Chapter XXXV: Book XIII (7)

z^2 x^2 z^2 y^2
--- - --- = 1, --- - --- = 1,
c^2 a^2 c^2 b^2

having a common transverse axis along z'Oz; the section by any plane z
= [+-][gamma] parallel to that of xy is the ellipse

x^2 y^2 [gamma]^2
--- + --- = --------- - 1,
a^2 b^2 c^2

provided [gamma]^2 > c^2, and the surface, consisting of two distinct
portions or sheets, may be considered as generated by a variable
ellipse moving parallel to itself along the hyperbolas as directrices.

38. _Differential Geometry of Curves._--For convenience consider the
coordinates (x, y, z) of a point on a curve in space to be given as
functions of a variable parameter [theta], which may in particular be
one of themselves. Use the notation x', x" for dx/d[theta],
d^2x/d[theta]^2, and similarly as to y and z. Only a few formulae will
be given. Call the current coordinates ([xi], [eta], [zeta]).

The _tangent_ at (x, y, z) is the line tended to as a limit by the
connector of (x, y, z) and a neighbouring point of the curve when the
latter moves up to the former: its equations are

([xi] - x)/x' = ([eta] - y)/y' = ([zeta] - z)/z'.

The _osculating plane_ at (x, y, z) is the plane tended to as a limit
by that through (x, y, z) and two neighbouring points of the curve as
these, remaining distinct, both move up to (x, y, z): its one equation
is

([xi] - x)(y'z" - y"z') + ([eta] - y)(z'x" - z"x') + ([zeta] - z)
(x'y" - x"y') = 0.

The _normal plane_ is the plane through (x, y, z) at right angles to
the tangent line, i.e. the plane

x'([xi] - x) + y'([eta] - y) + z'([zeta] - z) = 0.

It cuts the osculating plane in a line called the _principal normal_.
Every line through (x, y, z) in the normal plane is a normal. The
normal perpendicular to the osculating plane is called the _binormal_.
A tangent, principal normal, and binormal are a convenient set of
rectangular axes to use as those of reference, when the nature of a
curve near a point on it is to be discussed.

Through (x, y, z) and three neighbouring points, all on the curve,
passes a single sphere; and as the three points all move up to (x, y,
z) continuing distinct, the sphere tends to a limiting size and
position. The limit tended to is the sphere of closest contact with
the curve at (x, y, z); its centre and radius are called the centre
and radius of _spherical curvature_. It cuts the osculating plane in a
circle, called the _circle of absolute curvature_; and the centre and
radius of this circle are the centre and radius of absolute curvature.
The centre of absolute curvature is the limiting position of the point
where the principal normal at (x, y, z) is cut by the normal plane at
a neighbouring point, as that point moves up to (x, y, z).

39. _Differential Geometry of Surfaces._--Let (x, y, z) be any chosen
point on a surface [f](x, y, z) = 0. As a second point of the surface
moves up to (x, y, z), its connector with (x, y, z) tends to a
limiting position, a tangent line to the surface at (x, y, z). All
these tangent lines at (x, y, z), obtained by approaching (x, y, z)
from different directions on a surface, lie in one plane

dP[f] dP[f] dP[f]
----- ([xi] - x) + ----- ([eta] - y) + ----- ([zeta] - z) = 0.
dPx dPy dPz

This plane is called the _tangent plane_ at (x, y, z). One line
through (x, y, z) is at right angles to the tangent plane. This is the
normal

/dP[f] /dP[f] /dP[f]
([xi] - x) / ----- = ([eta] - y) / ----- = ([zeta] - z) = / -----.
/ dPx / dPy / dPz

The tangent plane is cut by the surface in a curve, real or imaginary,
with a node or double point at (x, y, z). Two of the tangent lines
touch this curve at the node. They are called the "chief tangents"
(_Haupt-tangenten_) at (x, y, z); they have closer contact with the
surface than any other tangents.

In the case of a quadric surface the curve of intersection of a
tangent and the surface is of the second order and has a node, it must
therefore consist of two straight lines. Consequently a quadric
surface is covered by two sets of straight lines, a pair through every
point on it; these are imaginary for the ellipsoid, hyperboloid of two
sheets, and elliptic paraboloid.

A surface of any order is covered by two singly infinite systems of
curves, a pair through every point, the tangents to which are all
chief tangents at their respective points of contact. These are called
_chief-tangent curves_; on a quadric surface they are the above
straight lines.

40. The tangents at a point of a surface which bisect the angles
between the chief tangents are called the _principal tangents_ at the
point. They are at right angles, and together with the normal
constitute a convenient set of rectangular axes to which to refer the
surface when its properties near the point are under discussion. At a
special point which is such that the chief tangents there run to the
circular points at infinity in the tangent plane, the principal
tangents are indeterminate; such a special point is called an umbilic
of the surface.

There are two singly infinite systems of curves on a surface, a pair
cutting one another at right angles through every point upon it, all
tangents to which are principal tangents of the surface at their
respective points of contact. These are called _lines of curvature_,
because of a property next to be mentioned.

As a point Q moves in an arbitrary direction on a surface from
coincidence with a chosen point P, the normal at it, as a rule, at
once fails to meet the normal at P; but, if it takes the direction of
a line of curvature through P, this is instantaneously not the case.
We have thus on the normal two centres of curvature, and the distances
of these from the point on the surface are the two _principal radii of
curvature_ of the surface at that point; these are also the radii of
curvature of the sections of the surface by planes through the normal
and the two principal tangents respectively; or say they are the radii
of curvature of the normal sections through the two principal tangents
respectively. Take at the point the axis of z in the direction of the
normal, and those of x and y in the directions of the principal
tangents respectively, then, if the radii of curvature be a, b (the
signs being such that the coordinates of the two centres of curvature
are z = a and z = b respectively), the surface has in the
neighbourhood of the point the form of the paraboloid

x^2 y^2
z = --- + ---,
2a 2b

and the chief-tangents are determined by the equation 0 = x^2/2a +
y^2/2b. The two centres of curvature may be on the same side of the
point or on opposite sides; in the former case a and b have the same
sign, the paraboloid is elliptic, and the chief-tangents are
imaginary; in the latter case a and b have opposite signs, the
paraboloid is hyperbolic, and the chief-tangents are real.

The normal sections of the surface and the paraboloid by the same
plane have the same radius of curvature; and it thence readily follows
that the radius of curvature of a normal section of the surface by a
plane inclined at an angle [theta] to that of zx is given by the
equation

1 cos^2 [theta] sin^2 [theta]
----- = ------------- + -------------.
[rho] a b

The section in question is that by a plane through the normal and a
line in the tangent plane inclined at an angle [theta] to the
principal tangent along the axis of x. To complete the theory,
consider the section by a plane having the same trace upon the tangent
plane, but inclined to the normal at an angle [phi]; then it is shown
without difficulty (Meunier's theorem) that the radius of curvature of
this inclined section of the surface is = [rho] cos [phi].

AUTHORITIES.--The above article is largely based on that by Arthur
Cayley in the 9th edition of this work. Of early and important recent
publications on analytical geometry, special mention is to be made of
R. Descartes, _Geometrie_ (Leyden, 1637); John Wallis, _Tractatus de
sectionibus conicis nova methodo expositis_ (1655, _Opera
mathematica_, i., Oxford, 1695); de l'Hospital, _Traite analytique des
sections coniques_ (Paris, 1720); Leonhard Euler, _Introductio in
analysin infinitorum_, ii. (Lausanne, 1748); Gaspard Monge,
"Application d'algebre a la geometrie" (_Journ. Ecole Polytech._,
1801); Julius Plucker, _Analytisch-geometrische Entwickelungen_, 3
Bde. (Essen, 1828-1831); _System der analytischen Geometrie_ (Berlin,
1835); G. Salmon, _A Treatise on Conic Sections_ (Dublin, 1848; 6th
ed., London, 1879); Ch. Briot and J. Bouquet, _Lecons de geometrie
analytique_ (Paris, 1851; 16th ed., 1897); M. Chasles, _Traite de
geometrie superieure_ (Paris, 1852); Wilhelm Fiedler, _Analytische
Geometrie der Kegelschnitte_ nach G. Salmon frei bearbeitet (Leipzig,
5te Aufl., 1887-1888); N.M. Ferrers, _An Elementary Treatise on
Trilinear Coordinates_ (London, 1861); Otto Hesse, _Vorlesungen aus
der analytischen Geometrie_ (Leipzig, 1865, 1881); W.A. Whitworth,
_Trilinear Coordinates and other Methods of Modern Analytical
Geometry_ (Cambridge, 1866); J. Booth, _A Treatise on Some New
Geometrical Methods_ (London, i., 1873; ii., 1877); A. Clebsch-F.
Lindemann, _Vorlesungen uber Geometrie_, Bd. i. (Leipzig, 1876, 2te
Aufl., 1891); R. Baltser, _Analytische Geometrie_ (Leipzig, 1882);
Charlotte A. Scott, _Modern Methods of Analytical Geometry_ (London,
1894); G. Salmon, _A Treatise on the Analytical Geometry of three
Dimensions_ (Dublin, 1862; 4th ed., 1882); Salmon-Fiedler,
_Analytische Geometrie des Raumes_ (Leipzig, 1863; 4te Aufl., 1898);
P. Frost, _Solid Geometry_ (London, 3rd ed., 1886; 1st ed., Frost and
J. Wolstenholme). See also E. Pascal, _Repertorio di matematiche
superiori, II. Geometria_ (Milan, 1900), and articles now appearing in
the _Encyklopadie der mathematischen Wissenschaften_, Bd. iii. 1, 2.
(E. B. El.)

V. LINE GEOMETRY

Line geometry is the name applied to those geometrical investigations in which the straight line replaces the point as element. Just as ordinary geometry deals primarily with points and systems of points, this theory deals in the first instance with straight lines and systems of straight lines. In two dimensions there is no necessity for a special line geometry, inasmuch as the straight line and the point are interchangeable by the principle of duality; but in three dimensions the straight line is its own reciprocal, and for the better discussion of systems of lines we require some new apparatus, e.g., a system of coordinates applicable to straight lines rather than to points. The essential features of the subject are most easily elucidated by analytical methods: we shall therefore begin with the notion of line coordinates, and in order to emphasize the merits of the system of coordinates ultimately adopted, we first notice a system without these advantages, but often useful in special investigations.

In ordinary Cartesian coordinates the two equations of a straight line
may be reduced to the form y = rx + s, z = tx + u, and r, s, t, u may
be regarded as the four coordinates of the line. These coordinates
lack symmetry: moreover, in changing from one base of reference to
another the transformation is not linear, so that the degree of an
equation is deprived of real significance. For purposes of the general
theory we employ homogeneous coordinates; if x1y1z1w1 and x2y2z2w2 are
two points on the line, it is easily verified that the six
determinants of the array

|x1y1z1w1|
|x2y2z2w2|

are in the same ratios for all point-pairs on the line, and further,
that when the point coordinates undergo a linear transformation so
also do these six determinants. We therefore adopt these six
determinants for the coordinates of the line, and express them by the
symbols l, [lambda], m, [mu], n, [nu] where l = x1w2 - x2w1, [lambda]
= y1z2 - y2z1, &c. There is the further advantage that if a1b1c1d1 and
a2b2c2d2 be two planes through the line, the six determinants

|a1b1c1d1|
|a2b2c2d2|

are in the same ratios as the foregoing, so that except as regards a
factor of proportionality we have [lambda] = b1c2 - b2c1, l = c1d2 -
c2d1, &c. The identical relation l[lambda] + m[mu] + n[nu] = o reduces
the number of independent constants in the six coordinates to four,
for we are only concerned with their mutual ratios; and the quadratic
character of this relation marks an essential difference between point
geometry and line geometry. The condition of intersection of two lines
is

l[lambda]' + l'[lambda] + m[mu]' + m'[mu] + n[nu]' + n'[nu] = 0

where the accented letters refer to the second line. If the
coordinates are Cartesian and l, m, n are direction cosines, the
quantity on the left is the mutual moment of the two lines.

Since a line depends on four constants, there are three distinct types
of configurations arising in line geometry--those containing a
triply-infinite, a doubly-infinite and a singly-infinite number of
lines; they are called Complexes, Congruences, and Ruled Surfaces or
Skews respectively. A _Complex_ is thus a system of lines satisfying
one condition--that is, the coordinates are connected by a single
relation; and the degree of the complex is the degree of this equation
supposing it to be algebraic. The lines of a complex of the nth degree
which pass through any point lie on a cone of the nth degree, those
which lie in any plane envelop a curve of the nth class and there are
n lines of the complex in any plane pencil; the last statement
combines the former two, for it shows that the cone is of the nth
degree and the curve is of the nth class. To find the lines common to
four complexes of degrees n1, n2, n3, n4, we have to solve five
equations, viz. the four complex equations together with the quadratic
equation connecting the line coordinates, therefore the number of
common lines is 2n1n2n3n4. As an example of complexes we have the
lines meeting a twisted curve of the nth degree, which form a complex
of the nth degree.

A _Congruence_ is the set of lines satisfying two conditions: thus a
finite number m of the lines pass through any point, and a finite
number n lie in any plane; these numbers are called the degree and
class respectively, and the congruence is symbolically written (m, n).

The simplest example of a congruence is the system of lines
constituted by all those that pass through m points and those that lie
in n planes; through any other point there pass m of these lines, and
in any other plane there lie n, therefore the congruence is of degree
m and class n. It has been shown by G.H. Halphen that the number of
lines common to two congruences is mm' + nn', which may be verified by
taking one of them to be of this simple type. The lines meeting two
fixed lines form the general (1, 1) congruence; and the chords of a
twisted cubic form the general type of a (1, 3) congruence; Halphen's
result shows that two twisted cubics have in general ten common
chords. As regards the analytical treatment, the difficulty is of the
same nature as that arising in the theory of curves in space, for a
congruence is not in general the complete intersection of two
complexes.

A _Ruled Surface_, _Regulus_ or _Skew_ is a configuration of lines
which satisfy three conditions, and therefore depend on only one
parameter. Such lines all lie on a surface, for we cannot draw one
through an arbitrary point; only one line passes through a point of
the surface; the simplest example, that of a quadric surface, is
really two skews on the same surface.

The degree of a ruled surface _qua_ line geometry is the number of its
generating lines contained in a linear complex. Now the number which
meets a given line is the degree of the surface _qua_ point geometry,
and as the lines meeting a given line form a particular case of linear
complex, it follows that the degree is the same from whichever point
of view we regard it. The lines common to three complexes of degrees,
n1n2n3, form a ruled surface of degree 2n1n2n3; but not every ruled
surface is the complete intersection of three complexes.

Linear complex.

In the case of a complex of the first degree (or linear complex) the
lines through a fixed point lie in a plane called the polar plane or
nul-plane of that point, and those lying in a fixed plane pass through
a point called the nul-point or pole of the plane. If the nul-plane of
A pass through B, then the nul-plane of B will pass through A; the
nul-planes of all points on one line l1 pass through another line l2.
The relation between l1 and l2 is reciprocal; any line of the complex
that meets one will also meet the other, and every line meeting both
belongs to the complex. They are called conjugate or polar lines with
respect to the complex. On these principles can be founded a theory of
reciprocation with respect to a linear complex.

This may be aptly illustrated by an elegant example due to A. Voss.
Since a twisted cubic can be made to satisfy twelve conditions, it
might be supposed that a finite number could be drawn to touch four
given lines, but this is not the case. For, suppose one such can be
drawn, then its reciprocal with respect to any linear complex
containing the four lines is a curve of the third class, i.e. another
twisted cubic, touching the same four lines, which are unaltered in
the process of reciprocation; as there is an infinite number of
complexes containing the four lines, there is an infinite number of
cubics touching the four lines, and the problem is poristic.

The following are some geometrical constructions relating to the
unique linear complex that can be drawn to contain five arbitrary
lines:

To construct the nul-plane of any point O, we observe that the two
lines which meet any four of the given five are conjugate lines of the
complex, and the line drawn through O to meet them is therefore a ray
of the complex; similarly, by choosing another four we can find
another ray through O: these rays lie in the nul-plane, and there is
clearly a result involved that the five lines so obtained all lie in
one plane. A reciprocal construction will enable us to find the
nul-point of any plane. Proceeding now to the metrical properties and
the statical and dynamical applications, we remark that there is just
one line such that the nul-plane of any point on it is perpendicular
to it. This is called the central axis; if d be the shortest distance,
[theta] the angle between it and a ray of the complex, then d tan
[theta] = p, where p is a constant called the pitch or parameter. Any
system of forces can be reduced to a force R along a certain line, and
a couple G perpendicular to that line; the lines of nul-moment for the
system form a linear complex of which the given line is the central
axis and the quotient G/R is the pitch. Any motion of a rigid body can
be reduced to a screw motion about a certain line, i.e. to an angular
velocity [omega] about that line combined with a linear velocity u
along the line. The plane drawn through any point perpendicular to the
direction of its motion is its nul-plane with respect to a linear
complex having this line for central axis, and the quotient u/[omega]
for pitch (cf. Sir R.S. Ball, _Theory of Screws_).

The following are some properties of a configuration of two linear
complexes:

The lines common to the two-complexes also belong to an infinite
number of linear complexes, of which two reduce to single straight
lines. These two lines are conjugate lines with respect to each of the
complexes, but they may coincide, and then some simple modifications
are required. The locus of the central axis of this system of
complexes is a surface of the third degree called the cylindroid,
which plays a leading part in the theory of screws as developed
synthetically by Ball. Since a linear complex has an invariant of the
second degree in its coefficients, it follows that two linear
complexes have a lineo-linear invariant. This invariant is
fundamental: if the complexes be both straight lines, its vanishing is
the condition of their intersection as given above; if only one of
them be a straight line, its vanishing is the condition that this line
should belong to the other complex. When it vanishes for any two
complexes they are said to be in _involution_ or _apolar_; the
nul-points P, Q of any plane then divide harmonically the points in
which the plane meets the common conjugate lines, and each complex is
its own reciprocal with respect to the other. As regards a
configuration of these linear complexes, the common lines from one
system of generators of a quadric, and the doubly infinite system of
complexes containing the common lines, include an infinite number of
straight lines which form the other system of generators of the same
quadric.

General line coordinates.

If the equation of a linear complex is Al + Bm + Cn + D[lambda] +
E[mu] + F[nu] = 0, then for a line not belonging to the complex we may
regard the expression on the left-hand side as a multiple of the
moment of the line with respect to the complex, the word moment being
used in the statical sense; and we infer that when the coordinates are
replaced by linear functions of themselves the new coordinates are
multiples of the moments of the line with respect to six fixed
complexes. The essential features of this coordinate system are the
same as those of the original one, viz. there are six coordinates
connected by a quadratic equation, but this relation has in general a
different form. By suitable choice of the six fundamental complexes,
as they may be called, this connecting relation may be brought into
other simple forms of which we mention two: (i.) When the six are
mutually in involution it can be reduced to x1^2 + x2^2 + x3^2 + x4^2
+ x5^2 + x6^2 = 0; (ii.) When the first four are in involution and the
other two are the lines common to the first four it is x1^2 + x2^2 +
x3^2 + x4^2 - 2x5x6 = 0. These generalized coordinates might be
explained without reference to actual magnitude, just as homogeneous
point coordinates can be; the essential remark is that the equation of
any coordinate to zero represents a linear complex, a point of view
which includes our original system, for the equation of a coordinate
to zero represents all the lines meeting an edge of the fundamental
tetrahedron.

The system of coordinates referred to six complexes mutually in
involution was introduced by Felix Klein, and in many cases is more
useful than that derived directly from point coordinates; e.g. in the
discussion of quadratic complexes: by means of it Klein has developed
an analogy between line geometry and the geometry of spheres as
treated by G. Darboux and others. In fact, in that geometry a point is
represented by _five_ coordinates, connected by a relation of the same
type as the one just mentioned when the five fundamental spheres are
mutually at right angles and the equation of a sphere is of the first
degree. Extending this to four dimensions of space, we obtain an exact
analogue of line geometry, in which (i.) a point corresponds to a
line; (ii.) a linear complex to a hypersphere; (iii.) two linear
complexes in involution to two orthogonal hyperspheres; (iv.) a linear
complex and two conjugate lines to a hypersphere and two inverse
points. Many results may be obtained by this principle, and more still
are suggested by trying to extend the properties of circles to spheres
in three and four dimensions. Thus the elementary theorem, that, given
four lines, the circles circumscribed to the four triangles formed by
them are concurrent, may be extended to six hyperplanes in four
dimensions; and then we can derive a result in line geometry by
translating the inverse of this theorem. Again, just as there is an
infinite number of spheres touching a surface at a given point, two of
them having contact of a closer nature, so there is an infinite number
of linear complexes touching a non-linear complex at a given line, and
_three_ of these have contact of a closer nature (cf. Klein, _Math.
Ann._ v.).

Sophus Lie has pointed out a different analogy with sphere geometry.
Suppose, in fact, that the equation of a sphere of radius r is

x^2 + y^2 + z^2 + 2ax + 2by + 2cz + d = 0,

so that r^2 = a^2 + b^2 + c^2 - d; then introducing the quantity e to
make this equation homogeneous, we may regard the sphere as given by
the six coordinates a, b, c, d, e, r connected by the equation a^2 +
b^2 + c^2 - r^2 - de = 0, and it is easy to see that two spheres
touch, if the polar form 2aa1 + 2bb1 + 2cc1 - 2rr1 - de1 - d1e
vanishes. Comparing this with the equation x1^2 + x2^2 + x3^2 + x4^2 -
2x5x6 = 0 given above, it appears that this sphere geometry and line
geometry are identical, for we may write a = x1, b = x2, c = x3, r =
x4(/[delta] - 1), d = x5, e = 1/2x6; but it is to be noticed that a
sphere is really replaced by two lines whose coordinates only differ
in the sign of x4, so that they are polar lines with respect to the
complex x4 = 0. Two spheres which touch correspond to two lines which
intersect, or more accurately to two pairs of lines (p, p') and (q,
q'), of which the pairs (p, q) and (p', q') both intersect. By this
means the problem of describing a sphere to touch four given spheres
is reduced to that of drawing a pair of lines (t, t') (of which t
intersects one line of the four pairs (pp'), (qq'), (rr'), (ss'), and
t' intersects the remaining four). We may, however, ignore the
accented letters in translating theorems, for a configuration of lines
and its polar with respect to a linear complex have the same
projective properties. In Lie's transformation a linear complex
corresponds to the totality of spheres cutting a given sphere at a
given angle. A most remarkable result is that lines of curvature in
the sphere geometry become asymptotic lines in the line geometry.

Some of the principles of line geometry may be brought into clearer
light by admitting the ideas of space of four and five dimensions.

Thus, regarding the coordinates of a line as homogeneous coordinates
in five dimensions, we may say that line geometry is equivalent to
geometry on a quadric surface in five dimensions. A linear complex is
represented by a hyperplane section; and if two such complexes are in
involution, the corresponding hyperplanes are conjugate with respect
to the fundamental quadric. By projecting this quadric
stereographically into space of four dimensions we obtain Klein's
analogy. In the same way geometry in a linear complex is equivalent to
geometry on a quadric in four dimensions; when two lines intersect the
representative points are on the same generator of this quadric.
Stereographic projection, therefore, converts a curve in a linear
complex, i.e. one whose tangents all belong to the complex, into one
whose tangents intersect a fixed conic: when this conic is the
imaginary circle at infinity the curve is what Lie calls a minimal
curve. Curves in a linear complex have been extensively studied. The
osculating plane at any point of such a curve is the nul-plane of the
point with respect to the complex, and points of superosculation
always coincide in pairs at the points of contact of stationary
tangents. When a point of such a curve is given, the osculating plane
is determined, hence all the curves through a given point with the
same tangent have the same torsion.

Non-linear complexes.

The lines through a given point that belong to a complex of the nth
degree lie on a cone of the nth degree: if this cone has a double line
the point is said to be a singular point. Similarly, a plane is said
to be singular when the envelope of the lines in it has a double
tangent. It is very remarkable that the same surface is the locus of
the singular points and the envelope of the singular planes: this
surface is called the singular surface, and both its degree and class
are in general 2n(n - 1)^2, which is equal to four for the quadratic
complex.

The singular lines of a complex F = 0 are the lines common to F and
the complex

[delta]F [delta]F [delta]F [delta]F [delta]F [delta]F
-------- --------------- + -------- ----------- + -------- ----------- = 0.
[delta]l [delta][lambda] [delta]m [delta][mu] [delta]n [delta][nu]

As already mentioned, at each line l of a complex there is an infinite
number of tangent linear complexes, and they all contain the lines
adjacent to l. If now l be a singular line, these complexes all reduce
to straight lines which form a plane pencil containing the line l.
Suppose the vertex of the pencil is A, its plane a, and one of its
lines [xi], then l' being a complex line near l, meets [xi], or more
accurately the mutual moment of l', and is of the second order of
small quantities. If P be a point on l, a line through P quite near l
in the plane a will meet [xi] and is therefore a line of the complex;
hence the complex-cones of all points on l touch a and the
complex-curves of all planes through l touch l at A. It follows that l
is a double line of the complex-cone of A, and a double tangent of the
complex-curve of a. Conversely, a double line of a cone or curve is a
singular line, and a singular line clearly touches the curves of all
planes through it in the same point. Suppose now that the consecutive
line l' is also a singular line, A' being the allied singular point,
a' the singular plane and [xi]' any line of the pencil (A', a') so
that [xi]' is a tangent line at l' to the complex: the mutual moments
of the pairs l', [xi] and l, [xi] are each of the second order; hence
the plane a' meets the lines l and [xi]' in two points very near A.
This being true for all singular planes, near a the point of contact
of a with its envelope is in A, i.e. the locus of singular points is
the same as the envelope of singular planes. Further, when a line
touches a complex it touches the singular surface, for it belongs to a
plane pencil like (Aa), and thus in Klein's analogy the analogue of a
focus of a hyper-surface being a bitangent line of the complex is also
a bitangent line of the singular surface. The theory of cosingular
complexes is thus brought into line with that of confocal surfaces in
four dimensions, and guided by these principles the existence of
cosingular quadratic complexes can easily be established, the analysis
required being almost the same as that invented for confocal cyclides
by Darboux and others. Of cosingular complexes of higher degree
nothing is known.

Following J. Plucker, we give an account of the lines of a quadratic
complex that meet a given line.

The cones whose vertices are on the given line all pass through eight
fixed points and envelop a surface of the fourth degree; the conics
whose planes contain the given line all lie on a surface of the fourth
class and touch eight fixed planes. It is easy to see by elementary
geometry that these two surfaces are identical. Further, the given
line contains four singular points A1, A2, A3, A4, and the planes into
which their cones degenerate are the eight common tangent planes
mentioned above; similarly, there are four singular planes, a1, a2,
a3, a4, through the line, and the eight points into which their conics
degenerate are the eight common points above. The locus of the pole of
the line with respect to all the conics in planes through it is a
straight line called the _polar line_ of the given one; and through
this line passes the polar plane of the given line with respect to
each of the cones. The name polar is applied in the ordinary
analytical sense; any line has an infinite number of polar complexes
with respect to the given complex, for the equation of the latter can
be written in an infinite number of ways; one of these polars is a
straight line, and is the polar line already introduced. The surface
on which lie all the conics through a line l is called the Plucker
surface of that line: from the known properties of (2, 2)
correspondences it can be shown that the Plucker surface of l cuts l1
in a range of the same cross ratio as that of the range in which the
Plucker surface of l1 cuts l. Applying this to the case in which l1 is
the polar of l, we find that the cross ratios of (A1, A2, A3, A4) and
(a1, a2, a3, a4) are equal. The identity of the locus of the A's with
the envelope of the a's follows at once; moreover, a line meets the
singular surface in four points having the same cross ratio as that of
the four tangent planes drawn through the line to touch the surface.
The Plucker surface has eight nodes, eight singular tangent planes,
and is a double line. The relation between a line and its polar line
is not a reciprocal one with respect to the complex; but W. Stahl has
pointed out that the relation is reciprocal as far as the singular
surface is concerned.

Quadratic complexes.

To facilitate the discussion of the general quadratic complex we
introduce Klein's canonical form. We have, in fact, to deal with two
quadratic equations in six variables; and by suitable linear
transformations these can be reduced to the form

a1x1^2 + a2x2^2 + a3x3^2 + a4x4^2 + a5x5^2 + a6x6^2 = 0
x1^2 + x2^2 + x3^2 + x4^2 + x5^2 + x6^2 = 0

subject to certain exceptions, which will be mentioned later.

Taking the first equation to be that of the complex, we remark that
both equations are unaltered by changing the sign of any coordinate;
the geometrical meaning of this is, that the quadratic complex is its
own reciprocal with respect to each of the six fundamental complexes,
for changing the sign of a coordinate is equivalent to taking the
polar of a line with respect to the corresponding fundamental complex.
It is easy to establish the existence of six systems of bitangent
linear complexes, for the complex l1x1 + l2x2 + l3x3 + l4x4 + l5x5 +
l6x6 = 0 is a bitangent when

l2^2 l3^2 l4^2 l5^2 l6^2
l1 = 0, and ------- + ------- + ------- + ------- + ------- = 0
a2 - a1 a3 - a1 a4 - a1 a5 - a1 a6 - a1

and its lines of contact are conjugate lines with respect to the first
fundamental complex. We therefore infer the existence of six systems
of bitangent lines of the complex, of which the first is given by

x2^2 x3^2 x4^2 x5^2 x6^2
x1 = 0, ------- + ------- + ------- + ------- + ------- = 0.
a2 - a1 a3 - a1 a4 - a1 a5 - a1 a6 - a1

Each of these lines is a bitangent of the singular surface, which is
therefore completely determined as being the focal surface of the (2,
2) congruence above. It is thence easy to verify that the two
complexes [Sigma]ax^2 = 0 and [Sigma]bx^2 = 0 are cosingular if b_r =
a_r[lambda] + [mu]/a_r[nu] + [rho].

The singular surface of the general quadratic complex is the famous
quartic, with sixteen nodes and sixteen singular tangent planes, first
discovered by E.E. Kummer.

We cannot give a full account of its properties here, but we deduce at
once from the above that its bitangents break up into six (2, 2)
congruences, and the six linear complexes containing these are
mutually in involution. The nodes of the singular surface are points
whose complex cones are coincident planes, and the complex conic in a
singular tangent plane consists of two coincident points. This
configuration of sixteen points and planes has many interesting
properties; thus each plane contains six points which lie on a conic,
while through each point there pass six planes which touch a quadric
cone. In many respects the Kummer quartic plays a part in three
dimensions analogous to the general quartic curve in two; it further
gives a natural representation of certain relations between
hyperelliptic functions (cf. R.W.H.T. Hudson, _Kummer's Quartic_,
1905).

Classification of quadratic complexes.

As might be expected from the magnitude of a form in six variables,
the number of projectivally distinct varieties of quadratic complexes
is very great; and in fact Adolf Weiler, by whom the question was
first systematically studied on lines indicated by Klein, enumerated
no fewer than forty-nine different types. But the principle of the
classification is so important, and withal so simple, that we give a
brief sketch which indicates its essential features.

We have practically to study the intersection of two quadrics F and F'
in six variables, and to classify the different cases arising we make
use of the results of Karl Weierstrass on the equivalence conditions
of two pairs of quadratics. As far as at present required, they are as
follows: Suppose that the factorized form of the determinantal
equation Disct (F + [lambda]F') = 0 is

([lambda] - [alpha])^(s1 + s2 + s3 ...)
([lambda] - [beta])^(t1 + t2 + t3 + ...) ...

where the root [alpha] occurs s1 + s2 + s3 ... times in the
determinant, s2 + s3 ... times in every first minor, s3 + ... times in
every second minor, and so on; the meaning of each exponent is then
perfectly definite. Every factor of the type ([lambda] - [alpha])^s is
called an _elementartheil_ (elementary divisor) of the determinant,
and the condition of equivalence of two pairs of quadratics is simply
that their determinants have the same elementary divisors. We write
the pair of forms symbolically thus [(s1s2 ...), (t1t2 ...), ...],
letters in the inner brackets referring to the same factor. Returning
now to the two quadratics representing the complex, the sum of the
exponents will be six, and two complexes are put in the same class if
they have the same symbolical expression; i.e. the actual values of
the roots of the determinantal equation need not be the same for both,
but their manner of occurrence, as far as here indicated, must be
identical in the two. The enumeration of all possible cases is thus
reduced to a simple question in combinatorial analysis, and the actual
study of any particular case is much facilitated by a useful rule of
Klein's for writing down in a simple form two quadratics belonging to
a given class--one of which, of course, represents the equation
connecting line coordinates, and the other the equation of the
complex. The general complex is naturally [111111]; the complex of
tangents to a quadric is [(111), (111)] and that of lines meeting a
conic is [(222)]. Full information will be found in Weiler's memoir,
_Math. Ann._ vol. vii.

The detailed study of each variety of complex opens up a vast subject;
we only mention two special cases, the harmonic complex and the
tetrahedral complex.

The harmonic complex, first studied by Battaglini, is generated in an
infinite number of ways by the lines cutting two quadrics
harmonically. Taking the most general case, and referring the quadrics
to their common self-conjugate tetrahedron, we can find its equation
in a simple form, and verify that this complex really depends only on
seventeen constants, so that it is not the most general quadratic
complex. It belongs to the general type in so far as it is discussed
above, but the roots of the determinant are in involution. The
singular surface is the "tetrahedroid" discussed by Cayley. As a
particular case, from a metrical point of view, we have L.F. Painvin's
complex generated by the lines of intersection of perpendicular
tangent planes of a quadric, the singular surface now being Fresnel's
wave surface. The tetrahedral or Reye complex is the simplest and best
known of proper quadratic complexes. It is generated by the lines
which cut the faces of a tetrahedron in a constant cross ratio, and
therefore by those subtending the same cross ratio at the four
vertices. The singular surface is made up of the faces or the vertices
of the fundamental tetrahedron, and each edge of this tetrahedron is a
double line of the complex. The complex was first discussed by K.T.
Reye as the assemblage of lines joining corresponding points in a
homographic transformation of space, and this point of view leads to
many important and elegant properties. A (metrically) particular case
of great interest is the complex generated by the normals to a family
of confocal quadrics, and for many investigations it is convenient to
deal with this complex referred to the principal axes. For example,
Lie has developed the theory of curves in a Reye complex (i.e. curves
whose tangents belong to the complex) as solutions of a differential
equation of the form (b - c)xdydz + (c - a)ydzdx + (a - b)zdxdy = 0,
and we can simplify this equation by a logarithmic transformation.
Many theorems connecting complexes with differential equations have
been given by Lie and his school. A line complex, in fact, corresponds
to a Mongian equation having [oo]^3 line integrals.

Congruences.

As the coordinates of a line belonging to a congruence are functions
of two independent parameters, the theory of congruences is analogous
to that of surfaces, and we may regard it as a fundamental inquiry to
find the simplest form of surface into which a given congruence can be
transformed. Most of those whose properties have been extensively
discussed can be represented on a plane by a birational
transformation. But in addition to the difficulties of the theory of
algebraic surfaces, a subject still in its infancy, the theory of
congruences has other difficulties in that a congruence is seldom
completely represented, even by two equations.

A fundamental theorem is that the lines of a congruence are in general
bitangents of a surface; in fact, since the condition of intersection
of two consecutive straight lines is ld[lambda] + dmd[mu] + dnd[nu] =
0, a line l of the congruence meets two adjacent lines, say l1 and l2.
Suppose l, l1 lie in the plane pencil (A1a1) and l, l2 in the plane
pencil (A2a2), then the locus of the A's is the same as the envelope
of the a's, but a2 is the tangent plane at A1 and a1 at A2. This
surface is called the focal surface of the congruence, and to it all
the lines l are bitangent. The distinctive property of the points A is
that two of the congruence lines through them coincide, and in like
manner the planes a each contain two coincident lines. The focal
surface consists of two sheets, but one or both may degenerate into
curves; thus, for example, the normals to a surface are bitangents of
the surface of centres, and in the case of Dupin's cyclide this
surface degenerates into two conics.

In the discussion of congruences it soon becomes necessary to
introduce another number r, called the rank, which expresses the
number of plane pencils each of which contains an arbitrary line and
two lines of the congruence. The order of the focal surface is 2m(n -
1) - 2r, and its class is m(m - 1) - 2r. Our knowledge of congruences
is almost exclusively confined to those in which either m or n does
not exceed two. We give a brief account of those of the second order
without singular lines, those of order unity not being especially
interesting. A congruence generally has singular points through which
an infinite number of lines pass; a singular point is said to be of
order r when the lines through it lie on a cone of the rth degree. By
means of formulae connecting the number of singular points and their
orders with the class m of quadratic congruence Kummer proved that the
class cannot exceed seven. The focal surface is of degree four and
class 2m; this kind of quartic surface has been extensively studied by
Kummer, Cayley, Rohn and others. The varieties (2, 2), (2, 3), (2, 4),
(2, 5) all belong to at least one Reye complex; and so also does the
most important class of (2, 6) congruences which includes all the
above as special cases. The congruence (2, 2) belongs to a linear
complex and forty different Reye complexes; as above remarked, the
singular surface is Kummer's sixteen-nodal quartic, and the same
surface is focal for six different congruences of this variety. The
theory of (2, 2) congruences is completely analogous to that of the
surfaces called cyclides in three dimensions. Further particulars
regarding quadratic congruences will be found in Kummer's memoir of
1866, and the second volume of Sturm's treatise. The properties of
quadratic congruences having singular lines, i.e. degenerate focal
surfaces, are not so interesting as those of the above class; they
have been discussed by Kummer, Sturm and others.

Ruled surfaces.

Since a ruled surface contains only [infinity]^1 elements, this theory
is practically the same as that of curves. If a linear complex
contains more than n generators of a ruled surface of the nth degree,
it contains all the generators, hence for n = 2 there are three
linearly independent complexes, containing all the generators, and
this is a well-known property of quadric surfaces. In ruled cubics the
generators all meet two lines which may or may not coincide; these two
cases correspond to the two main classes of cubics discussed by Cayley
and Cremona. As regards ruled quartics, the generators must lie in one
and may lie in two linear complexes. The first class is equivalent to
a quartic in four dimensions and is always rational, but the latter
class has to be subdivided into the elliptic and the rational, just
like twisted quartic curves. A quintic skew may not lie in a linear
complex, and then it is unicursal, while of sextics we have two
classes not in a linear complex, viz. the elliptic variety, having
thirty-six places where a linear complex contains six consecutive
generators, and the rational, having six such places.

The general theory of skews in two linear complexes is identical with
that of curves on a quadric in three dimensions and is known. But for
skews lying in only one linear complex there are difficulties; the
curve now lies in four dimensions, and we represent it in three by
stereographic projection as a curve meeting a given plane in n points
on a conic. To find the maximum deficiency for a given degree would
probably be difficult, but as far as degree eight the space-curve
theory of Halphen and Nother can be translated into line geometry at
once. When the skew does not lie in a linear complex at all the theory
is more difficult still, and the general theory clearly cannot advance
until further progress is made in the study of twisted curves.

REFERENCES.--The earliest works of a general nature are Plucker, _Neue
Geometrie des Raumes_ (Leipzig, 1868); and Kummer, "Uber die
algebraischen Strahlensysteme," _Berlin Academy_ (1866). Systematic
development on purely synthetic lines will be found in the three
volumes of Sturm, _Liniengeometrie_ (Leipzig, 1892, 1893, 1896); vol.
i. deals with the linear and Reye complexes, vols. ii. and iii. with
quadratic congruences and complexes respectively. For a highly
suggestive review by Gino Loria see _Bulletin des sciences
mathematiques_ (1893, 1897). A shorter treatise, giving a very
interesting account of Klein's coordinates, is the work of Koenigs,
_La Geometrie reglee et ses applications_ (Paris, 1898). English
treatises are C.M. Jessop, _Treatise on the Line Complex_ (1903);
R.W.H.T. Hudson, _Kummer's Quartic_ (1905). Many references to memoirs
on line geometry will be found in Hagen, _Synopsis der hoheren
Mathematik_, ii. (Berlin, 1894); Loria, _Il passato ed il presente
delle principali teorie geometriche_ (Milan, 1897); a clear resume of
the principal results is contained in the very elegant volume of
Pascal, _Repertorio di mathematiche superiori_, ii. (Milan, 1900).
Another treatise dealing extensively with line geometry is Lie,
_Geometrie der Beruhrungstransformationen_ (Leipzig, 1896). Many
memoirs on the subject have appeared in the _Mathematische Annalen_; a
full list of these will be found in the index to the first fifty
volumes, p. 115. Perhaps the two memoirs which have left most
impression on the subsequent development of the subject are Klein,
"Zur Theorie der Liniencomplexe des ersten und zweiten Grades," _Math.
Ann._ ii.; and Lie, "Uber Complexe, insbesondere Linien- und
Kugelcomplexe," _Math. Ann._ v. (J. H. Gr.)

VI. NON-EUCLIDEAN GEOMETRY

The various metrical geometries are concerned with the properties of the various types of congruence-groups, which are defined in the study of the _axioms_ of _geometry_ and of their immediate consequences. But this point of view of the subject is the outcome of recent research, and historically the subject has a different origin. Non-Euclidean geometry arose from the discussion, extending from the Greek period to the present day, of the various assumptions which are implicit in the traditional Euclidean system of geometry. In the course of these investigations it became evident that metrical geometries, each internally consistent but inconsistent in many respects with each other and with the Euclidean system, could be developed. A short historical sketch will explain this origin of the subject, and describe the famous and interesting progress of thought on the subject. But previously a description of the chief characteristic properties of elliptic and of hyperbolic geometries will be given, assuming the standpoint arrived at below under VII. _Axioms of Geometry_.

First assume the equation to the absolute (cf. _loc. cit._) to be w^2 - x^2 - y^2 - z^2 = 0. The absolute is then real, and the geometry is hyberbolic.

The distance (d12) between the two points (x1, y1, z1, w1) and (x2,
y2, z2, w2) is given by

cosh (d12/[gamma]) = (w1w2 - x1x2 - y1y2 - z1z2)/[(w1^2 - x1^2 - y1^2 - Z1^2)
(w2^2 - x2^2 - y2^2 - z2^2)]1/2 (1)

The only points to which the metrical geometry applies are those
within the region enclosed by the quadric; the other points are
"improper ideal points." The angle ([theta]12) between two planes, l1x
+ m1y + n1z + r1w = 0 and l2x + m2y + n2z + r2w = 0, is given by

cos [theta]12 = (l1l2 + m1m2 + n1n2 - r1r2)/{(l1^2 + m1^2 + n1^2 - r1^2)
(l2^2 + m2^2 + n2^2 - r2^2)}^1/2 (2)

These planes only have a real angle of inclination if they possess a
line of intersection within the actual space, i.e. if they intersect.
Planes which do not intersect possess a shortest distance along a line
which is perpendicular to both of them. If this shortest distance is
[delta]12, we have

cosh ([delta]12/[gamma]) = (l1l2 + m1m2 + n1n2 - r1r2)/(l1^2 + m1^2 + n1^2 - r1^2)
(l2^2 + m2^2 + n2^2 - r2^2)^1/2 (3)

Thus in the case of the two planes one and only one of the two,
[theta]12 and [delta]12, is real. The same considerations hold for
coplanar straight lines (see VII. _Axioms of Geometry_). Let O (fig.
67) be the point (0, 0, 0, 1), OX the line y = 0, z = 0, OY the line z
= 0, x = 0, and OZ the line x = 0, y = 0. These are the coordinate
axes and are at right angles to each other. Let P be any point, and
let [rho] be the distance OP, [theta] the angle POZ, and [phi] the
angle between the planes ZOX and ZOP. Then the coordinates of P can be
taken to be

sinh ([rho]/[gamma]) sin [theta] cos [phi], sinh ([rho]/[gamma]) sin [theta]
sin [phi], sinh ([rho]/[gamma]) cos[theta], cosh ([rho]/[gamma]).

If ABC is a triangle, and the sides and angles are named according to
the usual convention, we have

sinh (a/[gamma])/sin A = sinh (b/[gamma])/sin B = sinh (c/[gamma])/sin C, (4)

and also

cosh (a/[gamma]) = cosh (b/[gamma]) cosh (c/[gamma]) -
sinh (b/[gamma]) sinh (c/[gamma]) cos A, (5)