c}^{\rm VIII},$$
ought to {\it concur\/} in some {\it one point\/}: which
accordingly it is easy to see that they do, namely in the
point~${\sc a}'$; in fact, with the recent signification of
$a, \ldots$ and $a', \ldots$, we have the symbolic equations,
$$(a') - (a) = (b') - (b) = (c') - (c)
= (0 \, 1 \, 1) = ({\sc a}');$$
and
$$(\beta') - (\beta) = (\gamma') - (\gamma)
= (0 \, 2 \, 2) = 2 ({\sc a}').$$
\bigbreak
[98.]
The {\it two sets of six points}, on these two traces, with one
point common, are thus the points in which a certain
{\it six-rayed pencil}, with ${\sc a}'$ for vertex, is
{\it cut\/} by the two traces as transversals; the
{\it symbols\/} of the six {\it rays\/} being the following:
$${\sc a}' {\sc a} {\sc d}_1
= [0 \, 1 \, \overline{1}];\quad
{\sc a}' {\sc b}_0 {\sc b}^{\rm V}
= [\overline{2} \, \overline{1} \, 1];\quad
{\sc a}' {\sc c}_0 {\sc c}_1^{\rm V}
= [\overline{2} \, 1 \, \overline{1}];$$
$${\sc a}' {\sc a}''
= [1 \, 0 \, 0];\quad
{\sc a}' {\sc b}_1^{\rm VII} {\sc b}_1^{\rm VIII}
= [1 \, \overline{1} \, 1];\quad
{\sc a}' {\sc c}^{\rm VII} {\sc c}^{\rm VIII}
= [1 \, 1 \, \overline{1}].$$
And from a mere inspection of these symbols, we can infer
(comp.~(33)) that the {\it first\/} and {\it fourth rays\/} are
the {\it common harmonic conjugates\/} of the {\it two pairs},
{\it second\/} and {\it third}, {\it fifth\/} and {\it sixth}; or
that they are the {\it double rays\/} of the {\it involution},
which those two {\it pairs of rays\/} determine: the theorem
[96.] being thus, in a new way, confirmed.
\bigbreak
[99.]
We have now discussed the arrangements of the {\it points\/} on
those {\it nine\/} typical {\it lines\/}~$\Lambda_3$, whereof
each passes through not less than {\it four}, nor more than
{\it six}, of the 52 points in the plane
${\sc a} \, {\sc b} \, {\sc c}$; but we have still {\it three
other typical lines\/} to consider, namely the lines~$\Lambda_1$
and $\Lambda_2$, of which each passes through {\it at least seven
points}. Taking first, for this purpose, the typical
line~$\Lambda_{2,1}$, namely ${\sc a} {\sc a}'$, which contains
{\it only seven\/} points, whereof the ternary symbols have been
assigned in [55.], and the literal symbols there given may be
retained, we shall, for the moment, reserve the consideration of
the two points~${\sc p}_{2,3}$; but shall introduce a new and
auxiliary point~${\sc p}_{3,1}$ on the same line, which may be
thus denoted:
$${\sc a}^{\rm X} = (1 \, 2 \, 2)
= {\sc a} {\sc a}' \cdot {\sc b} {\sc c}'''
\cdot {\sc c} {\sc b}''';$$
and which may be said to {\it represent\/} or {\it typify\/} a
{\it first group of third construction}, containing {\it fifteen
points}, {\it one\/} on each of the {\it fifteen
lines\/}~$\Lambda_{2,1}$; although, in the present Paper, we can
only {\it allude\/} to {\it such\/} new points~${\sc p}_3$, and
cannot {\it here\/} attempt to {\it enumerate}, or even to
{\it classify\/} them.
\bigbreak
[100.]
We have thus again {\it six\/} points, at this stage, to
consider, namely the points
${\sc a}$, ${\sc a}'$, ${\sc d}_1$, ${\sc a}'''$,
${\sc a}_0$, ${\sc a}^{\rm X}$;
and their symbols easily show that they are connected by the
{\it three\/} following harmonic equations,
$$({\sc a} \, {\sc a}' \, {\sc d}_1 \, {\sc a}''')
= ({\sc a} \, {\sc d}_1 \, {\sc a}' \, {\sc a}_0)
= ({\sc a}' \, {\sc a} \, {\sc d}_1 \, {\sc a}^{\rm X})
= -1;$$
from which it follows, by [85.], that the {\it two triads of
points},
$${\sc a} {\sc a}' {\sc d}_1
\quad\hbox{and}\quad
{\sc a}^{\rm X} {\sc a}''' {\sc a}_0,$$
{\it are triads in involution\/}: with, of course, all the
properties which have been proved, in recent paragraphs of this
Paper, to belong generally to {\it any two\/} such triads. As a
verification, it may be mentioned that, with the particular
arrangement [91.] of the five initial points
${\sc a} \, \ldots \, {\sc e}$, if we determine two new points
${\sc p}$,~${\sc p}'$, of {\it third\/} construction, by the
formul{\ae},
$${\sc p} = (2 \, 1 \, 4)
= {\sc b} {\sc c}''' \cdot {\sc c} {\sc a}''',\quad
{\sc p}' = (2 \, 4 \, 1)
= {\sc c} {\sc b}''' \cdot {\sc b} {\sc a}''',$$
it can be proved that each of the five successive intervals
(comp.~[92.]) between the six points,
$${\sc a},\enspace
{\sc a}''',\enspace
{\sc d}_1,\enspace
{\sc a}^{\rm X},\enspace
{\sc a}',\enspace
{\sc a}_0,$$
subtends the third part of a right angle at each of these two new
auxiliary points, ${\sc p}$ and ${\sc p}'$. But with
{\it other\/} initial configurations, the {\it coordinates\/} of
these two {\it new vertices\/} would be different, because they
are connected with {\it angles}, which are not generally
{\it projective\/} [90.]; althrough, as has been already
remarked, there would always be {\it some\/} new
points~${\sc p}$, or rather a {\it circle\/} of such, possessing
the property in question.
\bigbreak
[101.]
We may however enunciate generally, and without reference to any
such particular {\it arrangement\/} of the five initial points,
this {\it Theorem\/}:---
\medbreak
``{\it On any one of the fifteen lines~$\Lambda_{2,1}$, of second
construction, and first group, the given point~${\sc p}_0$, and
the two derived points of first construction~${\sc p}_1$, compose
a triad, the triad in involution to which\/} [85.] {\it consists
of the point\/}~${\sc p}_{3,1}$, {\it of third construction and
first group, and of the two points~${\sc p}_{2,2}$, of second
construction and second group, upon that line\/};'' with
{\it seven involutions of segments\/} (comp.~[84.])
{\it included\/} under this general relation.
\medbreak
For example, on the line~${\sc a} {\sc a}'$, the {\it three
segments\/}
${\sc a} {\sc a}^{\rm X}$, ${\sc a}' {\sc a}'''$, ${\sc d}_1 {\sc a}_0$
form always an {\it involution\/} of the {\it ordinary\/} kind,
with its {\it double points imaginary\/}; the {\it three other
sets\/} of segments,
${\sc a} {\sc a}^{\rm X}$, ${\sc a}' {\sc a}_0$, ${\sc d}_1 {\sc a}'''$;
${\sc a}' {\sc a}'''$, ${\sc a} {\sc a}_0$, ${\sc d}_1 {\sc a}^{\rm X}$;
and
${\sc d}_1 {\sc a}_0$, ${\sc a} {\sc a}'''$, ${\sc a}' {\sc a}^{\rm X}$,
form {\it each\/} an involution, with {\it real double points\/};
the points ${\sc a}$,~${\sc a}^{\rm X}$ are the {\it real foci\/}
of a {\it fifth\/} involution, determined by the {\it two
pairs\/} of segments ${\sc a}' {\sc d}_1$, and
${\sc a}''' {\sc a}_0$; the points ${\sc a}'$,~${\sc a}'''$ are,
in like manner, the real double points of that {\it sixth\/}
involution, which the {\it two other pairs},
${\sc a}$,~${\sc d}_1$, and ${\sc a}_0$,~${\sc a}^{\rm X}$,
determine: and finally, ${\sc d}_1$ and ${\sc a}_0$ are such
points, for the {\it seventh\/} involution, determined by
${\sc a} {\sc a}'$, ${\sc a}''' {\sc a}^{\rm X}$.
\bigbreak
[102.]
Introducing now the consideration of the two lately {\it reserved
points\/}~${\sc p}_{2,3}$ [99.] of {\it second construction\/}
and {\it third group\/} [45.], upon the typical
line~$\Lambda_{2,1}$, we may derive them from the
point~${\sc p}_0$, the two points~${\sc p}_1$, and the two
points~${\sc p}_{2,2}$ upon that line~${\sc a} {\sc a}'$, by the
two following harmonic equations:
$$({\sc a} \, {\sc a}''' \, {\sc a}' \, {\sc a}^{\rm IV})
= ({\sc a} \, {\sc a}_0 \, {\sc d}_1 \, {\sc a}_1^{\rm IV})
= -1;$$
or by these two others,
$$({\sc a} \, {\sc a}' \, {\sc a}_0 \, {\sc a}^{\rm IV})
= ({\sc a} \, {\sc d}_1 \, {\sc a}''' \, {\sc a}^{\rm IV})
= -1,$$
which may indeed by inferred from the two former, with the help
of the relations between the six points previously considered:
for, in general, if $a b c$, $a' b' c'$ be collinear
triads in involution, and if $d$ and $d'$ be the harmonic
conjugates of $b'$ and $c'$, with respect to the two pairs,
$a b$, $a c$, they are also the harmonic conjugates of $b$ and
$c$, with respect to the two {\it other\/} pairs, $a c'$, $a b'$;
or in symbols,
$$(a \, b \, c' \, d)
= (a \, c \, b' \, d')
= -1,
\quad\hbox{if}\quad
(a \, b' \, b \, d) = (a \, c' \, c \, d') = -1,$$
when the three harmonic equations [81.] exist. We have also,
generally, under these conditions, the equation
$$(a \, d \, a' \, d') = -1;$$
for example, on the line ${\sc a} {\sc a}'$, we have
$$({\sc a} \, {\sc a}^{\rm IV} \, {\sc a}^{\rm X} \, {\sc a}_1^{\rm IV})
= -1.$$
\bigbreak
[103.]
It is scarcely worth while to remark that the 15
lines~$\Lambda_{2,1}$ of the net, as being all {\it syntypical},
are all {\it homographically divided\/}; although it may just be
noticed, as a verification, that the six lines,
$${\sc b} {\sc c},\quad
{\sc b}' {\sc c}',\quad
{\sc b}''' {\sc c}''',\quad
{\sc b}_0 {\sc c}_0,\quad
{\sc b}^{\rm IV} {\sc c}^{\rm IV},\quad
{\sc b}_1^{\rm IV} {\sc c}_1^{\rm IV},$$
which connect corresponding points on the two other lines of the
same group in the given plane, namely
${\sc b} {\sc b}' {\sc d}_1$ and ${\sc c} {\sc c}' {\sc d}_1$,
{\it concur\/} in one point~${\sc a}''$. But it may not be
without interest to observe, that ${\sc a}^{\rm X}$ is the
{\it common harmonic conjugate\/} of ${\sc a}$, with respect to
{\it each\/} of the three pairs,
${\sc a}' {\sc d}_1$, ${\sc a}''' {\sc a}_0$,
${\sc a}^{\rm IV} {\sc a}_1^{\rm IV}$;
which {\it three pairs},\footnote*{That the {\it two first\/} of
these three pairs belong to an involution, with those two double
points, was seen in [101.].}
or segments, form thus an {\it involution}, with ${\sc a}$ and
${\sc a}^{\rm X}$ for its {\it double points}. We have therefore
this {\it Theorem\/}:---
\medbreak
``{\it On each of the fifteen lines~$\Lambda_{2,1}$, the three
pairs of derived points, of first and second constructions},
namely the {\it pair~${\sc p}_1$, the pair~${\sc p}_{2,2}$, and
the pair ${\sc p}_{2,3}$, compose an involution, one double point
of which is the given point~${\sc p}_0$; the other double point
being the point~${\sc p}_{3,1}$, of third construction and first
group, upon the line.}''
\bigbreak
[104.]
We have thus discussed the arrangements of the points
${\sc p}_0$, ${\sc p}_1$, ${\sc p}_2$, on each of the {\it ten\/}
typical lines which connect not {\it fewer\/} than {\it four},
and not {\it more\/} than {\it seven\/} of them; but there are
still {\it two other\/} typical lines to be considered, belonging
to the groups $\Lambda_1$ and $\Lambda_{2,2}$, whereof one, as
${\sc b} {\sc c}$, passes through {\it eight\/} points [54.]; and
the other, as ${\sc b}' {\sc c}'$, has {\it ten\/} points upon it
[56.]. Beginning with the first, we easily find that the two
sets of points
${\sc a}' {\sc b} {\sc c}$ and
${\sc a}'' {\sc a}_1^{\rm V} {\sc a}^{\rm V}$,
are {\it triads in involution\/} [85.]; the latter set being thus
deducible from the former: while the two other points upon the
line may be determined by the condition that they satisfy this
{\it other involution of two triads},
${\sc a}'' {\sc b} {\sc c}$,
${\sc a}' {\sc a}_1^{\rm VI} {\sc a}^{\rm VI}$.
With the {\it initial arrangement\/} [91.], the line
${\sc a}^{\rm VI} {\sc a}_1^{\rm VI}$
is {\it trisected\/} in ${\sc b}$ and ${\sc c}$, and its
{\it middle part\/}~${\sc b} {\sc c}$ is likewise
{\it trisected\/} in ${\sc a}^{\rm V}$ and ${\sc a}_1^{\rm V}$;
while {\it each\/} line is {\it bisected\/} in ${\sc a}'$, and
{\it cut at infinity\/} in ${\sc a}''$. And in general we may
enunciate these {\it two Theorems\/}:---
\medbreak
I. ``{\it On every line of first construction, the
point~${\sc p}_1$ and the two points~${\sc p}_0$ form a triad,
the triad in involution with which consists of the point
${\sc p}_{2,1}$, and the two points~${\sc p}_{2,4}$.}''
\medbreak
II. ``{\it On every such line~$\Lambda_1$, the triad formed by
the point~${\sc p}_{2,1}$, and the two points~${\sc p}_0$, is in
involution with a triad which consists of the point~${\sc p}_1$
and the two points ${\sc p}_{2,5}$.}''
\bigbreak
[105.]
Besides these {\it two involutions of triads}, we have {\it two
distinct involutions\/} of the {\it ordinary\/} kind, into
{\it each\/} of which {\it all the eight points enter\/}; two
being {\it double points\/} in each. For we have these {\it two
other Theorems}, deducible, indeed from the two former, but
perhaps deserving to be separately stated:---
\medbreak
III. ``{\it On every line of first construction, the two given
points are foci of an involution of six points, in which the
points ${\sc p}_1$, ${\sc p}_{2,1}$, are one pair of conjugates,
while the two other pairs are of the common form,
${\sc p}_{2,4}$, ${\sc p}_{2,5}$.}''
For example, ${\sc a}^{\rm V}$, ${\sc a}^{\rm VI}$ are such a
pair, on the line ${\sc b} {\sc c}$.
\medbreak
IV. ``{\it On every such line~$\Lambda_1$, the points
${\sc p}_1$, ${\sc p}_{2,1}$, are the double points of a second
involution of six points, obtained by pairing the two points of
each of the three other groups.}''
\bigbreak
[106.]
Finally, as regards the {\it remaining typical line\/}
${\sc b}' {\sc c}'$, which connects {\it two
points\/}~${\sc p}_1$, and passes through {\it eight
points\/}~${\sc p}_2$, if we reserve for a moment the
consideration of the {\it last pair}, ${\sc p}_{2,8}$, or
${\sc a}^{\rm IX}$ and ${\sc a}_1^{\rm IX}$, we have a
{\it system of eight points upon that line, homographic with the
recent system of eight points on the line\/}~${\sc b} {\sc c}$;
being indeed the {\it intersections\/} of the line
${\sc b}' {\sc c}'$ with the {\it eight-rayed pencil},
${\sc a} \mathbin{.} {\sc a}' {\sc b} {\sc c} {\sc a}'' {\sc a}_1^{\rm V}
{\sc a}^{\rm V} {\sc a}_1^{\rm VI} {\sc a}^{\rm VI}$,
when taken in the order
${\sc a}''' {\sc c}' {\sc b}' {\sc a}'' {\sc a}_1^{\rm VIII}
{\sc a}^{\rm VIII} {\sc a}_1^{\rm VII} {\sc a}^{\rm VII}$.
No description of the arrangement of these latter {\it points\/}
is therefore at this stage required: but as regards the
{\it pencil}, it may be remarked that, by [104.], the 1st, 2nd,
and 3rd {\it rays\/} form a {\it triad of lines, in involution\/}
[85.] {\it with the triad\/} formed by the 4th, 5th and 6th; and
that the {\it triad\/} of the 2nd, 3rd and 4th rays is, in the
same new sense, in {\it involution\/} with the {\it triad\/} of
the 7th, 8th, and 1st: from which {\it double involution of
triads}, the {\it five last rays\/} may be {\it derived}, if the
{\it three first\/} are {\it given}. We have also by [105.] a
{\it double involution of the rays}, considered as {\it paired\/}
with {\it each other}, or with {\it themselves\/}: thus the
second and third rays are the {\it double rays\/} of an
involution (or the {\it usual\/} kind), in which the first is
conjugate to the fourth, the fifth to the seventh, and the sixth
to the eighth; while the first and fourth rays are the double rays
of {\it another\/} involution, in which the second and third, the
fifth and sixth, and the seventh and eighth are conjugate.
\bigbreak
[107.]
It only remains to assign the arrangement of the {\it two last
points of second construction}, ${\sc p}_{2,8}$, with respect to
the {\it other points\/} ${\sc p}_1$, ${\sc p}_2$, on a
line~$\Lambda_{2,2}$, or to some {\it three\/} of them; or to
show how ${\sc a}^{\rm IX}$ and ${\sc a}_1^{\rm IX}$ can be
{\it derived},\footnote*{This point~${\sc a}^{\rm IX}$ may also,
by [81.], be determined on the {\it seventh trace}, or
{\it seventh typical line\/} [74.], as the {\it harmonic
conjugate\/} of ${\sc a}'$, with respect to ${\sc c}_0$ and
${\sc c}_1^{\rm V}$.}
for example, from ${\sc b}'$,~${\sc c}'$, and ${\sc a}''$: which
derivation may easily be effected, on the plan already described
for the fifth and sixth typical traces. In fact, if we denote
the six points
${\sc a}'' \, {\sc c}' \, {\sc b}' \, {\sc a}''' \,
{\sc a}_1^{\rm IX} \, {\sc a}^{\rm IX}$
by
$a \, b \, c \, \alpha \, \beta \, \gamma$,
we have the three harmonic equations of [94.]; and if, by one of
the modes of {\it perspective}, or {\it projection}, mentioned in
[95.], which answers to the initial arrangement [91.], we throw
off the first point~${\sc a}''$ to {\it infinity}, the finite
line ${\sc a}^{\rm IX} {\sc a}_1^{\rm IX}$ is then
{\it quadrisected\/}: being {\it itself bisected\/} at
${\sc a}'''$, while ${\sc c}'$ and ${\sc b}'$ {\it bisect its
halves}. In general, we shall have again the equations [94.], if
we otherwise represent the six lately mentioned points on
${\sc b}' {\sc c}'$ by
$\alpha \, \beta \, \gamma \, a \, b \, c$;
and thus it is seen that {\it those six points\/} are {\it always
homographic, in every state\/} of the figure, or {\it net}, with
the six points
${\sc a}'' \, {\sc b}_1^{\rm VII} \, {\sc c}^{\rm VII}
\, {\sc a} \, {\sc b}_0 \, {\sc c}_0$
on the {\it fifth trace\/} ${\sc a} {\sc a}''$, and with the six
points
${\sc a}'' \, {\sc b}_1^{\rm VIII} \, {\sc c}^{\rm VIII}
\, {\sc d}_1 \, {\sc b}^{\rm V} \, {\sc c}_1^{\rm V}$
on the {\it sixth trace\/} ${\sc d}_1 {\sc a}''$; in fact they
are, if taken in a suitable order, the points in which the
{\it six-rayed pencil\/} [98.], with ${\sc a}'$ for vertex, is
cut by the line~${\sc b}' {\sc c}'$.
\bigbreak
[108.]
We have thus shown for each of the {\it twelve typical lines\/}
[74.], in the plane ${\sc a} {\sc b} {\sc c}$, how {\it all the
points but three}, upon that line, may be derived {\it from those
three\/} by a {\it system of harmonic equations}, not
{\it necessarily\/} employing any point~${\sc p}_3$, or other
{\it foreign\/}\footnote\dag{This {\it non-requirement\/} of
{\it foreign points\/} is the only remarkable thing here: for the
{\it anharmonic function\/} of {\it every group\/} of {\it four
collinear net-points\/} is necessarily {\it rational\/}; and
whenever $(a \, b \, c \, d) =$ any positive or negative quotient
of {\it whole numbers}, it is {\it always possible\/} to deduce
the {\it fourth point\/}~$d$ from the {\it three\/} points
$a$,~$b$,~$c$, by {\it some system\/} of {\it auxiliary points},
derived successively from them through {\it some system of
harmonic equations}.}
or merely {\it auxiliary point\/}: although it appeared that
something was gained, in respect to elegance and clearness, by
introducing, on the line ${\sc a} {\sc a}'$, such a
point~${\sc a}^{\rm x}$ [99.]; or by considering generally, on
any one of the fifteen lines $\Lambda_{2,1}$, a
point~${\sc p}_{3,1}$ of {\it third construction}, belonging to
what may perhaps deserve to be regarded as a {\it first group\/}
[103.] of the points~${\sc p}_3$, in any future {\it extension\/}
[1.] of the results of the present Paper.
\bigbreak
\centerline{{\sc Part~V.}---{\it Applications to the Net,
continued: Distribution of the Given or Derived Points,}}
\centerline{{\it in a Plane of Second Construction, and of First
or Second Group.}}
\nobreak\bigskip
[109.]
It will be necessary to be much more concise, in our remarks on
the distribution of the {\it net-points\/} in {\it planes\/} of
{\it second construction\/}; but a few general remarks may here
be offered, from which it will appear that each plane~$\Pi_{2,1}$
contains {\it forty-seven\/} of the 305 points
${\sc p}_0$,~${\sc p}_1$,~${\sc p}_2$; and that each
plane~$\Pi_{2,2}$ contains {\it forty-three\/} of those points;
with many cases of {\it collineation\/} for each.
\bigbreak
[110.]
We saw in [33.], that each plane~$\Pi_{2,1}$ contains two
lines~$\Lambda_{2,1}$, which intersect in a point~${\sc p}_0$,
and may be regarded as the diagonals of a quadrilateral, of which
the four sides are lines~$\Lambda_{2,2}$. It contains,
therefore, as has been seen, one point~${\sc p}_0$, and four
points~${\sc p}_1$; but it is found to contain also 42
points~${\sc p}_2$, arranged in {\it six groups}, as follows.
\bigbreak
[111.]
There are 2 points~${\sc p}_{2,1}$, namely the intersections of
opposite sides of the quadrilateral; thus, in what we have called
the {\it second typical plane\/} [33.], the sides
${\sc b}_1 {\sc c}_1$, ${\sc c}_2 {\sc b}_2$
intersect in the point~${\sc a}''$; and the sides
${\sc c}_1 {\sc c}_2$, ${\sc b}_2 {\sc b}_1$
in ${\sc d}_1'$ (62).
\bigbreak
[112.]
The plane contains also 8 points~${\sc p}_{2,2}$; namely,
{\it two\/} on each of the {\it two diagonals}, and {\it one\/}
on each of the {\it four sides\/}; and it contains 4
points~${\sc p}_{2,3}$, namely two on each diagonal: but it
contains {\it no\/} point of either of the two groups
${\sc p}_{2,4}$, ${\sc p}_{2,5}$, as a comparison of their
{\it types\/} sufficiently proves, or as may be inferred from the
{\it laws\/} of their construction [46.] [47.].
\bigbreak
[113.]
The same plane contains 12 points ${\sc p}_{2,6}$; namely two on
each side of the quadrilateral; and four others, in which the
plane is intersected by four lines~$\Lambda_{2,2}$; as the
{\it types\/} sufficiently prove. But to show, geometrically,
{\it why\/} there should be {\it only four such intersections},
conducting thus to new points~${\sc p}_{2,6}$ in the plane, let
the five inscribed pyramids [28.] be denoted by the symbols
${\sc a}' \, \ldots \, {\sc e}'$; then the six edges of the
pyramid~${\sc a}'$ are found to intersect the present
plane~$\Pi_{2,1}$ in points already considered, namely in the two
points~${\sc p}_{2,1}$, of {\it meetings of opposite sides}, and
in those four points~${\sc p}_{2,2}$, which are situated {\it on
the diagonals\/} of the quadrilateral; they give therefore
{\it no new points}. Also, each {\it side\/} of the same
quadrilateral is an {\it edge\/} of one of the {\it four other
pyramids} ${\sc b}' \, \ldots \, {\sc e}'$; but there remains,
for each such pyramid, an {\it opposite edge\/}: and these are
the {\it four lines, out of the plane}, which {\it intersect
it\/} in the {\it four points\/}~${\sc p}_{2,6}$, additional to
the {\it eight points\/}~${\sc p}_{2,6}$, which are ranged, two
by two, {\it upon the sides}. There are thus {\it twelve
points\/} of the {\it group\/}~${\sc p}_{2,6}$, in any one
plane~$\Pi_{2,1}$; and we have now exhausted the intersections of
that plane with the lines~$\Lambda_{2,2}$; and also, as it will
be found, with the lines $\Lambda_{2,1}$, and $\Lambda_1$.
\bigbreak
[114.]
But there remain {\it eight\/} points~${\sc p}_{2,7}$, and
{\it eight\/} points~${\sc p}_{2,8}$, in the plane now
considered; namely {\it two\/} of {\it each group}, on each of
the {\it four sides\/} of the quadrilateral. There are,
therefore, 16 such points; which, with the 12
points~${\sc p}_{2,6}$; the 4 points~${\sc p}_{2,3}$; the 8
points~${\sc p}_{2,2}$; the 2 points~${\sc p}_{2,1}$; the 4
points~${\sc p}_1$; and the one point~${\sc p}_0$, make up (as
has been said in [109.]) a system of 47 points, {\it given or
derived}, in any one of the fifteen planes~$\Pi_{2,1}$.
It may be remarked that with the initial arrangement [91.] of the
five given points, the four points
${\sc b}' \, {\sc c}' \, {\sc b}_2 \, {\sc c}_2$,
in a new plane~$\Pi_{2,1}$, are corners of a {\it square}, which
has the point~${\sc e}$ for its {\it centre\/}; and that thus the
Figure, of the 47 points in such a plane, may be thrown into a
clear and elegant perspective.
\bigbreak
[115.]
As regards the distribution in a plane~$\Pi_{2,2}$, such as the
{\it Third Typical Plane\/} [34.], it may here be sufficient to
observe, that besides containing {\it three
lines\/}~$\Lambda_{2,2}$, namely the {\it sides of a triangular
face\/} [34.] of one of the {\it five inscribed pyramids\/}
[28.], and {\it three points\/}~${\sc p}_1$, which are the
{\it corners\/} of that {\it triangle}, and serve to
{\it determine the plane\/} [1.], it contains also {\it forty
points\/}~${\sc p}_2$, which are arranged in {\it groups}, as
follows. {\it Each\/} of the {\it four first groups}, of
{\it second construction},
${\sc p}_{2,1}, \ldots \, {\sc p}_{2,4}$,
gives {\it three points\/} to the plane; the {\it fifth group\/},
${\sc p}_{2,5}$, furnishes only {\it one point\/}; and the
{\it sixth}, {\it seventh} and {\it eighth\/} groups,
${\sc p}_{2,6}, \ldots \, {\sc p}_{2,8}$,
supply {\it six}, {\it twelve}, and {\it nine\/} points,
respectively. Of these 40 points~${\sc p}_2$,
{\it twenty-four\/} are ranged, eight by eight, {\it on the three
sides\/} of the triangle, as was to be expected from [56.]; and
the existence of {\it at least\/} 27 {\it points},
${\sc p}_1$,~${\sc p}_2$, in a plane $\Pi_{2,2}$, might thus have
been at once foreseen. But we have also to consider the
{\it traces\/}, on that plane, of the 52 {\it lines},
$\Lambda_1$,~$\Lambda_2$, which are not contained therein. Of
these lines, it is found that 36 {\it intersect the sides\/} of
the triangle, and give therefore {\it no new points}. But the
{\it sixteen other lines\/} intersect the {\it plane}, in so many
{\it new\/} and {\it distinct points\/}; and thus the {\it total
number\/} [109.], of {\it forty-three derived points},
${\sc p}_1$,~${\sc p}_2$, in a plane~$\Pi_{2,2}$, which contains
{\it no given point\/}~${\sc p}_0$, is made up.
\bigbreak
[116.]
Without attempting here to enumerate the cases of
{\it collineation}, in either of the two typical planes~$\Pi_2$,
we may just remark, that while the traces of four of the
planes~$\Pi_1$ on the typical plane~$\Pi_{2,1}$ are the four
sides, and the traces of four others are the diagonals, of the
quadrilateral already mentioned, the trace of a ninth
plane~$\Pi_1$, namely ${\sc a} {\sc b} {\sc c}$, on that
plane~$\Pi_{2,1}$ has been already considered, as the
trace~${\sc a} {\sc a}''$ of the latter on the former; but that
the trace of the {\it tenth plane\/}~$\Pi_1$, namely
${\sc a} {\sc d} {\sc e}$, or
$[0 \, 1 \, \overline{1} \, 0 \, 0]$, on
${\sc a} {\sc b}_1 {\sc c}_2 {\sc c}_1 {\sc b}_2$,
or on
$[0 \, 1 \, 1 \, \overline{1} \, \overline{1}]$,
is a {\it new line}, ${\sc a} {\sc d}'_1$; which passes thus
through one point~${\sc p}_0$ and one point~${\sc p}_{2,1}$, and
also through two points~${\sc p}_{2,2}$, namely
$(0 \, 1 \, 1 \, 2 \, 0)$ and $(0 \, 1 \, 1 \, 0 \, 2)$,
and through two points~${\sc p}_{2,6}$, namely
$(2 \, 0 \, 0 \, 1 \, \overline{1})$ and
$(2 \, 0 \, 0 \, \overline{1} \, 1)$:
being, however, {\it syntypical\/} with the formerly considered
trace~${\sc a} {\sc a}''$, and therefore leading to no new
harmonic or anharmonic relations.
\bigbreak
[117.]
As a specimen of a case of collineation which conducts to such
{\it new relations}, let us take the four following
points~${\sc p}_2$, in the second typical plane,
$$a = (0 \, 1 \, 1 \, 2 \, 0),\quad
b = (0 \, 0 \, 2 \, 1 \, 1),\quad
c = (0 \, 2 \, 0 \, 3 \, \overline{1}),\quad
d = (0 \, \overline{1} \, 3 \, 0 \, 2),$$
whereof the two first are points~${\sc p}_{2,2}$, and the two
last are points~${\sc p}_{2,8}$; and of which the symbols satisfy
the equations,
$$(c) = 2(a) - (b),\quad
(d) = - (a) + 2(b);
\quad\hbox{whence}\quad
(a \, d \, b \, c) = 4.$$
{\it These four points}, therefore, with which it is found that
{\it no other\/} given or derived point of the system
${\sc p}_0$,~${\sc p}_1$,~${\sc p}_2$ is {\it collinear}, do
{\it not\/} form a {\it harmonic group\/}; and consequently we
{\it cannot construct the fourth point},~$d$, when the {\it three
other\/} points, $a$,~$b$,~$c$, are {\it given}, by means of
{\it harmonic relations alone\/} (comp.~[108.]), unless we
introduce some {\it auxiliary point}, or points, $e,\ldots$,
which shall be at lowest of the {\it third construction}. But if
we write
$$e = (1 \, 2 \, 0 \, 2 \, 0)
\equiv (0 \, 1 \, \overline{1} \, 1 \, \overline{1}),\quad
f = (\overline{1} \, 0 \, 2 \, 2 \, 0)
\equiv (0 \, 1 \, 3 \, 3 \, 1),$$
so that $e$ is a point~${\sc p}_{3,1}$ [99.], while $f$ may be
said to be a point~${\sc p}_{3,2}$, we find that these two
{\it new\/} or {\it auxiliary\/} points, $e$,~$f$, are the
{\it double points\/} of the {\it involution}, determined by the
{\it two pairs\/}, $ab$, $cd$; because we have the two harmonic
equations,
$$(a \, e \, b \, f) = (c \, e \, d \, f) = -1.$$
And because we have also,
$$(c \, a \, b \, e) = (a \, b \, d \, e) = -1,$$
we need only employ the {\it one\/} auxiliary point~$e$,
considered as the harmonic conjugate of $a$, with respect to $b$
and $c$; and then determine the fourth point~$d$, as the
harmonic conjugate of $a$, with respect to $b$ and $e$. It may
be added that $a b e$ and $d c f$ are {\it triads in
involution\/} [85.]; so that if $e$ be projected to infinity, the
finite line $c d $ is {\it trisected\/} at $a$ and $b$.
\bigbreak
\centerline{{\sc Part~VI.}---{\it On some other Relations of
Complanarity, Collinearity, Concurrence, or Homology,}}
\centerline{{\it for Geometrical Nets in Space.}}
\nobreak\bigskip
[118.]
Although we have not proposed, in the present Paper, to
{\it enumerate}, or even to {\it classify}, any points, lines, or
planes, beyond what we have called the {\it Second
Construction\/} [1.], yet {\it some\/} such points, lines, and
planes have offered themselves naturally to our consideration:
and we intend, in this {\it Sixth Part}, to consider a few
others, chiefly in connexion with relations of {\it homology}, of
triangles or pyramids which have been already mentioned.
\bigbreak
[119.]
It was remarked in [29.], that the thirty lines~$\Lambda_{2,2}$
are the sides of {\it ten triangles\/}~${\sc t}_2$, of
{\it second construction}, which are certain {\it inscribed
homologues\/} of ten {\it other\/} triangles~${\sc t}_1$, of
{\it first\/} construction [26.]; the {\it ten\/} corresponding
{\it centres\/} of homology being the ten points~${\sc p}_1$.
For example, the triangle ${\sc a}' {\sc b}' {\sc c}'$ is
inscribed in ${\sc a} {\sc b} {\sc c}$, and is {\it homologous\/}
thereto, the point~${\sc d}_1$ being their {\it centre\/} of
homology; because we have the three relations of
{\it intersection},
$${\sc a}' = {\sc d}_1 {\sc a} \cdot {\sc b} {\sc c},
\hbox{ \&c.};$$
or because, ${\sc a}'$ being a point of ${\sc b} {\sc c}$, \&c.,
the {\it three joining lines\/} ${\sc a} {\sc a}'$, \&c.,
{\it concur\/} in the point~${\sc d}_1$.
\bigbreak
[120.]
Proceeding to determine the {\it axis\/} of this homology, or the
right line which is the locus of the points of intersection of
corresponding sides, we easily see that it is the line
${\sc a}'' {\sc b}'' {\sc c}''$; because we had
${\sc a}'' = {\sc b} {\sc c} \cdot {\sc b}' {\sc c}'$, \&c.
And because an analogous result must take place in {\it each\/}
of the {\it ten planes\/}~$\Pi_1$, we see that {\it the ten
points~${\sc p}_{2,1}$ are ranged, three by three, on
ten lines~$\Lambda_{3,1}$, in the ten planes\/}~$\Pi_1$; namely
on the {\it axes of homology\/} of the {\it ten pairs of
triangles} ${\sc t}_1$,~${\sc t}_2$, in those ten planes: which
axes are the lines,
$${\sc d}_1' {\sc a}_1' {\sc a}_2', \hbox{ \&c.};\quad
{\sc c}_1' {\sc b}_1' {\sc a}'', \hbox{ \&c.};\quad
{\sc c}_2' {\sc b}_2' {\sc a}'', \hbox{ \&c.};
\quad\hbox{and}\quad
{\sc a}'' {\sc b}'' {\sc c}'';$$
each point~${\sc p}_{2,1}$ being thus {\it common\/} to
{\it three\/} of them, because it is common to those
{\it three planes\/}~$\Pi_1$, which contain the line~$\Lambda_1$
whereupon it is situated. Each point~${\sc p}_{2,1}$ is also the
{\it common intersection\/} of this last line with
{\it three lines\/}~$\Lambda_{2,2}$; we have for example, the
{\it formul{\ae} of concurrence},
$${\sc a}''
= {\sc b} {\sc c} \cdot {\sc b}' {\sc c}'
\cdot {\sc b}_1 {\sc c}_1 \cdot {\sc b}_2 {\sc c}_2.$$
\bigbreak
[121.]
The line ${\sc a}'' {\sc b}'' {\sc c}''$ was seen to be the
{\it common trace\/} of {\it two planes\/}~$\Pi_{2,2}$, namely of
${\sc a}_1 {\sc b}_1 {\sc c}_1$ and ${\sc a}_2 {\sc b}_2 {\sc c}_2$,
on the plane~$\Pi_1$, namely ${\sc a} {\sc b} {\sc c}$, in which
it is situated; and a similar result must evidently hold good for
{\it each\/} of the {\it ten lines\/}~$\Lambda_{3,1}$. But we
may add that the {\it three triangles\/}
${\sc a} {\sc b} {\sc c}$,
${\sc a}_1 {\sc b}_1 {\sc c}_1$,
${\sc a}_2 {\sc b}_2 {\sc c}_2$,
in the plane of {\it each\/} of which the line
${\sc a}'' {\sc b}'' {\sc c}''$
is contained, are {\it homologous, two by two}, and have this line
for the {\it common axis of homology\/} of each of their
{\it three pairs\/}; having however {\it three distinct
centres\/} of homology, namely ${\sc d}_1'$ for second and third,
${\sc d}$ for third and first, and ${\sc e}$ for first and
second: with (as we need not again repeat) analogous results for
the {\it other\/} lines $\Lambda_{3,1}$, of which {\it group\/}
we here take the line ${\sc a}'' {\sc b}'' {\sc c}''$ as
{\it typical}. It may be remarked that the {\it four centres},
recently determined, are {\it collinear}, and compose an
{\it harmonic group\/}; and that the {\it inscribed\/} triangle
${\sc a}' {\sc b}' {\sc c}'$ is also {\it homologous\/} with
{\it each\/} of the two triangles
${\sc a}_1 {\sc b}_1 {\sc c}_1$, ${\sc a}_2 {\sc b}_2 {\sc c}_2$,
although not {\it complanar\/} with {\it either\/}; the line
${\sc a}'' {\sc b}'' {\sc c}''$ being {\it still\/} the
{\it common axis\/} of homology; while the {\it two centres}, of
these last two homologies, are the two given points, ${\sc d}$
and ${\sc e}$.
\bigbreak
[122.]
The {\it six points\/}~${\sc p}_{2,2}$, in the
plane~${\sc a} {\sc b} {\sc c}$, have been seen to range
themselves, according to their {\it two ternary types\/} [41.],
into {\it two sets of three}, which are the {\it corners\/} of
{\it two new triangles\/}; {\it one\/} of these, namely
${\sc a}''' {\sc b}''' {\sc c}'''$, being an
{\it inscribed homologue\/} of ${\sc a}' {\sc b}' {\sc c}'$;
while the {\it other}, namely ${\sc a}_0 {\sc b}_0 {\sc c}_0$, is
an {\it exscribed homologue\/} of ${\sc a} {\sc b} {\sc c}$; and
these two triangles are also homologous to {\it each other\/}:
the {\it line\/} ${\sc a}'' {\sc b}'' {\sc c}''$ being still the
{\it common axis}, and the {\it point\/}~${\sc d}_1$ being the
{\it common centre\/} of homology. And the same thing holds good
for any one of these four triangles,
${\sc a}_0 {\sc b}_0 {\sc c}_0$,
${\sc a} {\sc b} {\sc c}$,
${\sc a}' {\sc b}' {\sc c}'$,
${\sc a}''' {\sc b}''' {\sc c}'''$,
in the plane~$\Pi_1$ here considered, as compared with the
triangle
${\sc a}_1^{\rm IV} {\sc b}_1^{\rm IV} {\sc c}_1^{\rm IV}$,
whereof the corners are those three points~${\sc p}_{2,3}$, which
are {\it not\/} ranged on the line
${\sc a}'' {\sc b}'' {\sc c}''$,
as the three {\it other\/} points~${\sc p}_{2,3}$, namely
${\sc a}^{\rm IV}$,~${\sc b}^{\rm IV}$,~${\sc c}^{\rm IV}$, have
been seen to be.
\bigbreak
[123.]
It was remarked in [28.], that each of the {\it five
pyramids\/}~${\sc r}_2$ is not only {\it inscribed\/} in the
corresponding pyramid~${\sc r}_1$ [26.], but it is also
{\it homologous\/} therewith; the {\it centre\/} of their
homology being a point~${\sc p}_0$: thus the point~${\sc e}$ is
such a centre, for the two pyramids
${\sc a} {\sc b} {\sc c} {\sc d}$ and
${\sc a}_1 {\sc b}_1 {\sc c}_1 {\sc d}_1$,
or for those which we have lettered as ${\sc e}$ and ${\sc e}'$
[26.] [113.]. The {\it planes\/}
${\sc b} {\sc c} {\sc d}$,
${\sc b}_1 {\sc c}_1 {\sc d}_1$,
of two corresponding {\it faces}, intersect in the
{\it line\/}~${\sc c}_2' {\sc b}_2' {\sc a}''$; the planes
${\sc c} {\sc a} {\sc d}$,
${\sc c}_1 {\sc a}_1 {\sc d}_1$,
in
${\sc a}_2' {\sc c}_2' {\sc b}''$; the planes
${\sc a} {\sc b} {\sc d}$,
${\sc a}_1 {\sc b}_1 {\sc d}_1$,
in
${\sc b}_2' {\sc a}_2' {\sc c}''$; and the planes
${\sc a} {\sc b} {\sc c}$,
${\sc a}_1 {\sc b}_1 {\sc c}_1$,
in ${\sc a}'' {\sc b}'' {\sc c}''$. Hence it is easy to infer
that {\it these six points\/}~${\sc p}_{2,1}$, namely
$${\sc a}'',\enspace
{\sc b}'',\enspace
{\sc c}'',\enspace
{\sc a}_2',\enspace
{\sc b}_2',\enspace
{\sc c}_2',$$
are all situated {\it in one plane\/}, which is the {\it plane of
homology\/} of the {\it two pyramids} ${\sc e}$ and ${\sc e}'$,
and which we shall denote by $[{\sc e}]$; its {\it quinary
symbol\/} being
$$[{\sc e}] = [ 1 \, 1 \, 1 \, 1 \, \overline{4}],$$
which may also serve as a {\it type\/} of the {\it group\/}
$[{\sc a}] \, \ldots \, [{\sc e}]$. And in fact, the quinary
symbols of the six points all satisfy the {\it equation\/}
(comp.~[19.],
$$x + y + z + w = 4v.$$
\bigbreak
[124.]
It may be noted that the {\it two planes\/} of homology,
$[{\sc d}]$ and $[{\sc e}]$, have the {\it line\/}
${\sc a}'' {\sc b}'' {\sc c}''$ for their {\it common trace\/} on
the plane~${\sc a} {\sc b} {\sc c}$; and that the traces of the
{\it three other planes\/} of the same group,
$[{\sc a}]$, $[{\sc b}]$, $[{\sc c}]$, which have
$$[\overline{4} \, 1 \, 1],\quad
[1 \, \overline{4} \, 1],\quad
[1 \, 1 \, \overline{4}],$$
for their {\it ternary symbols}, pass respectively through the
points ${\sc a}^{\rm X}$,~${\sc b}^{\rm X}$,~${\sc c}^{\rm X}$,
(comp.~[99.]), and coincide with the lines
${\sc b}_1^{\rm IV} {\sc c}_1^{\rm IV}$, \&c., or with the
{\it sides\/} of the last mentioned {\it triangle\/} [122.]. And
it follows from [123.], that {\it the ten
points\/}~${\sc p}_{2,1}$ are ranged {\it six by six}, and that
the {\it ten lines\/}~$\Lambda_{3,1}$ are ranged {\it four by
four}, in {\it five planes\/}~$\Pi_{3,1}$; namely in the five
planes $[{\sc a}] \, \ldots \, [{\sc e}]$ of {\it homology of
pyramids}. But {\it these last laws\/} of arrangement, of points
and lines, must be considered as included in results which have
been comparatively long known, respecting
{\it transversal\/}\footnote*{Compare the second note to [1.].}
{\it lines and planes in space}.
\bigbreak
[125.]
Instead of {\it inscribing\/} a pyramid ${\sc e}'$ in the
pyramid~${\sc e}$, we may propose to {\it exscribe\/} to the
latter a {\it new\/} pyramid
${\sc a}^\backprime {\sc b}^\backprime {\sc c}^\backprime {\sc d}^\backprime$,
or ${\sc e}^\backprime$, which shall be {\it homologous\/} with
it, the given point~${\sc e}$ being still the {\it centre\/} of
homology. In other words, the {\it four new planes\/}
${\sc b}^\backprime {\sc c}^\backprime {\sc d}^\backprime,\ldots,
{\sc a}^\backprime {\sc b}^\backprime {\sc c}^\backprime$,
or
${\sc e}_{\sc a}$,~${\sc e}_{\sc b}$,~${\sc e}_{\sc c}$,~${\sc e}_{\sc d}$,
are to pass {\it through the four given points\/}
${\sc a}$,~${\sc b}$,~${\sc c}$,~${\sc d}$; and the {\it four new
lines\/}
${\sc a} {\sc a}^\backprime$,
${\sc b} {\sc b}^\backprime$,
${\sc c} {\sc c}^\backprime$,
${\sc d} {\sc d}^\backprime$
are to {\it concur}, in the {\it fifth given point\/}~${\sc e}$.
The solution of this problem is found to be expressed by the
following quinary symbols for the four sought planes:
$$[{\sc e}_{\sc a}] = [0 \, 1 \, 1 \, 1 \, \overline{3}],
\quad \ldots \quad
[{\sc e}_{\sc d}] = [1 \, 1 \, 1 \, 0 \, \overline{3}].$$
In fact, the pyramid~${\sc e}^\backprime$, with these four planes
for {\it faces\/} is evidently {\it exscribed\/} to the
pyramid~${\sc a} {\sc b} {\sc c} {\sc d}$, or ${\sc e}$; and
because its {\it corners\/} may be represented by these other
quinary symbols,
$${\sc a}^\backprime = (3 \, 0 \, 0 \, 0 \, 1),
\quad \ldots \quad
{\sc d}^\backprime = (0 \, 0 \, 0 \, 3 \, 1),$$
the condition of {\it concurrence\/} is satisfied. We may remark
that the plane~$[{\sc e}]$ of [123.] is the plane of homology of
the two last pyramids ${\sc e}$ and ${\sc e}^\backprime$; and
that this {\it exscribed pyramid\/}~${\sc e}^\backprime$ is
homologous also to the {\it inscribed\/} pyramid~${\sc e}'$, the
point~${\sc e}$ being still the {\it centre}, and the
plane~$[{\sc e}]$ the {\it plane\/} of their homology.
\bigbreak
[126.]
It may be remarked that the {\it common trace\/} of the two
planes ${\sc e}_{\sc d}$ and ${\sc d}_{\sc e}$, on the
plane~${\sc a} {\sc b} {\sc c}$, is the line
${\sc a}'' {\sc b}'' {\sc c}''$; to {\it construct}, then, the
{\it exscribed pyramid\/} ${\sc e}^\backprime$, we may construct
the plane~${\sc e}_{\sc d}$ of {\it one\/} of its {\it faces}, by
connecting the {\it point\/}~${\sc d}$ with the line
${\sc a}'' {\sc b}'' {\sc c}''$; and similarly for the rest. Or
if we wish to determine separately the {\it new point}, or
corner,~${\sc d}^\backprime$, which {\it corresponds\/} to the
given point~${\sc d}$, we may do so, by the {\it anharmonic
equation},
$$({\sc d} \, {\sc d}_1 \, {\sc e} \, {\sc d}^\backprime)
= 3;$$
for which may be substituted\footnote*{Compare the note to
[108.].}
the system of the {\it two\/} following {\it harmonic\/}
equations:
$$({\sc d} \, {\sc d}_1 \, {\sc e} \, {\sc f})
= ({\sc d} \, {\sc d}^\backprime \, {\sc d}_1 \, {\sc f})
= -1;$$
where ${\sc f}$ is an auxiliary point, namely ${\sc d}_1'$.
\bigbreak
\centerline{{\sc Part~VII.}---{\it On the Homography and
Rationality of Nets in Space; and on a Connexion of}}
\centerline{{\it such Nets with Surfaces of the Second Order.}}
\nobreak\bigskip
[127.]
In general, {\it all geometric nets in space\/} are
{\it homographic figures\/}; {\it corresponding points, lines},
and {\it planes}, being those which have the {\it same\/} or
({\it congruent\/}) {\it quinary symbols}, in whatever manner we
may pass from one to another system of {\it five initial points},
${\sc a} \, \ldots \, {\sc e}$; whereof it is still supposed that
{\it no four are complanar}. {\it All\/} points, lines, and
planes of any such {\it Net\/} are evidently {\it rational}, in
the sense [8] already defined, with respect to the initial
system; and conversely it is not difficult to prove that every
{\it rational point, line}, or {\it plane}, in space, {\it is a
net-point}, {\it net-line}, or {\it net-plane}, whatever that
initial system of five points may be. It follows that although
{\it no irrational point, line}, or {\it plane}, can possibly
{\it belong\/} to the {\it net}, with respect to which it
{\it is\/} thus irrational, yet it can be {\it indefinitely
approached to}, by points, lines, or planes which {\it do\/} so
belong: a remarkable and interesting theorem, which appears to
have been first discovered by {\it M\"{o}bius\/};\footnote*{See
page 295 of the {\it Barycentric Calculus}. As regards the
theory of {\it homographic figures}, chapter~{\sc xxv}.\ of the
{\it G\'{e}om\'{e}trie Sup\'{e}rieure\/} of M.~Chasles may be
consulted with advantage. But with respect to {\it anharmonic
ratio}, generally, it must be remarked that Professor
{\it M\"{o}bius\/} was thoroughly familiar with its theory and
practice, when he published in 1827; although he called it by
the longer but perhaps more expressive name of
Doppelschnittsverh\"{a}ltniss ({\it ratio bissectionalis\/}). It
may be added that he denotes by
$({\sc a}, {\sc c}, {\sc b}, {\sc d})$,
what I write as
$({\sc a} \, {\sc b} \, {\sc c} \, {\sc d})$.}
to whom indeed, as has been already said, the {\it conception of
the net\/} is due, but whose {\it analysis\/} differs essentially
from that employed in the present Paper.
\bigbreak
[128.]
As regards the {\it passage from one net in space to another},
let the {\it quinary symbols\/} of some five given points
${\sc p}_1 \, \ldots \, {\sc p}_5$, whereof no four are in one
plane, be with respect to the {\it given\/} initial system
${\sc a} \, \ldots \, {\sc e}$ the following:---
$${\sc p}_1 = (x_1 \, \ldots \, v_1),
\quad\ldots\quad
{\sc p}_5 = (x_5 \, \ldots \, v_5);$$
and let $a' \, \ldots \, e'$ and $u'$ be six coefficients,
determined so as to satisfy the {\it quinary equation\/} [5.],
$$a' ({\sc p}_1) + b' ({\sc p}_2) + c' ({\sc p}_3)
+ d' ({\sc p}_4) + e' ({\sc p}_5)
= - u' ({\sc u}),$$
or the five ordinary equations which it includes, namely,
$$a' x_1 + \cdots + e' x_5 = \cdots =
a' v_1 + \cdots + e' v_5 = - u'.$$
Let ${\sc p}'$ be any sixth point of space, such that
$$({\sc p}')
= x a' ({\sc p}_1) + y b' ({\sc p}_2) + z c' ({\sc p}_3)
+ w d' ({\sc p}_4) + v e' ({\sc p}_5)
+ u ({\sc u});$$
then {\it this sixth point~${\sc p}'$ can be derived from the
five points ${\sc p}_1 \, \ldots \, {\sc p}_5$ by the same
constructions, as those by which the point
${\sc p} = (x \, y \, z \, w \, v)$
is derived from the five given points\/}
${\sc a} \, {\sc b} \, {\sc c} \, {\sc d} \, {\sc e}$.
For example, if we take the five points,
$${\sc a}_1 = (1 \, 0 \, 0 \, 0 \, 1),\quad
{\sc b}_1 = (0 \, 1 \, 0 \, 0 \, 1),\quad
{\sc c}_1 = (0 \, 0 \, 1 \, 0 \, 1),\quad
{\sc d}_1 = (0 \, 0 \, 0 \, 1 \, 1),\quad
{\sc e} = (0 \, 0 \, 0 \, 0 \, 1),$$
we have the symbolic equation,
$$({\sc a}_1) + ({\sc b}_1) + ({\sc c}_1) + ({\sc d}_1)
- 3 ({\sc e})
= ({\sc u});$$
if then we write $v' = x + y + z + w - 3v$, the point
$(x \, y \, z \, w \, v')$ is derived from
${\sc a}_1 \, {\sc b}_1 \, {\sc c}_1 \, {\sc d}_1 \, {\sc e}$,
by the same constructions as $(x \, y \, z \, w \, v)$ from
${\sc a} \, {\sc b} \, {\sc c} \, {\sc d} \, {\sc e}$.
In particular, ${\sc d}$ is related to
${\sc a}_1 \, {\sc b}_1 \, {\sc c}_1 \, {\sc d}_1 \, {\sc e}$,
as the point ${\sc p} = (0 \, 0 \, 0\, 3 \, 1)$ is related to
${\sc a} \, {\sc b} \, {\sc c} \, {\sc d} \, {\sc e}$;
but this point~${\sc p}$ satisfies the anharmonic equation,
$({\sc d} \, {\sc d}_1 \, {\sc e} \, {\sc p}) = +3$;
if then
${\sc e}_1 = {\sc d}_1 {\sc e} \cdot {\sc a}_1 {\sc b}_1 {\sc c}_1
= (0 \, 0 \, 0 \, \overline{1} \, 2)$,
we must have the corresponding equation
$({\sc d}_1 {\sc e}_1 {\sc e} {\sc d}) = +3$:
which is accordingly found to exist and furnishes a
{\it construction\/} for {\it exscribing a pyramid\/}
${\sc a} {\sc b} {\sc c} {\sc d}$
{\it to a given pyramid\/}
${\sc a}_1 {\sc b}_1 {\sc c}_1 {\sc d}_1$,
with which it is to be {\it homologous}, and to have a {\it given
point\/}~${\sc e}$ for the {\it centre\/} of their homology,
agreeing with the construction assigned in [126.] for a similar
problem of {\it exscription}. And in general, {\it from any five
given points of a net}, whereof no four are complanar, we can (as
was first shown by {\it M\"{o}bius\/}) {\it return, by linear
constructions, to the five initial points\/}
${\sc a} \, \ldots \, {\sc e}$; and therefore can, in this way,
{\it reconstruct the net}.
\bigbreak
[129.]
If we content ourselves with {\it quaternary\/} (or
{\it anharmonic\/}) {\it coordinates\/} [12.], or suppose (as we
may) that $v = 0$, the {\it equation of a surface of the second
order\/} takes the form,
$$0 = f(x \, y \, z \, w)
= \alpha x^2 + \beta y^2 + \gamma z^2 + \delta w^2
+ 2 (\epsilon y z + \zeta z x + \eta x y)
+ 2 w (\theta x + \iota y + \kappa z);$$
and if the ten coefficients $\alpha \, \ldots \, \kappa$, or
their ratios, be determined by the condition that the surface
shall pass {\it through nine given net-points}, those
{\it coefficients\/} may then be replaced by {\it whole numbers},
and the {\it surface\/} may be said to be {\it rationally related
to the given net}, or to the {\it initial system\/}
${\sc a} \, \ldots \, {\sc e}$, or briefly to be (comp.~[8.]) a
{\it Rational Surface}. For example, if the nine points be
${\sc a} \, {\sc b} \, {\sc c} \, {\sc d} \, {\sc e}
\, {\sc c}' \, {\sc a}' \, {\sc c}_2 \, {\sc a}_2$,
so that, besides passing through~${\sc e}$, the surface has the
gauche quadrilateral ${\sc a} {\sc b} {\sc c} {\sc d}$
superscribed upon it, the equation is
$$\hbox{I}\ldots\quad 0 = f = xz - yw;$$
and if they be
${\sc a}$, ${\sc b}$, ${\sc a}'$, ${\sc b}'$,
${\sc a}_2$, ${\sc b}_2$, ${\sc a}_1$,
${\sc a}^{\rm VII} = (1 \, 2 \, \overline{1} \, 0)$,
and
${\sc f} = (1 \, 2 \, 0 \, \overline{1})$, so that this new
point~${\sc f}$, like ${\sc a}^{\rm VII}$, belongs to the group
${\sc p}_{2,6}$, the equation of the surface is then found to be,
$$\hbox{II}\ldots\quad 0 = f
= w^2 + z^2 - (w + z)(x + y) - 2xy.$$
\bigbreak
[130.]
In general, whether the surface of the second order be
{\it rational\/} or not, it results from the principles of a
former communication that any point
${\sc p} = (x \, y \, z \, w)$
of space is the {\it pole\/} of the {\it plane\/}
$\Pi = [X \, Y \, Z \, W]$,
if $X \, \ldots \, W$ be the {\it derivatives},
$$X = D_x f,\quad
Y = D_y f,\quad
Z = D_z f,\quad
W = D_w f;$$
hence, in particular, the {\it pole of the plane~$[{\sc e}]$ of
homology\/} of the three pyramids
${\sc e}$,~${\sc e}'$,~${\sc e}^\backprime$,
[26.] [113.] [125.], of which plane the {\it quaternary symbol\/}
[12.] is $[1 \, 1 \, 1 \, 1]$, is the point~${\sc k}$ determined
by the equations,
$$X = Y = Z = W,
\quad\hbox{or}\quad
D_x f = D_y f = D_z f = D_w f;$$
and if the point~${\sc e}$ be the mean point of the pyramid
${\sc a} {\sc b} {\sc c} {\sc d}$,
the {\it plane\/} $[{\sc e}]$ is then {\it infinitely distant},
and this {\it point\/}~${\sc k}$ is the {\it centre of the
surface}.
\bigbreak
[131.]
For example, in the case of the I${}^{\rm st}$ surface [129.],
this {\it pole\/}~${\sc k}$ is the point
$(1 \, \overline{1} \, 1 \, \overline{1})
\equiv (2 \, 0 \, 2 \, 0 \, 1)$,
which belongs to the group~${\sc p}_{3,1}$; and because it is
{\it on\/} the plane~$[{\sc e}]$, that plane {\it touches\/} the
surface in that point: so that when the point~${\sc e}$ is the
{\it mean\/} point of the pyramid
${\sc a} {\sc b} {\sc c} {\sc d}$,
the surface becomes a ruled {\it paraboloid}. In the case of the
II${}^{\rm nd}$ surface [129.], the pole~${\sc k}$ of $[{\sc e}]$
is always the point $(1 \, 1 \, 0 \, 0)$, or ${\sc c}'$; this
point~${\sc c}'$ becomes therefore the {\it centre\/} of the
surface, when ${\sc e}$ is the {\it mean\/} point of the pyramid;
and the five following lines,
$${\sc a} {\sc b},\quad
{\sc a}' {\sc b}_1^{\rm VII},\quad
{\sc b}' {\sc a}^{\rm VII},\quad
{\sc a}_2 {\sc f},
\quad\hbox{and}\quad
{\sc b}_2 {\sc g},$$
where ${\sc g}$ is the new point $(2 \, 1 \, 0 \, \overline{1})$
of the group~${\sc p}_{2,6}$, which are {\it always chords
through\/} ${\sc c}'$, become in that case {\it diameters}. It
may be added that, with the initial arrangement [91.], the
surface last considered becomes the {\it sphere}, which is
described with ${\sc a} {\sc b}$ for diameter; and that it
{\it always\/} passes through the auxiliary point~${\sc p}$, of
{\it third\/} construction, which was mentioned in [100.].
\bigbreak
[132.]
We have then here an {\it example}, of a surface of the second
order, which was {\it determined\/} so as to pass [129.] through
{\it nine net-points}
$${\sc a},\enspace
{\sc b},\enspace
{\sc a}',\enspace
{\sc b}',\enspace
{\sc a}_2,\enspace
{\sc b}_2,\enspace
{\sc a}_1,\enspace
{\sc a}^{\rm VII},
\hbox{ and }
{\sc f},$$
but which has been subsequently {\it found\/} to pass
{\it also\/} through at least {\it four other points of the net},
namely
$${\sc b}_1,\enspace
{\sc b}_1^{\rm VII},\enspace
{\sc g},
\hbox{ and }
{\sc p}.$$
This is, however, only a very particular {\it case\/} of a much
more general {\it Theorem}, with the enunciation of which I shall
conclude the present Paper, regretting sincerely that it has
already extended to a length, so much exceeding the usual limits
of communications designed for the
{\it Proceedings\/}\footnote*{{\it Some\/} of the early
formul{\ae} of this Paper are unavoidably repeated from a
communication of the preceding Session (1859--60), but with
extended significations, as connected now with a {\it quinary
calculus}. And in a not yet published volume, entitled
``{\it Elements of Quaternions},'' the subject of {\it Nets in
Space\/} is incidentally discussed, as an illustration of the
{\it Method of Vectors}. But it will be found that the present
Paper is far from being a mere reprint of the Section on Nets, in
the unpublished work thus referred to: many new {\it theorems\/}
having been introduced, and the {\it plan\/} of the treatment
generally being different, although the {\it notations\/} have,
on the whole, been retained. Besides it was thought that Members
of the Academy might like to see the subject treated, in their
Proceedings, without any express reference to
{\it quaternions\/}: with which indeed the {\it nets\/} have not
any {\it necessary\/} connexion.}
of the Academy, but hoping that some at least of its processes
and results will be thought not wholly uninteresting:---
\medbreak
``{\it If a Surface of the Second Order be determined by the
condition of passing through nine given points of a Geometrical
Net in Space, it passes through indefinitely many others: and
every Point upon the Surface, which is not a point of the Net,
can be included within a Geodetic Triangle on that surface, of
which the corners are net-points, and of which the sides can be
made as small as we may desire.}''
\medbreak
In fact, the {\it surface\/} is a {\it rational\/} one [129.], or
the coefficients of its equation may be made whole numbers; and
therefore {\it every rational line\/} [8.], from any {\it one net
point}, or rational point, upon it, if not happening to
{\it touch\/} the surface, is easily proved to meet it
{\it again}, in {\it another rational point\/}: whence, with the
aid of a lately mentioned principle [127.], the theorem evidently
follows.
\bye