Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 2, 317-328.

Equiform Bundle Motions in E_{3} with Spherical Trajectories II

Anton Gfrerrer Johann Lang

Technische Universität Graz, Institut für Geometrie Kopernikusgasse 24, 8010 Graz, Austria

Abstract. The 3-parameter group of Euclidean bundle motions moves every point of the 3-space on a ball centered in the bundle vertex. In this paper we investigate motions of the 4-parameter group of equiform bundle transformations with the property, that some of the points have spherical orbits i. e. orbits lying in a plane or on a ball. In the first part of this paper we introduced a “principle of transfer” which turned out to be a usefull tool for a systematical approach to these problems. The second part contains a detailed investigation of some motions presented in the first part. We also give parameter representations of these examples.

5. One-parametric spherical motions

According to section 4 we have to investigate those motions which belong to second order curves l := L ∩ B_{6} on the surface B_{6}. These motions will keep every point of a certain bundle plane ε on its orbit sphere. The centres of all the orbit spheres lie in a bundle plane ε′ (Theorem 4). We can assume that ε is the coordinate plane [z = 0] in Σ and that ε′ is [z′ = 0] in Σ′ as well. In L we use a projective coordinate system with U_{1}, U_{2}, U_{3} as fundamental points (see section 4). Then l satisfies the equation

l … ∑_{i,j=1, i<j}^{3} (x_{i} y_{j} − x_{j} y_{i}) · (B_{i} C_{j} − B_{j} C_{i}) · α_{i} α_{j} = 0.  (32)

The second order curve l is regular, iff none of the coefficients of (32) vanishes (regular case). Otherwise l consists of a pair of distinct straight lines (singular case).

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1998 Heldermann Verlag

A. Gfrerrer, J. Lang: Equiform Bundle Motions in E_{3} … II 318

We restrict the collinear mappings (see Remark 4)

\{ \begin{matrix}P (W) & ⟶ & O(OX) \\ U & ⟶ & OX \end{matrix} \} , \{ \begin{matrix}P (W) & ⟶ & O(OM′) \\ U & ⟶ & OM′ \end{matrix} \}

to the plane L and obtain collinear mappings from L to the line pencils O(OX) ⊂ ε and O(OM′) ⊂ ε′. The defect of these mappings is equal to 1. We compute for each of the mappings the kernel point K_{X}, K_{M′}:

K_{X} … α_{1}: α_{2}: α_{3} = [B_{2}B_{3}(x_{2}y_{3} − x_{3}y_{2})] : [B_{3}B_{1}(x_{3}y_{1} − x_{1}y_{3})] : [B_{1}B_{2}(x_{1}y_{2} − x_{2}y_{1})] , (33)
K_{M′} … α_{1}: α_{2}: α_{3} =[x_{2}x_{3}(B_{2}C_{3}−B_{3}C_{2})] : [x_{3}x_{1}(B_{3}C_{1}−B_{1}C_{3})] : [x_{1}x_{2}(B_{1}C_{2}−B_{2}C_{1})] . (34)

According to (32) both kernel points are contained in l.

Fig. 1a Fig. 1b

Regular case: Because^{1} of O(OX) \overline{∧} K_{X}(x) \overline{∧} K_{M′}(m) \overline{∧} O(OM′) the line pencils O(OX), O(OM′) are projectively related. In this chain of projectivities corresponding lines of K_{X}(x) \overline{∧} K_{M′}(m) intersect each other in points of l (fig. 1a). The bundle rays through X_{1}, X_{2}, X_{3} are distinct and so are the bundle rays through M′_{1}, M′_{2}, M′_{3}. The projectivity is uniquely determined by these two triples of rays.

Singular case: l is separated into two straight lines a, b. In this case there is at least one coefficient in (32), which is equal to zero: Either two of the points – say X_{1}, X_{2} – or two of the centres – say M′_{1}, M′_{2} – lie on the same bundle ray^{2}. A short computation shows: There is just one of the points K_{X}, K_{M′} on each of the lines a, b (fig. 1b). So we have: The straight line a going through K_{X}, is of type 1 according to our classification (section 3). The line b through K_{M′} is of type 2. Thus the motion regarded is part of a two-parameter motion of type 1 and in the same time part of another two-parameter motion, which is of type 2. The fact, that both lines have one point G in common (fig. 1b) means that the two two-parameter motions have to satisfy a common sphere condition. So we get the following classification of the motions of our singular case:

1\overline{∧} is used as a symbol for a projective mapping.

2We will abandon the case that both of these conditions are satisfied simultaneously, because it only bears a Euclidean bundle motion.

1. The points of a circle k through O are moved on concentric balls (with centre M′). One point of k even runs on a circle n′, whose axis contains M′.

2. The points of a circle k through O are moved on concentric balls (with centre M′). One point N of the tangent t of k in O runs of a circle n′, whose axis contains M′.

3. The points of a circle k are moved on concentric balls (centre M′). One point of the tangent t of k in O runs on a straight line contained in a plane orthogonal to OM′.

4. The points of a straight line g are moved in a plane γ′, one of these points even runs on a circle contained in γ′.

5. The points of a straight line g are moved in a plane γ′. One of them even runs on a straight line orbit in γ′.

6. An example

After this theoretical classification we think, that the reader deserves an example. We now consider an equiform motion in a plane π with a global fixed point O. We will call it

− a straight line motion, if all points have straight line orbits.

− a circular motion, if all points have circle orbits.

It can easily be seen^{3} that a plane equiform motion with a global fixed point O is a straight line motion (circular motion), if there exists one straight line (circle) orbit. Such motions are the starting point for the following

Theorem 6. a) The spatial continuation of a plane straight line motion gives a Darbouxmotion^{4}: All the points of the moving space Σ have plane trajectories; the points outside of the plane π – except those of the bundle ray z orthogonal to π – have orbits on hyberbolas contained in planes orthogonal to π. The minor axes of those hyperbolas are in π (see fig. 2a).

b) The spatial continuation of a plane circular motion gives a Bricard-motion (see fig. 2b). All the points of the moving space Σ have trajectories on balls with centres in π. The points outside of the plane π – except those of the bundle ray z orthogonal to π – have orbits on spatial algebraic curves of order 4 of the first kind. Furthermore: The points of a circle k through O, tangent to z in O, have their orbits on concentric balls.

Proof. The plane π and the orthogonal bundle ray z keep their positions during the whole motion. Thus every point X has an orbit on a cone of revolution Γ_{X} with vertex O and axis z. Lines parallel to z keep their direction.

a) A point X ∉ π moves on Γ_{X} and on the other hand in the plane ε_{X} containing the orbit line of the orthogonal projection X^{n} of X on π (see fig. 2a). This is why the orbit of X is the intersection of Γ_{X} and ε_{X}, a hyperbola with minor axis in π.

b) A point X ∉ π moves on Γ_{X} and on the other hand on the cylinder of revolution ζ_{X} containing the orbit circle of the orthogonal projection X^{n} of X on π (see fig. 2b). This is why the orbit of X is part of the intersection curve of Γ_{X} and ζ_{X}, which is symmetric with respect to π. Second order surfaces of revolution with parallel axes have always intersection curves which are spherical: One can easily prove by calculation that the pencil of second order surfaces through the given intersection curve will always contain one sphere.

3see L. Burmester [4] and H. Schaal [11]

4see A. Karger [7]

Fig. 2a

Fig. 2b

Because of the common symmetry with respect to π the centre of the sphere containing the intersection curve Γ_{X} ∩ ζ_{X} is in π. Γ_{X} and ζ_{X} do have an additional common symmetry plane π̄ going through both axes of revolution. Be X and Y two points on a circle k which contacts z in O. Both X and Y will at the same moment have their positions in π̄. This happens twice; we call these positions X_{0}, Y_{0} and X_{1}, Y_{1}. The centres M_{X}, M_{Y} of the orbit spheres of X, Y are the intersections of the midperpendiculars of X_{0}X_{1} and Y_{0}Y_{1} (lying in π̄) with π (see fig. 3). Because of the equality of the ratios (X_{0}, X_{1}; O) and (Y_{0}, Y_{1}; O) and because of Thales’ Theorem we get: M_{X} = M_{Y}.□

Fig. 3

The motion described in Theorem 6 a) can be represented by

γ(t_{1}) … ( \begin{matrix}x′ \\ y′ \\ z′ \end{matrix} ) = \frac{1}{cos t_{1}} · ( \begin{matrix}cos t_{1} & − sin t_{1} & 0 \\ sin t_{1} & cos t_{1} & 0 \\ 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}x \\ y \\ z \end{matrix} )  (35)

Theorem 6 b) describes a motion with a parametric representation

γ(t_{1}) … ( \begin{matrix}x′ \\ y′ \\ z′ \end{matrix} ) = (λ cos t_{1} ± \sqrt{(λ − 1)^{2} − λ^{2} sin^{2} t_{1}} ) · ( \begin{matrix}cos t_{1} & − sin t_{1} & 0 \\ sin t_{1} & cos t_{1} & 0 \\ 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}x \\ y \\ z \end{matrix} ) (36)

Here λ is a constant determining the angle 2α under which any orbit circle is seen from the bundle centre O; we have^{5} λ = \frac{1}{cos^{2} α}.

In the first part of this paper we started from linear intersections of the surface B_{6} ⊂ P (W) and investigated the corresponding motions. Conversely, with the use of our principle of transfer, any motion determines a corresponding manifold on B_{6}, which is a complete linear intersection. We now will look for these manifolds on B_{6} belonging to the motions of Theorem 6. a) The motion regarded be the spatial continuation of a plane straight line motion. A point X … (x, y, 0)^{t} of the xy-plane has a straight line orbit in this plane. Any of these straight line trajectories is the axis of a linear pencil of planes. So X satisfies a one parameter set of sphere conditions (i.e. plane conditions); the coefficients according to (2) are A: B: C: D: E = 0: x: y: d: −2(x^{2} + y^{2}), d ∈ ℜ. The corresponding points on B_{6} – with respect to the mapping b – form a manifold

5λ is a real constant even in the case that O is an interior point of all orbit circles. The angle α itself is then a purely imaginary number.

Ψ … (ω_{0}: …: ω_{10})^{t} = (0: x^{2}: xy: 0: xy: y^{2}: 0: dx: dy: 0: −2 (x^{2} + y^{2}))^{t}  (37)

homogenously parametrized by x, y, d. Points X … (x, y, z = / 0)^{t} have orbits in z-parallel planes

A: B: C: D: E = 0: x: y: 0: −2(x^{2} + y^{2}).

The corresponding point set Ψ̄ on B_{6} is

Ψ̄ … (ω_{0}: …: ω_{10})^{t} = (0: x^{2}: xy: xz: xy: y^{2}: yz: 0: 0: 0: −2 (x^{2} + y^{2}))^{t}  (38)

with x, y, z as homogenous parameters.

b) We now regard the spatial continuation of a plane circular motion. A point X … (x, y, 0)^{t} in the xy-plane has a circle as orbit, which of course lies in ∞^{1} balls given by A: B: C: D: E = 1: −λx: −λy: d: (2λ − 1) (x^{2} + y^{2}), d ∈ ℜ. Hence the manifold on B_{6} is

\begin{matrix}Φ … (ω_{0}: …: ω_{10})^{t} = \\ = (−(x^{2} + y^{2}): λx^{2}: λxy: 0: λxy: λy^{2}: 0: −dx: −dy: 0: (1 − 2λ) (x^{2} + y^{2}))^{t} \end{matrix}  (39)

Finally the points X … (x, y, z = / 0)^{t} have orbit spheres

A: B: C: D: E = 1: −λx\frac{x^{2} + y^{2} + z^{2}}{x^{2} + y^{2}} : −λy \frac{x^{2} + y^{2} + z^{2}}{x^{2} + y^{2}} : 0: (2λ − 1) (x^{2} + y^{2} + z^{2}).

The corresponding point set Φ̄ on B_{6} is

\begin{matrix}Φ̄ … (ω_{0}: …: ω_{10})^{t} = \\ = (−(x^{2} + y^{2}): λx^{2}: λxy: λxz: λxy: λy^{2}: λyz: 0: 0: 0: (1 − 2λ) (x^{2} + y^{2}))^{t}. \end{matrix}  (40)

The shape of the 2-surfaces Φ, Φ̄ (see fig. 4). A homogenous linear condition

β x + γ y + δ d = 0  (41)

yields a second order curve on Φ ⊂ B_{6}, in case of δ = 0 it collapses into a straight line. This is why the surface Φ must contain a 2-parametric set of second order curves l(β: γ: δ). Such a second order curve on B_{6} gives the condition:^{6} The points of the [xy]-plane have their orbits on spheres with centres in the bundle-plane β x + γ y + δ d = 0. Specially the second order curve l(0: 0: 1) belongs to the bundle plane z = 0.

6compare with conclusion b) of Theorem 4

Fig. 4

The ∞^{1} equations β x + γ y = 0 give us the straight lines

\begin{matrix}e(β: γ) … (ω_{0}: …: ω_{10})^{t} = \\ = (−(β^{2} + γ^{2}): λγ^{2}: −λβγ: 0: −λβγ: λβ^{2}: 0: 0: 0: 0: (1 − 2λ)(β^{2} + γ^{2}))^{t} + \\ + λd (0: 0: 0: 0: 0: 0: 0: −γ^{2}: βγ: 0: 0)^{t}. \end{matrix}  (42)

Each of them connects a point U_{1}(β: γ) of the second order curve mentioned above

\begin{matrix}l(0: 0: 1) … (ω_{0}: …: ω_{10})^{t} = \\ = (−(β^{2} + γ^{2}): λγ^{2}: −λβγ: 0: −λβγ: λβ^{2}: 0: 0: 0: 0: (1 − 2λ)(β^{2} + γ^{2}))^{t} \end{matrix}  (43)

with a point U_{2}(β: γ) of the directing line on Φ represented by

g … (ω_{0}: …: ω_{10})^{t} = (0: 0: 0: 0: 0: 0: 0: −γ: β: 0: 0)^{t}.  (44)

The mapping \{ \begin{matrix}l(0: 0: 1) & → & g \\ U_{1} & → & U_{1} \end{matrix} \} is a projectivity. The lines e(β: γ) are of type 1 (according

to our classification in section 3). In our principle of transfer everyone of them represents the points of a bundle ray in the xy-plane. These points are moved on circles of the x′y′-plane and as a consequence each of them keeps on lying on a pencil of spheres (with centres on a z′-parallel axis) throughout the whole motion. The directing line g itself is of type 2 (see section 3). It belongs to the condition, that the points of the xy-plane have plane trajectories (i. e. are moved in the x′y′-plane). The 2-surface Φ lies in the 4-dimensional subspace H of P (W) spanned by g and l(0: 0: 1). The quadratic hypercones with the equations ω_{1} ω_{5} − ω_{2} ω_{4} = 0 and ω_{4} ω_{8} − ω_{5} ω_{7} = 0 (see (9)) have intersections order 2. They intersect in a 2-surface Φ of order 4.

Analogously we can find a 2-parameter family of second order curves l̄(ρ: σ: τ) on Φ̄, which are determined by the linear homogenous conditions

ρ x + σ y + τ z = 0  (45)

with τ = / 0. In accordance with conclusion b) of Theorem 4 (see part I of this paper) l̄(ρ: σ: τ) belongs to the condition: The points of the bundle plane (45) of the moving space move on spheres with centres in the x′y′-plane. For τ = 0, however, (45) yields a one parameter family of straight lines

\begin{matrix}ē(ρ: σ) … (ω_{0}: …: ω_{10})^{t} = \\ = (−(ρ^{2} + σ^{2}): λσ^{2}: −λρσ: 0: −λρσ: λρ^{2}: 0: 0: 0: 0: (1 − 2λ) (ρ^{2} + σ^{2}))^{t} + \\ + λz(0: 0: 0: σ^{2}: 0: 0: 0: −σρ: 0: 0: 0)^{t}. \end{matrix}  (46)

This line ē(ρ: σ) connects the point Ū_{1} (ρ: σ) of

\begin{matrix}l̄(0: 0: 1) … (ω_{0}: …: ω_{10})^{t} = \\ = (−(ρ^{2} + σ^{2}): λσ^{2}: −λρσ: 0: −λρσ: λρ^{2}: 0: 0: 0: 0: (1 − 2λ) (ρ^{2} + σ^{2}))^{t} \end{matrix}  (47)

with the point U_{2}(ρ: σ) of the directing line lying on

ḡ … (ω_{0}: …: ω_{10})^{t} = (0: 0: 0: σ: 0: 0: 0: −ρ: 0: 0: 0)^{t}.  (48)

Again such pairs of points (U_{1}, U_{2}) are connected with a line ē(ρ: σ) ⊂ Φ̄, which do correspond in a projectivity. The lines ē(ρ: σ) are of type 2; each of them belongs to the condition: The points of a circle contacting the z-axis in the point O, are moved on concentric balls with centres in the x′y′-plane. The directing line ḡ is, however, of type 1 and it says: The points of the z-axis are moved on this line.

As above we can conclude that Φ̄ is a 4^{th} order 2-surface just as Φ is. This 2-surface Φ̄ lies in a 4-dimensional subspace H ⊂ P (W) spanned by ḡ and l̄(0: 0: 1). H and H̄ are different and their intersection is the plane L: Comparing (43) with (47) shows immediately l(0: 0: 1) = l̄(0: 0: 1) ⊂ L and furthermore: l(0: 0: 1) is the complete intersection of the two surfaces Φ and Φ̄ and therefore Φ ∪ Φ̄ is the intersection of the 6-dimensional space H + H̄ with B_{6}. The consideration of the 2-surfaces Ψ and Ψ̄ (see(37) and (38)) is analogous to the one above. We summarize:

Theorem 7. The b-images of the motions of Theorem 6a) ( 6b) ) are reducible 2-surfaces consisting of two ruled surfaces Ψ, Ψ̄ (Φ, Φ̄). Each of the ruled surfaces has the following properties:

− It is algebraic of order 4.

− It contains a 2-parameter-set of conic sections.

− It spans a 4-dimensional space.

− It contains a directing line.

− The ruled surface can be generated by a projectivity between the directing line and one of its conic sections.

7. Two-parametric spherical motions

We will not investigate the trivial motion of type 1, section 3 (one point is moved on a circle or a straight line). In the following we will give parametric representations of the motions of type 2, which do deserve more attention.

Type 2a: The points of a circle k, which lies in a bundle plane ε and contains the bundle centre O, are moved on spheres with common centre M′ = / O. Then all points in ε have orbit spheres with centres on the bundle ray OM′; the points of the tangent t of k in O are moved in planes orthogonal to OM′.

Fig. 5

First we consider the case that all these planes are identical: This case occurs iff t is orthogonal to OM′ during the whole motion. We assume that OM′ is the x′-axis of Σ′. Be z′ = z = t in the “starting position” (see fig. 5). Furthermore let the circle k be contained in the xz-plane of the moving system. The point X ∈ k which is opposite to O lies on the x-axis. In the starting position X is in the x′y′-plane. We take the circular motion in the x′y′-plane keeping X on a circle around M′ and we regard its spatial continuation (see (36)). The points of k run on balls with centre M′ (see Theorem 6b)). If we superpose this motion with a rotation with axis x′ = OM′, we gain the 2-parameter motion we were looking for. A parametric representation is given by

γ(t_{1}, t_{2}) … ( \begin{matrix}x′ \\ y′ \\ z′ \end{matrix} ) =

= (λ cos t_{1} ± \sqrt{(λ − 1)^{2} − λ^{2} sin^{2} t_{1}}) · ( \begin{matrix}cos t_{1} & − sin t_{1} & 0 \\ cos t_{2} sin t_{1} & cos t_{2} cos t_{1} & − sin t_{2} \\ sin t_{2} sin t_{1} & sin t_{2} cos t_{1} & cos t_{2} \end{matrix} ) ( \begin{matrix}x \\ y \\ z \end{matrix} ) .

(49)

General case: Any two of the parallel orbit planes of the points of t are distinct. In the starting position we can demand that the coordinate system of Σ and Σ′ are identical (see fig. 6). We put t and the bundle ray of the orbit sphere centres into the x-axis. In the starting position we can also assume that the centre of the circle k lies on z. The radius of k be r = \frac{1}{2}. We first consider the motion satisfying the following two conditions

1. A point X ∈ t (let X … (1, 0, 0)) runs on a line e′ parallel to z′.

2. The point P opposite to O on k is moved on a sphere with centre M′ … (m′, 0, 0), M′ ∈ x′.

Fig. 6

Be t_{1} the angle between x′ and OX′, where X′ denotes the actual position of X on e′. We now superpose the straight line motion, where X moves on e′, with a rotation around OX (rotation angle β) and get a 2-parameter motion. The condition that the orbit of P is a ball with centre M′, picks out a one parameter motion. The angle β can be calculated for every moment t_{1}. The result is:

cos β(t_{1}) = \frac{tan t_{1}}{2m′} .  (50)

This way we have got a one parameter motion where not only the point P, but any point of k is forced to move on a ball as well. A parametric representation is:

( \begin{matrix}x^{⋆} \\ y^{⋆} \\ z^{⋆} \end{matrix} ) = \frac{1}{cos t_{1}} · ( \begin{matrix}cos t_{1} & − sin β sin t_{1} & − cos β sin t_{1} \\ 0 & cos β & − sin β \\ sin t_{1} & sin β cos t_{1} & cos β cos t_{1} \end{matrix} ) ( \begin{matrix}x \\ y \\ z \end{matrix} ) =: A(t_{1}) ( \begin{matrix}x \\ y \\ z \end{matrix} ) . (51)

The superposition of (51) with the rotation around the x-axis

( \begin{matrix}x′ \\ y′ \\ z′ \end{matrix} ) = ( \begin{matrix}1 & 0 & 0 \\ 0 & cos t_{2} & − sin t_{2} \\ 0 & sin t_{2} & cos t_{2} \end{matrix} ) ( \begin{matrix}x \\ y \\ z \end{matrix} ) =: B(t_{2}) ( \begin{matrix}x^{⋆} \\ y^{⋆} \\ z^{⋆} \end{matrix} )  (52)

Fig. 7

finally gives the parametric representation of the spherical 2-parameter motion regarded:

γ(t_{1}, t_{2}) … ( \begin{matrix}x′ \\ y′ \\ z′ \end{matrix} ) = B(t_{2}) · A(t_{1}) ( \begin{matrix}x \\ y \\ z \end{matrix} ) .  (53)

Type 2b: A non-bundle-ray straight line g of the moving space Σ is moved in a plane ε′ fixed in Σ′. If ε′ was a bundle plane, we would get an axial 2-parameter motion. This rather simple case is suppressed here. We choose our coordinate systems Σ and Σ′ the way that g is represented by

g … ( \begin{matrix}x \\ y \\ z \end{matrix} ) = ( \begin{matrix}1 \\ 0 \\ 0 \end{matrix} ) + λ ( \begin{matrix}0 \\ 0 \\ 1 \end{matrix} )

and ε′ by x′ = const (see fig. 7). All the points of the xz-plane run in planes orthogonal to the x′-axis. In order to gain a parametric representation we first regard that straight line motion in the x′y′-plane, where the point G = g ∩ x runs on the intersection line e′ of [x′y′] and ε′ (see also (35)). The points of g are moved on hyperbolas in ε′ (see Theorem 6(a)). We superpose this motion by a rotation with axis x′ and get the parametric representation

γ(t_{1}, t_{2}) … ( \begin{matrix}x′ \\ y′ \\ z′ \end{matrix} ) = \frac{1}{cos t_{1}} · ( \begin{matrix}cos t_{1} & − sin t_{1} & 0 \\ cos t_{2} sin t_{1} & cos t_{2} cos t_{1} & − sin t_{2} \\ sin t_{2} sin t_{1} & sin t_{2} cos t_{1} & cos t_{2} \end{matrix} ) ( \begin{matrix}x \\ y \\ z \end{matrix} ) .  (54)

8. Final considerations

The systematic approach used in this paper has allowed to describe one- and two-parametric equiform bundle motions satisfying an infinity of sphere conditions in the sense of Remark 1.

It cannot be definitely ruled out, that in some case a linear subspace of P (V) with a dimension less than 7 gives an intersection with Γ_{4} which is at least one-dimensional. This obviously depends on its spatial position with regard to Γ_{4}. The corresponding b-images are intersections of B_{6} with linear subspaces of dimensions^{7} greater than 2. These are just the cases which are not to be gained by the considerations above. An investigation of these cases is nevertheless imaginable in the course of the principle of transfer used here, but the amount of calculation would surely increase, as some more sophisticated algebraic-geometric considerations would be necessary.

A list of references is given in the first part of this paper.

Received April 8, 1996, revised version August 28, 1997

7In section 6 we actually gave an example of a one-parameter motion with a b-image that turned out to be the intersection of B_{6} with a 6-dimensional subspace of P (W).