K H Hunt

Professor, F R E Crossley

Coefficient of Restitution Interpreted as Damping in Vibroimpact

come

Introduction

Section 1.1. This analysis is confined to compact solid bodies (normally regarded as elastic) impacting one another at fairly low velocities, e.g., below 20 ips (50 em/sec), for reasonable shapes of impacting profiles when the material is, say, hardened steel. No account is taken of the possible effect of reflected elastic shock waves on the forces in the zone of contact considered. Long rods impacted axially, and beams or plates impacted transversely, are therefore not regarded as "compact solid bodies" [1j,I though it may be possible to incorporate the present conclusions as a part of a system which includes other vibratory effects as well.

Section 1.2. The classical definition of "coefficient of restitution" e is, in simple one-dimensional impact between two rigid bodies in pure translation, the ratio of their relative speed after impact to that before, so tha.t v 0 = ev 1 , [START_REF] Goldsmith | Impa ct[END_REF] v; and v 0 being the relative speeds respectively "in" and "out." In [START_REF] Goldsmith | Impa ct[END_REF] there is a good account of many experiments conducted with different materials, usually solid spheres, to determine values for e. Yet in these experiments, now rather antiquated, the effects of surface curvature on e are not isolated. Nevertheless, for given impacting bodies e invariably decreases as Vi increases. For high Vi one or both bodies can permanently deform plastically, but, as 1 Numbers in brackets designate References at end of paper. stated in Section 1.1, this eventuality is not taken account of here, because the typical impact in the elastic range now to be analyzed is that which leaves both bodies unchanged in dimension and properties. Surely, then, one might conclude that, if the bodies are unchanged, e must he unity and the impact perfectly reversible; hut it need not he, because, while most of the elastic strain energy is restored, a proportion, which increases as Vi increases, is dissipated in heat (random molecular vibrations) and, perhaps, in microscopic slip between grains if the material in question is crystalline. For a limited range of low vi, and for most materials with a linear elastic range, [1, p. 258], it appears that one can write with tolerable accuracy [START_REF] Kobrinskii | Mechanisms With Elastic Connections and Impact Systems (translated from the Russian by R. Lennox-Napier)[END_REF] For steel, bronze, or ivory, a will have a value somewhere between 0.002 and 0.008 sec/in. (i.e., 0.08-0.32 sec/m), but the linear law of equation ( 2) cannot apply above some limiting value of vi which depends on the material and on the geometry of the surfaces at or near the zone of contact.

Section 1.3. If one of the bodies is taken to be stationary, the loss, over a total single-impact sequence, of the kinetic energy of the moving body (of mass m) is

1 ( 2 2 D.E = 2 m v 1 -v 0 ),
and, using successively equations ( 1) and (2),

1 2 ( t:.E = 2 mv 1 1 1 2{ = 2 mv 1 1 e2)
Now, since a is much less than 1 '(and because it is empirically determined we cannot expect perfect exactitude anyway), one may Section 1.4. Some workers in vibroimpact, e.g., [START_REF] Kobrinskii | Mechanisms With Elastic Connections and Impact Systems (translated from the Russian by R. Lennox-Napier)[END_REF], and others concerned with impact dampers, for instance, have (sometimes with remarkable success) considered impact to be instantaneous, and have used a constant e irrespective of v;. In such work the "approach," namely, the extent by which the center's of mass of the two impacting bodies move toward and away from one another as a result of the localized deformation near the zone of contact, is ignored, and this practice may be justifiable when the interval of time between successive contacts is substantially longer than the duration of one contact. In [START_REF] Dubowsky | Dynamic Analysis of Mechanical Systems With Clearances, Part 1: Formation of Dynamic Model; Part 2: Dynamic Response[END_REF], however, neither the time of contact nor the approach are ignored; the equation governing the period of contact during a single impact is taken to be a half period of damped harmonic vibration during which, in the absence of any other excitation, the standard form of linear equation

m x + ex + kx = 0 (4)
(in which c is a constant coefficient of viscous damping and k a linear spring-constant presumed to match the force-approach law) is taken to apply. In effect equation (4) matches precisely the ideal Kelvin-Voigt viscoelastic solid [START_REF] Fliigge | Viscoelasticity[END_REF]. In the Appendix the consequences of the assumptions inherent in equation ( 4) are developed qualitatively, and the conclusions reached confirm the validity of the analysis which now follows.

Section 1.5. According to equation ( 4) the damping force, c .:i:, is greatest when x is highest, that is to say at the precise moment of impact when .:i: = v;. This damping term c.:i:, and the spring term kx, can be combined to produce a form of hysteresis loop shown as OABCDO in Fig. 1(a), the shape of which is, as it turns out, close to a half ellipse, provided that the damping .is small. This half-ellipse hysteresis loop is in conflict with the expected shape of a loop that arises from a compressive load being applied to and removed from a body within its elastic range at a slow rate. A loop of this more realistic form is sketched in Fig. 1(b). Because the applied load is compressive, through a pair of profiles that touch but are not otherwise joined together, no tensile force is admissible; i.e., since the "force" in Fig. 1(b) is everywhere compressive (namely, above the Ox-axis) the loop cannot be allowed to extend into the region below the Ox-axis. Moreover, the compressive force at the moment the load is appljed is zero. These arguments lead to the form of hysteresis loop drawn in Fig. 1(b), the loop coming to a sharp point at the origin. The half ellipse in Fig. 1(a), that arises simply from the direct intrusion of equation ( 4), is sufficiently different from the loop of Fig.

I (b)
to raise doubts about its validity as a model of what actually occurs, because (a) , for the region 0 < x < OC of the load-removal portion, it hypothesises a tensile force (which would tend to lengthen the period of contact and reduce the speed at which the impacting body leaves), and (b), at the moment of impact, there is a "shock" application of a supposed force that arises entirely from the damping term c .:i: in equation ( 4) . Section 1.6. True, when the impacting bodies are highly resilient (namely, when c is very much less than the critical damping coefficient), these two anomalous effects may be so small as to cause no concern to someone establishing a mathematical model suitable for analogue or digital simulation. It is however demonstrated below, equation ( 16), that a much better moqel can quite easily be set up which, in contrast to the one just described, shows promise of being usefully adaptable both to bodies with nonlinear elastic properties and to coefficients of restitution substantially less than unity.

Section 1.7. Fig. 1(a) fails in another respect to meet the required conditions; the half-ellipse hysteresis loop remains to all intents and purposes identically proportioned with respect to t he rest of the diagram whatever the maximum approach Xm. The area of the loop, which represents the energy loss DE, would then (for resilient bodies) be closely proportional to xm2, which, since

1 2 ~ 1 k 2 2 mvi ~ 2 xm,
may be taken as being proportional to u? Yet equation (3) requires DE to be much more nearly proportional to v;3 t han to v; 2 . Section 1.8. Equation ( 4) has the advantage of simplicity. As it stands, for an isolated half period of impact-duration, it is linear, and in the well-known classical form. But, since the overall pattern of impact (including .relative movements across clearances, etc.) is far from linear-at best it is piecewise linear-one may justifiably propose some improvement on the highly idealized Kelvin-Voigt model, particularly now that there are available siml!lation techniques that readily accept nonlinearities and still give high accuracy. Clearly the hysteresis loop in Fig. 1(a) should be abandoped and replaced by one looking more like that in Fig. 1(b). Its shape should be consistent with damping forces being zero both when x = 0 (.:i: = v; or .:i: = -vo) and when x = Xm (.X = 0). A damping term of the form A(x)P(x)q satifies these boundary condit ions, and a way is described next for determining suit able values for A, p , and q.

Section 1.9. The force-approach law based on the Hertzian theory for localized normal and frictionless contact between linearly elastic bodies is not itself linear (as required by equation ( 4)) but is nonlinear and of the form F = kx312. The theory, described in [START_REF] Goldsmith | Impa ct[END_REF][START_REF] Timoshenko | Theory of Elasticity[END_REF] and by many other authorities, cannot br. regarded as valid unless the zone of contact is of small dimensions in comparison with the radii of curvature of the impacting bodies in its vicinity, and unless it has very large radii of curvature in comparison with its linear dimensions. For two bodies of steel (elastic modulus 30 X 10 6 lb/in. 2 (2.068 X 10 N/m 2 ) and Poisson's ratio 0.30) impacting in spherical regions whose radii of curvature are Rt and R 2 , the forceapproach law becomes

F = Ktx312 (J.. + __!_) -2 Rt R2 (5)
where K 1 = 2.19 X 107 if R 1 , R 2 and x are measured in in. and Fin lb, or K 1 = 3.83 X 10 6 if R~, R 2 , and x are measured in em and F in newtons. The maximum shear stress (occurring at points on the common normal a little below the surfaces of the bodies) is of value

T = K2 v (it + i 2 ) 2 F. (6)
K 2 being 1.16 X 10 4 if r is to be read in lb/in. 2 ; or K 2 = 9.02 X 10 7 for -r in N/m 2 • In [1 and 5], and elsewhere, it is explained how equations ( 5) and ( 6) can be formulated for more general shapes of contacting contours, and the results here presented can with complete validity be extended if desired. Section 1.10. According to [START_REF] Harris | Rolling Bearing Analysis[END_REF], direct compressive contact stresses (of a highly localised nature) exceeding 200,000 lb/in. 2 (1.38 X 10 9 N/m 2 ) are not uncommon for conditions of continuous operation in rolling bearings, and in certain circumstances an acceptable limit may be as high as 500,000 lb/in. 2 (3.45 X 109 N/m 2 ). For ductile materials [START_REF] Goldsmith | Impa ct[END_REF] the most probable form of impact failure is in shear and, according to the Hertzian theory, maximum shear is about 0.31 times the maximum compressive stress, this factor being used in writing equation [START_REF] Harris | Rolling Bearing Analysis[END_REF]. So, in view of the foregoing, a limiting value T!im of 0.31 X 200,000 = 62,000 lb/in. 2 (4.27 X 10 8 N/m 2 ) appears to be reasonable, even conservative, for alloy steels with surface properties suitable not only for rolling bearings, but also for other joint-pairs with highly localized regions of contact such as those often found in cams, gears, etc. This limiting localized shear-stress value is, incidentally, 10 times higher than the unrealistically low limit used in part of [START_REF] Dubowsky | Dynamic Analysis of Mechanical Systems With Clearances, Part 1: Formation of Dynamic Model; Part 2: Dynamic Response[END_REF], and it is seen below to be reasonably compatible with impact speeds of the magnitudes mentioned in Section 1.1.

Secion l.ll. The 3/2-power law of equation ( 5) does not apply when, for instance, contact is between perfectly flat surfaces; then, at least ideally, the area of the zone of contact does not change during the time of impact, and a linear force-approach law would probably apply. For parallel cylinders in longitudinal contact along a narrow zone on either side of a line, an index somewhere between 1 and 1.5 could be expected to match the conditions fairly well, and this does not conflict greatly with various formulas to be found in [START_REF] Goldsmith | Impa ct[END_REF][START_REF] Dubowsky | Dynamic Analysis of Mechanical Systems With Clearances, Part 1: Formation of Dynamic Model; Part 2: Dynamic Response[END_REF][START_REF] Harris | Rolling Bearing Analysis[END_REF][START_REF] Timoshenko | The Strength of Gear Teeth[END_REF][START_REF] Lubkin | Handbook of Engineering Mechanics[END_REF][START_REF] Hugk | Untersuchung der Kontaktschwingungen an der Eingriffsstelle von Kurvenmechanismen[END_REF]. Harris [START_REF] Harris | Rolling Bearing Analysis[END_REF] emphasises that, unless conditions are very special, a uniform force per unit length of roller cannot be assumed. In any case a formula for cylinders in longitudinal contact must depend for its validity on extremely precise alignment and highly exact knowledge of the axial distribution of the applied load; even in precision engineering applications information on these two matters may be hard, or impossible, to guarantee. So, despite the doubts about these special geometries yielding valid formulas, a more general form of the force-approach law of equation ( 5), namely,

F = l1x" (7)
will be used in what follows, and it will be found that it is possible to use a variable damping coefficient in an equation derived by extension and generalization of equation ( 4). It is then seen that the damping factor A (see Section 1.8) can be made independent of the index n in equation ( 7).

2 Analytical Formulation Section 2.1. Consider first nonlinear elastic impact (e = 1) of a mass m with an impacting speed v; Fig. 2. The kinetic energy at contact is (lf2)mvi 2 , and, provided next that the proposition of x m x Fig. 3 small energy dissipation holds, this can be equated with reasonable accuracy, via equation [START_REF] Timoshenko | The Strength of Gear Teeth[END_REF], to the maximum elastic-strain energy stored, i.e., for a maximum approach Xm, or and

1 2 2mvi ( __ k_) X (n+l) n + 1 m ' ~ 2k (n+j) vi = m(n + 1) Xm --r:-. (8) (9)
For an intermediate position

x, 0 < x < Xm, .!.mv2 1 lx 2 = 2 mv/ - Fdx 0 and v = I 2k I X (n+j) -x<n+1l (10) m(n + 1) V m • Section 2.
2. Now in fact there is an (assumed relatively small) hysteretic energy loss represented by the area of a hysteresis loop, Fig. 3. This area can be equated to t:.E and, combining equations ( 3) and ( 9),

t:..E _!!___ (__E:._) 3/2 X ¥ -Jrnn+1 '" • (11)
If the notion of a nonconstant damping coefficient is accepted, one may also write, assuming q = 1 (see Section 1.8)

E = ¢cvdx

where the j' refers to the. circuit around the loop in Fig. 3. Now, with an energy loss assumed to be small, equation [START_REF] Hugk | Untersuchung der Kontaktschwingungen an der Eingriffsstelle von Kurvenmechanismen[END_REF] can be used with fair accuracy to give v and, since the loop's area is shared sufficiently equally between "inward" and "outward" paths, the cyclic integral can be written

rXm 2 Jo cvdx,
which in turn leads to

The next question is how to evaluate this integral to produce the nonconstant c in managable form. This question can quite easily be resolved if one introduces the constant )\ (see Section 1.8, making p = n), and thus writing

c = A.x". ( 13
)
Then equation ( 12) becomes integrable and reduces to

AE = ~~ 2!? (-1_ \(~\X ~ill vm n+1 n+1) 3)m (14)
This expression for D.E is required to be the same as that of equation (11). Thus one reaches the extremely simple conclusion that 3

A. = 2 a!?, (15)
Note that, as anticipated in Section 1.11, equation (15) does not contain the index n. Using it with an nth power force-approach law, the equation (applicable to a free damped half cycle of vibroimpact) is now not equation (4) but or

The index q (see Section 1.8) need not be taken as unity; several suitable choices could be made so long as they are compatible with equation (12) being easily integrable and leading to an expression in which Xm has a power of

3(n + 1) 2
For instance if q = 2, )\ comes out to ~ah!?m(n + 1); then instead of the second of equations ( 16) one has the damping factor )\ being no longer independent of n.

3 Closure Section 3.1. The authors have found it quite simple to incorporate equation ( 16) in digital computer simulation of a forced vibroimpact system, and in some instances the response is significantly at variance with that of an equivalent linearized model. Some specimen hysteresis loops, obtained from trial runs, are presented in Fig. 4. These all apply to a steel body weighing about one .4 .6 .8

1.0 XI0-3 APPROACH (in) Fig. 4 pound (4.448 N) having a convex spherical surface of radius R1 = 1 in. (2.54 em) impacting a concave spherical steel surface of radius R 2 = -2 in. ( -5.08 em). The undamped force-approach law, equation (5), then becomes with K 3 = 3.1 X 10 7 if x is in in., F inlb, or Ka = 3.4 X 10 7 for x in em, F in newtons (4.27 X 10 8 N/m 2 ) and, corresponding with a shear stress of 62,000 lb/in. 2 (see Section 1.10), the maximum allowable force, by equation [START_REF] Harris | Rolling Bearing Analysis[END_REF], is 600 lb (2690 N). As can be seen from Fig. 4, when the impacting speed is 16 ips (40 em/sec) this figure of 600 lb is exceeded, but the shear stress (equation ( 6)) is proportional to the cube root of normal force, and the figure of 62,000 lb/in. 2 (4.27 X 10 8 N/m 2 ) is not exceeded by more than about 15 percent. With the same impacting speed the shear stress can be reduced by altering the surface-geometries, e.g., by wrapping the concave surface more around the convex impacting mass. However, if the two impacting surfaces were to match one another too closely, the Hertzian analysis would no longer be valid (see Section 1.9, particularly the italicized portion); for this reason much of the modeling described in [START_REF] Dubowsky | Dynamic Analysis of Mechanical Systems With Clearances, Part 1: Formation of Dynamic Model; Part 2: Dynamic Response[END_REF] cannot be applied to this form of impactpair.

Section 3.2. Hysteresis loops similar to those in Fig. 4 have been drawn for a = 0.002, 0.004, 0.008, and 0.016 sec/in. As mentioned in Section 1.2, values of a somewhere between 0.002 and 0.008 sec/in. (0.08 -0.32 sec/m) appear to be reasonable. For a to have a value as low as 0.002 the material would have to be particularly' resilient. For a material with a as large as 0.016 sec/in. (0.64 sec/m) the hysteresis loop may, with large Vi, become so large as to call into question several of the assumptions made in the derivation of equation ( 15). Nevertheless, a hysteresis loop of moderately convincing shape is, even then, automatically produced, and it is tentatively put forward that the method here described might be usefully applied in modeling situations quite far removed from those initially regarded as applicable.

Section 3.3. The whole proposition here put forward devolves in the first place on the validity of equation ( 2), which in turn is supported by the evidence of many experimenters who worked with a wide variety of apparatus over the best part of a century; see the bibliography in reference [START_REF] Goldsmith | Impa ct[END_REF]. Even so one must not ignore the present need for up-to-date verification and extension of the concept of coefficient of restitution to new engineering materials. New experiments must not neglect, as earlier workers appear to have done, the dependence of coefficient of restitution on the geometry of the impacting surfaces. Also any experiment must be carefully contrived so that the cycle of impact is as simple as possible, so that there is no risk of making circuits of one or more smaller hysteresis loops within the main one during a single impact. The authors suspect that this may have given rise to discrepancies in past experiments, and indeed that it accounts for extraordinarily low coefficients of restitution quoted in [START_REF] Kobrinskii | Mechanisms With Elastic Connections and Impact Systems (translated from the Russian by R. Lennox-Napier)[END_REF]. Even though the notion of coefficient of restitution is inexact, the analysis here pursued shows that, in an engineering context at least, there is much to recommend it, and there is scope for further refinement of it. on its right side, returning to its central position, and then, at the instant when it leaves the right-hand hlock, impacting compressively the left-hand block. The force-approach hysteretic law applicable to one cycle of the instance of Fig. 5(h) is, when small quantities of energy are lost, far more realistically represented by l•'ig. 7. The left and right-hand halves of Fig. 7 can be separated by a distance If and then applied as they stand to Fig. 5{a) with more authenticity than two separated half ellipses, Fig. 1 (a).

When the force-approach law is nonlinear (as in Fig. 2), two loops of the form in Fig. 3 The regultant (i.e., the sum) of these two loops (for a complete double-impact cyde) is drawn in Fig. R(h). Because the amplitude here is greater than 1 }2\g\ there is still impact on both sides. When the amplitude is less than 1 }J!,1 it may be expected that something like Figs. 9(a) and (h) applies. Now, of course, there is no impact at all, the body being firmly sandwiched between the blocks all the time, and the resultant loop of Fig. 9{h) begins to look much more like the expected elliptical shape used as a hasis for Fig. 6, and indeed supports the validity of the classical approach via a Kelvin-Voigt model in situations where there is no impact at all.

The qualitative line of reasoning put forward in this Appendix therefore links the nonimpacting with the impacting situations and, though distinct from the main part of the paper, supports the analysis which is used in Section 2 to derive suitable differential equations for models that are considered to relate acceptably closely to reality.

  Fig.1

F 0 Fig. 2

 02 Fig.2

Fig. 7 (

 7 Fig. 7

XFig. 9 the

 9 Fig.[START_REF] Lubkin | Handbook of Engineering Mechanics[END_REF]

Acknowledgments

The criticisms and helpful comments of Prof. J. D. C. Crisp (Monash University) are much appreciated. The assistance of the National Science Foundation through Grant GK-32372 is gratefully acknowledged.

APPENDIX

Suppose a perfectly elastic body of mass m is in the gap between two perfectly elastic massive blocks leaving an overall clearance g. If the two compressive force-approach laws (to left and right) are both taken to be linear, the combined piecewise linear force-approach law is as shown in Fig. 5(a). Now with the gap closed up to be just zero, the body in its central position just touches both blocks (with zero compressive forces on both sides), and the combined force-approach law is as in Fig. 5(b). The central horizontal portion of Fig. 5(a) has vanished, and the two linear end sections line up with one another exactly.

The force-approach curve of Fig. 5(b) is, then, identical in shape with one that would result if the body were securely fastened to one of the blocks, the other block being removed, Fig. 5(c). Now, however, when the approach x is negative, the force is not compressive acting on the left of the body but tensile acting on the right. For free vibrations of the system in Fig. 5(c) the analogy with the classical spring-mass system immediately springs to mind. If, in actuality, a small amount of energy is dissipated each cycle in friction, why should one not reasonably postulate a thin hysteresis loop, roughly elliptical, spreading a little on either side of the original (nondissipative) force-approach line? This loop would steadily diminish in size (exponentially) corresponding with the exponential decay of the amplitude in the classical theory of damped vibrations, Fig. 6. And this common assumption is known to apply well in many instances of mechanical vibrations. But is it valid to superpose such a loop on the, apparently identical, linear force-approach line in Fig. 5(b)? The theme of the present paper is that such superposition is dubious, because the instance of Fig. 5(b) entails two impacts per cycle, whereas there are no impacts at all for Fig. 5

(c).

There is discontinuity between the positive and negative portions of Fig. 5(b), the body being in compressive contact first


Sheng Jiang

Angles Between Euclidean Subspaces

Keywords: Mathematics Subject Classifications (1991): 51M05, 51Ml6, 51K05 higher-dimensional angle, principal angles, Grassmann manifolds

The angle between two subspaces of dimensions p and q in a Euclidean space is considered by using exterior algebra. Some properties of angles are obtained. The relation between such a higher dimensional angle and the usual principal angles is also given. And finally, an application to Grassmann manifolds is briefly discussed.

Preliminaries

For convenience, we will use exterior algebra for computations, it makes the results clear and the proofs simpler. In this section we briefly state some basic facts about exterior or Grassmann algebra which are needed in our paper; for details see, for example, Bourbaki [6, ch. 3], or Flanders [7, ch. 2].

Let En be an n-dimensional real linear space endowed with a Euclidean inner product. For two vectors a and b, denote by a • b their inner product.

The symbol l'l En denotes the linear space consisting of all linear combinations with real coefficients of wedges of p vectors in En, 1 :::; p :::; n. Elements of f\P En are called p-vectors over En. A p-vector is said to be simple or decomposable, if it can be decomposed as a single wedge of p vectors in En.

An induced product in f\P En can be naturally introduced as follows.

For a pair of p-vectors define (a, (3) = det (ai

• bj)•
We can prove the following LEMMA 1. The linear space f\P En endowed with ( , ) is a Euclidean space. Proof Choosing in En an orthonormal basis { ej}, j = 1, 2, ... , n, we get in f\P En an induced orthonormal basis { E K}, J( = 1, 2, ... , (;), where the E K 's are the wedges of p different vectors ej (see Flanders [7, p. 14]). An arbitrary p-vector a can be decomposed as where the )..K's are real scalars. So Hence for any a E f\P En, (a, a) = 0 if and only if a = 0. Consequently (,) defines a Euclidean inner product. where equality holds if and only if a differs from (3 only in a scalar factor.

Remark. The length of a p-vector a is defined by a = 0 if and only if Ia I = 0.

Higher Dimensional Angle

Let a = a 1 1\ a 2 1\ • • •I\ ap be a nonzero decomposable p-vector. Then a corresponds to a p-dimensional subspace AP of En spanned by a1, a2, ... , ap. Conversely, for any basis of the vector subspace AP, the wedge of basis vectors equals ka, where k is a nonzero real number.

Further, suppose that 1 ::; p ::; q < n, and

(1)

(2)
If A and B are the subspaces corresponding to a and {3, respectively, and BJ.. is the orthogonal complement of B in En, then we have where af E B, aj E BJ.., j = 1,2, ... ,p.

Put

= a [I /\ a!f /\ • • • /\ aff, av v v v = a 1 /\ a 2 /\ • • • /\ aP , aM = a -aH -av.
Then we have a decomposition

a = aH + av +aM, (3)
where aH, av and aM are called the horizontal, vertical and mixed parts, respectively, of a.

Remark 1. The concept of the p-dimensional angle defined above is a natural generalization of classical angles such as the angles between two lines, a line and a plane, and between two planes. By Lemma 4, it is clear that(} is real and satisfies 0 ~ (} ~ 7r/2; further, (} = 1r /2 if and only if the subspace A is orthogonal to B. Remark 2. Whenp = q, formula (4) can be written as l(a,,B)I (} = arccos lal . 1!31' In this case, the p-dimensional angle(} between two decomposable p-vectors a and ,8 over En is equal to the usual Euclidean angle between a and ,8 as two vectors in the induced Euclidean space f\P En.

First Properties

Using the projection O:'H, we can easily generalize some well-known properties of usual angles to the higher dimensional case as follows.

THEOREM 1 (Reducibility). Let A and B be subspaces of En with dimensions p and q, respectively, where 1 < p ::::; q < n. Suppose that A n B f= {0}, and that the orthogonal complements of A n B in A and in B are A' and B', respectively.

Then the angle between A and B is equal to the angle between A' and B'.

THEOREM 2 (Three cosines). Suppose that 1 ~ p < q < n, AP and Bq are subspaces of En with dimensions p and q, respectively, and AP is not orthogonal to Bq. Suppose CP is the projection of AP in Bq, and DP is an arbitrary p-dimensional subspace of Bq. Denote by()',() and <P the angles between AP and DP, AP and Bq, and CP and DP, respectively. Then cos ()' = cos() cos ¢.

Proof Let a and 8 be the p-vectors corresponding to AP and DP, respectively. Denote by O:'H, av and aM the horizontal, vertical and mixed parts, respectively, = cos(} cos </J. 0 COROLLARY (Minimum). Suppose AP and Bq are subs paces of En with dimensions p and q, respectively, 1 :::; p < q < n. Then the angle (} between AP and Bq equals the minimum of the angle between AP and an arbitrary p-dimensional subspace DP of Bq.

Further Properties

Now we consider some more interesting properties of higher dimensional angles. is a metric space (see, for example, Blumenthal [START_REF] Blumenthal | Theory and Applications of Distance Geometry[END_REF]), inequality ( 5) is obtained.

For equality to hold in [START_REF] Hotelling | Relations between two sets of variates[END_REF], it is necessary and sufficient that the point B* lies on the metric segment A* C*. So there exist nonzero scalars h and k such that f3 = ha+ k1, where a, f3 and 1 are the decomposable p-vectors corresponding to the subspaces A, Band C, respectively. Since f3 is decomposable, we have

/3 1\ y = 0, y E B,
dimA' =dimE'= dimC' = p-r, A' n C' = A' n B' = B' n C' = { 0}.
Denote by a', (3' and 1' the (pr )-vectors corresponding to A', B' and C', respectively. From (6) it follows that ha' 1\ y' + k1' 1\ y' = 0, y' E B'.

One can choose a basis in En (which is not orthogonal in general) such that

a' = a e1 1\ • • • 1\ ep-n I / = Cep-r+l 1\ • • • 1\ e2p-2r,
where a and c are nonzero scalars. The vector y' can be decomposed as follows:

p-r p-r n-2p+2r Y 1 = L Uiei + L Vjep-r+j + L Wte2p-2r+t• i==l j==l t=l
So we obtain o THEOREM 4 (Complement). Suppose AP and Bq are subspaces of En with dimensions p and q, respectively, 1 :::; p < q < n, p :::; nq, Bj_ the orthogonal complement of Bin En. Denote by (J and (Jj_ the angles between A and Band between A and Bj_, respectively. Then

p~ ~r haL VjCI 1\ • • • 1\ ep-r 1\ ep-r+j + kc L Uiep-r+I 1\ • • • 1\ e2p-2r 1\ ei+ j=l i=l =0. n-2p+2r +ha L Wtel 1\ • • • 1\ ep-r 1\ e2p-2r+t+ t=l n-2p+2r +kc L Wtep-r+I 1\ • • • 1\ e2p-2r 1\ e2p-2r+t t==l Ifr < p-1, thenp-
cos 2 (J + cos 2 (Jj_ < 1 -' (7)
where equality holds if and only if either p = 1 or (J(Jj_ = O(p > 1 ).

Proof Inequality (7) follows from decomposition (3) and Lemma 4. Equality in [START_REF] Flanders | Differential Forms[END_REF] holds if and only if

C'tM = 0.
This occurs if and only if either p = 1, or p > 1 and a = a H or a = av. o Theorems 3 and 4 show some of the differences between the cases of higher and lower dimensions.

Principal Angles

Let AP and Bq be subspaces of En with dimensions p and q, respectively, 1 :::; p:::; q < n. The principal angles between AP and Bq, 0 :::; fh :::; The following theorem gives a simple relation between the higher dimensional angle (J and the principal angles 01, 02, ... , Op. M was denoted by cos{L,M} only as a symbol in [START_REF] Miao | On principal angles between subs paces in Rn[END_REF]. Now, Theorem 5 shows that this symbol cos{ L, M} is really the cosine of an angle.

Grassmann Manifolds

The set of all p-dimensional subspaces of En with suitable topology forms a Grassmann manifold G(p, n-p). The theory of angles between subspaces of En is closely connected with the geometry of Grassmann manifolds. In this way, several theorems of distance geometry can be used to obtain the corresponding results of differential geometry.

For example, by normalization, we can restrict p-vectors to have unit lengths. All these unit p-vectors form a hypersphere sN -I of 1\P En' where N = dim 1\P En = (;). Denote by G the submanifold of sN -I consisting of points corresponding to decomposable unit p-vectors. Then G is the isometrically immersed image of the Grassmann manifold G (p, n -p) into SN -t. Research on geometric properties of G is an interesting topic in differential geometry (see, for example, W. H. Chen [START_REF] Chen | The differential geometry of Grassmann manifolds as submanifolds (in Chinese)[END_REF], who pointed out that G is an algebraic submanifold). In order to obtain the immersed equations for G into sN -I' we choose an orthonormal basis { ej} in En' and then get an induced orthonormal basis { EK} in 1\P En, where Therefore a unit p-vector a can be represented as where The quantities )..K are the usual Cartesian coordinates in the induced Euclidean space EN or 1'-.P En, and the ui,iz .. •ip 's are called Grassmann coordinates of o:; they will change sign if any two subscripts are interchanged. Our immersed equations are just the conditions for a p-vector to be simple, which are known as Plucker equations and can be written out explicitly as follows: where subscripts { a1, a2, ... , ap} and {b1, b2, ... , bp} are two arbitrary arrangements of {1, 2, ... ,p}, and aj means that aj is omitted. (See Jiang [10, p. 84]; for some particular cases also see Cartan [11, pp. 18-20]).

On the other hand, according to Menger, a subset of a metric space is called (metrically) convex provided it contains for each two of its points at least one between-point (see Blumenthal [8,p. 41]). Now from Theorem 3 and its proof we infer the following result. THEOREM 6. The Grassmann manifold G (p, np) as a submanifold of a sphere is a metric space. When n > 3 and p > 1, it is not metrically convex.

Theorem 6 shows another difference between higher and lower dimensions.

0COROLLARY.

  For arbitrary a, (3 E N En, we have (a, a) ((3, (3) ~ (a, (3) 2 ,

  of a with respect to Bq. Then aH corresponds to CP, and we have Hence cosO'= l(a,<5)1 _l(aH,<5)1 lal • 1<51 lal • 1<51 laHI l(aH,<5)1 = ~ . laHI • 1<51

THEOREM 3 ( 5 )

 35 Triangle inequality). Let A, B and C be three different pdimensional subspaces of En, 1 < p < n. Denote by (} AB, (} AC and (} BC the angles between A and B, A and C, and B and C, respectively. Then (Equality in (5) holds if and only ifdim(AnBnC) = p-1 and the !-dimensional orthogonal complements A', B' and C' of An B n C in A, B and C, respectively, lie in one and the same plane and B' is placed in the smaller pair of vertical angles formed by A' and C'. Proof Denote N = dim f\P En. Let SN -l be the unit hypersphere in f\P En. Identifying opposite points of sN -1 , we obtain an elliptic space :EN -1 . Three different p-dimensional subspaces A, B and C of En correspond to three different points A*, B* and C* in :EN -1 . Then (} AB, (} BC and (} AC equal the distances between the points A* and B*, B* and C*, and A* and C*, respectively. Since :EN -1

  and hence ha/\y+k!/\y=O, yEB.

  An B n C), from fJ = ha + k1 we get dim( An C) = r.Let A', B' and C' be the orthogonal complements of An B n C in A, Band C, respectively. Then

  r 2: 2, we will get for all i,j and t. But y' 1= 0 and A' n B' = B' n G' = {0}, so we must have h = k = 0, which is impossible by assumption. Therefore we getdim( A n B n G) = p -1.Consequently, by Theorem 1, (JAB, (J AC and (J BC are equal to the angles between A' and B', A' and G', and B' and G', where A', B' and G' are the !-dimensional orthogonal complements of An B n C in A, Band G, respectively. The rest of the proof is trivial.

  02 :::; • • • :::; (}P :::; 1r /2, are given by =max{ l:l•l:l :a ..L am,b ..L bm,m = 1,2, ... ,i -1}, where a E AP, bE Bq (see Miao and Ben-Israel [4, p. 81]).

THEOREM 5 .

 5 Suppose the angle between AP and Bq is 0, and the principal angles between them are 01, 02, ... , Op. Then cos(} = cos fh cos (}2 . .. cos ()p• Proof By normalization, all these ai's and b/s can be assumed to be unit vectors. Put a= a1 1\ a2 1\ • • • I\ ap, then a is the p-vector corresponding to subspace AP, and (a, a) = 1. Noting that af = bi cos(}i, i = 1, 2, ... ,p, we get aH = cos Ot cos 02 • • •cos Opbt I\ b2 I\ • • • I\ bp. Therefore cosO = iaHi = cos Ot cos 02 • • •cos Op. 0 Remark. The product of cosines of principal angles between subspaces L and

p 2 :

 2 :) -1 )j+luajbz•••bp Ub 1 a 1 •••ar••ap = Ua 1 a 2 •••ap Ub 1 bz•••bp, j=l

  The angle(} between subspaces AP and Bq,p ~ q, is defined by

	Proof One easily verifies that
	(aH, av) = (aH, aM) = (av, O:'M) = 0.
	Ia HI (}=arccos~•
	By direct computation, we get the following lemmas.
	LEMMA 2. The decomposition (3) is invariant under transformations of basis in
	subspace A. Precisely, if in the linear subspace A we choose another basis {an, we have the corresponding p-vector a' and projections a'rr, a~ and a~, and
	a'= ka, a'rr = kaH , a~= kav , a~= kaM
	for some nonzero scalar k.

LEMMA 3. (a, aH) 2:0, where equality holds if and only ifaH = 0. 0 DEFINITION.

Acknowledgement

The author would like to thank the referee for many valuable suggestions and helpful criticisms.


, the famous novel by Agatha Christie, takes place between two spaces and combines two opposing forces: Europe with its homeward pull, and a hazy Orient, its vector Istanbul, the vanishing point as if travelling backwards. The story starts in Syria and finishes in Yugoslavia, covering a period of four days, passing through towns, stations, mountains, straits, frontiers, regions, countries and even continents, between Aleppo and Vincovci. The explicit area of reference is vast, but this is implicit; Murder on the Orient Express could have been a travel novel, but the journey is not geographically conventional.

The route is from Istanbul towards Europe. However, between the starting point of this enclosed--setting journey and the destination there is a hazy somewhere. The mythical train, streaking across the unfolding landscape, stops, frozen in time and space. Here, somewhere that is precisely nowhere, on the borders of the Orient and Europe, within the confined space of the train, the vendetta and the journey come together. We know that time never stops; we know less that neither does space. The immobilization of the Orient Express provides the literary possibility of simulating these joint impossibilities.

MURDER ON THE ORIENT EXPRESS, A LITERARY AND TRAIN JOURNEY FROM

ISTANBUL TO EUROPE ANNA MADOEUF "If you want to express That kind of gloom You feel alone in a double room […] Waiting at the Station For a near relation Puff, Puff, Puff, Puff The Istanbul Train…." Graham Greene, Orient-Express, 1979, p. 285 (cabaret song).

The title and setting of Murder on the Orient Express, one of Agatha Christie's best-known detective novels, published in 1934, are taken from this emblematic train 1 . The best-selling novel is both inspired by and part of the myth, contributing to the fantasy about the king of trains, the train of kings 2 . Many international celebrities have travelled on it, including the Emperor Haile Selassie, Josephine Baker, Leopold II, Mata Hari, Isadora Duncan, Lawrence of Arabia, Agatha Christie herself, and… James Bond 3 . The Orient Express, "the Maltese Falcon of international express trains" 4 , is without doubt the train that has been the source of the greatest literary inspiration, for example in the work of Joseph Kessel, Paul Morand, 1 Quotations from the book are taken from the edition published by Fontana 1959Fontana (16th impression, 1976). 2 A film of the novel was made by Sidney Lumet in 1974, with an all-star cast. More recently, a number of Agatha Christie's novels have been produced as comic-strip books, including Le Crime de l'Orient-Express, by François Rivière and Solidor, published by Emmanuel Proust in 2003. There is also a video game named after the train and which uses its décor as a setting. 3 In the film From Russia with Love (1963), based on the novel by Ian Fleming (1957). 4 As described by Douglas Kennedy, 1988, Beyond the Pyramids.

Colette, Graham Greene, Vladimir Nabokov, John Dos Passos, and Ernest Hemingway. In fact, the story of the Orient Express began at its inauguration, with the journalist Edmond About's account of the epic initial journey on 4 th October 1883, when it steamed out of the Gare de l'Est (known at the time as the Gare de Strasbourg) in Paris to the sound of Mozart's Turkish March, for a journey of approximately 80 hours and 3200 kilometres 5 . From its creation to the present day, it has been the setting and source of endless stories, from journalistic accounts to novels, from history to fiction, with themes of adventure, luxury, politics, dreams, exoticism, travel, or nostalgia for times past 6 .

The saga of the Orient Express, from the end of the 19th century to the middle of the 20th century, is one of fantastic and epic episodes, of minor events and moments of history. Passing through a number of unstable areas and time zones, the line and the train have been affected on many occasions by geopolitical upheavals 7 . The emblematic Orient Express, linking West and East, Europe and its borders, is a line drawn between two worlds. However, it is generally represented as travelling in a single direction -towards Constantinople, its outward destination. While the arrival of the railway undoubtedly changed the shape of the world in general, the Orient Express also helped shape the way the world is perceived by the West, pushing back or blurring the borders of Europe and the exotic East. Of course, "when we take the train, it's to go from one town to another" (Pérec 1974, 85), but taking the Orient Express was to go from a city in Europe to the target city of Stamboul. The Orient Express formalized the direction from Europe, perceived as the point of departure, the beginning, the centre of the world, an autonomous universe, to Stamboul, which was not only the terminus of the train but also the threshold of another world. The train also changed both the image and the 5 The Orient Express was the brainchild of a Belgian businessman, Georges Nagelmackers, who founded the Compagnie Internationale des Wagons Lits et des Grands Express Européens in 1884. The train was then called the Express d'Orient and was renamed the Orient Express in 1891. 6 As still shown today by the success of the exhibition "Il était une fois l'Orient-Express" at the Institut du Monde Arabe in Paris (April -August 2014). 7 It was in one of the train's carriages that both the armistice of 11th November 1918 and the capitulation of France on 22nd June 1940 were signed. The route has varied considerably, from the initial route through Strasbourg, Munich, Vienna, Budapest, Bucarest and Varna (crossing the Black Sea), to the most southerly route (following the opening of the Simplon tunnel) taken by the train on which Hercule Poirot travelled (Belgrade, Trieste, Venice, Milan, Lausanne, Paris Gare de Lyon).

geography of Stamboul-Constantinople, henceforward a city at the edge of Europe, the horizon of the Orient, a disembarkation point for travellers who suddenly arrived in the city in large numbers. For example, Pierre Loti, in 1890, deplored the new influx in Stamboul of European tourists, "disgorged by the Orient Express"8 . Associated with the train, luxurious hotels were built in the host cities, such as the mythical Pera Palace of Tepebasi in Constantinople, built in 1892 and inaugurated with great pomp by a grand ball in 1895. Finally, the very name of the train combines two apparently incompatible words, given that the Orient seemed to be a different world, subject to and associated with an idea of a specific time. It is maybe for this reason that the Orient Express is described as the train that "makes haste slowly". Above all, is the Orient Express not a mystification, a terminological imposture? The train will never keep its promises, being neither express nor doing more than brush the Orient.

The train journey predicted by Agatha Christie's novel takes connects Istanbul to Europe. However, between the city that triggers that fictional in camera and the place of destination, a blurred somewhere, shrouded in snow, will appear, in the Far East. The mythical train that moves along unfolding landscapes and places will suddenly stop. In the confined and tubular space of the Orient-Express, that sudden stop will expedite the crime promised in the title.

The plot of Murder on the Orient Express may be original, but the singularity of this narrative lies somewhere else, in the mechanisms of the staging of space, time and the illusion of movement.

My aim is thus to draw a parallel between that geo-literary mechanism end the creations of the Alexander Calder, carver (1898-1976) that were instigated and animated by that triptych (space/time/movement). Calder did indeed create constructions that were said to be mobile for some of them (as they were suspended in the air), and stabile for others (as if they were suspended on the ground), or even mobile and stabile at the same time (hybrid shapes).The stabile can only be imagined as a follow-up of the mobile, being in a way its antonym. The latter only exists through its instability, whereas the stabile ossifies in its own object a denial of fixity. Between the two possible shapes, all the asymmetrical tones of an unbalanced balance, and vice versa can be observed, from the most constructed to the most unstructured. These objects, mobile and stabile, also denote an obsession with time, in the questions it raises, in its manifestations, expressed in his way by Calder, the artist who "carves times", as Jacques Prévert said, (1971,3). The movement of these constructions is time, whose energy is created by mobility, whose evolution is composed of the interaction of stable and mobile things. This essay will hence make use of the qualities of these artefacts to imagine the action mechanisms, the places and the landscapes of Murder on the Orient Express as determining and interpretative configurations of spatial situations. For that aim, we need to travel back to the beginning of the 1930s, have a seat in that fictional train leaving from Stamboul and going to the suspense of the criminal plot imagined by Agatha Christie.

"Constantinople -Trieste -Calais"

In contrast to the pattern mentioned above, Agatha Christie's novel 9 is oriented and functions in the reverse direction, towards Europe, in other words it takes place on the homeward journey as the travellers set off from Stamboul 10 . The boarding point is the station of Sirkeci; the coaches are attached, the characters board, the train and the plot set off in unison. The construction and the plot are classic, based around a small self-centred and exclusive world (a group of westerners), confined and transported across a vast area, the ideal enclosed setting. It comes as no surprise that there are no Turks in Murder on the Orient Express, just as there are no Egyptians in Death on the Nile, another enclosed-setting mystery by Agatha Christie involving an exotic journey, this time involving a river cruise 11 . Setting the intrigue in a train does away with the need for natives; the characters can picture themselves and talk about themselves, highlighted against the foreign surroundings and amongst people who are different from them and who become mere shadowy figures. On the other hand, the foreign lands provide the perfect foil for the western characters and the scenes and intrigues in which they are involved. In the Orient, even if here it is subliminal and only hinted at, the imported/transposed characters stand out sharply, like transfers placed on a pre-drawn background. There are neither Orientals nor the Orient; both can be dispensed with, because the Orient Express itself conjures up the myth of the Orient, whose representation reached its zenith in the 1930s or was at least sufficiently eloquent to be able to do without its subject matter.

The detective Hercule Poirot boards the Simplon Orient Express bound for Europe, a 3,000-kilometre journey lasting three days, in a carriage bearing a plate indicating the destinations "Constantinople -Trieste -Calais" 12 . From then on, the plot takes place in a closed setting, and the train functions as a net ensnaring the characters, who are of different nationalities, social classes, places of residence, ages and fortunes. This multiplicity of identities is plausible in a train, but in the novel, the characters were all connected in some way, and, unbeknown to Hercule Poirot, they already knew each other.

The train envelops the passengers/characters, detaching them from the outside world, its torments and storms; the train is the only place, an absolute, a world in itself. At the same time, the train, together with its coaches, compartments, restaurant, staff (subsidiary characters there to serve the travellers), way of life, conventions and protocol, all retain strong references to social class and identity; rather than being held in abeyance, this aspect is highlighted and rendered almost mechanistic by the confined space and the inherent cramped conditions and lack of privacy.

(1938) between Jerusalem and Petra. They came to Baghdad (1952), published after the war, whose heroine is a young woman from London, is a spy story set in Baghdad.

12 The Simplon tunnel in Switzerland was built in 1906. After the First World War, the train was called the Simplon-Orient-Express, and its new route was London -Calais (or Boulogne) -Paris -Vallorbe -Lausanne -Simplon -Milan -Venice -Trieste -Zagreb. At Vinkovci, there were two alternative routes: Bucarest-Constanza-Odessa and Belgrade-Constantinople-Athens.

From station to station; an abstract, fleeting landscape

Like the dividing of the Red Sea as the Hebrews crossed over, the landscape appears and disappears hypnotically and majestically, simultaneously emerging and dividing on either side of the train that confronts, strikes and flees it in a single movement. Here, the unfolding landscape, composed of two asymmetrical and yet perfectly parallel horizons, is singularly devoid of depth of political or social meaning. It is detached, an absolute, simultaneously created and erased by the train along a given stretch of the track. However, it creates an atmosphere; landscape and time go by, images and moments linked in a perpetual fleeing procession. It is the stations that punctuate the stages of the journey, and also where the shuffling ballets and codified rituals of boarding and alighting occur, but they are not treated as features of the modernity of the time, obligatory thresholds of passage and arrival, but rather as non-places breaking the journey. Aleppo station, for example, is described as follows: "Nothing to see, of course. Just a long, poor-lighted platform with loud furious altercations in Arabic going on somewhere." 13 (p. 9). Belgrade is dismissed in a similarly terse manner:

The Simplon Orient Express arrived at Belgrade at a quarter to nine that evening. It was not due to depart again until 9.15, so Poirot descended to the platform. He did not, however, remain there long. The cold was bitter and though the platform itself was protected, heavy snow was falling outside. He returned to his compartment. (p. 27)

The two platforms, symbols of the cities quoted, are uninteresting and hostile: foreign voices and no lighting on one, cold and snow on the other. The stations are not used to introduce reminiscences or parallel erudite comments. They merely serve as props to re-set universal time, incidental, ill-defined spaces, apparently imperfect, which are neither exotic nor firmly standardized. The station is "outside and inside" the town (Sansot 1973, 85), but here, is it even part of the country to which it belongs? Sirkeci station in Istanbul, even though it is where the story starts, is not even mentioned; it is just the departure point, and the departure itself is only indicated by the movement of the train along the platform: "There was a sudden jerk. Both men swung round to the window, looking out at the long, lighted platform as it slid slowly past them" (p. 20). A station is just a station; it can even be reduced to a platform. Ultimately, a country is concentrated into a city, which in turn becomes a station, summed up in a platform, which becomes a track. In other words, it is a synthesis of the entire space, masterfully reduced to a geometric outline. And yet, Agatha Christie's story is based on a remarkable journey, starting in Syria on a Sunday morning and finishing in Croatia (then Yugoslavia) the following Wednesday. In only four days, they travel a dizzying 2,000 kilometres (from Aleppo to Vincovci) through stations, towns, plains, mountains, straits, regions, frontiers, countries and even continents; explicitly a vast area, but only implicitly hinted at. Murder on the Orient Express has all the appeal and appearance of a travel novel, but it is far from being such, or at least, it does not provide any geographical account.

As observed by Guy Laflèche when analysing the novel as material for a narrative grammar: "The geography is that of crossword puzzles" (1999,104). In fact, only nine places are named, but not described: six stations (Aleppo, Konya, two in Istanbul -one in Asia and one in Europe, Belgrade and Vincovci), a mountain range (the Taurus), a hotel (the Tokatlian in Pera), and "somewhere" (between Vincovci and Brod). The stations are simply mentioned as indications of place and time, setting the scene and providing a way of checking both the places along the route and the punctuality and progress of the train. The landscape is clearly there, but it is incomplete and the description is terse; it has no narrator and is inaudible and hence invisible. When it is beautiful, which only occurs once (the Taurus), we only learn about it indirectly, in a roundabout way, like a reflection in the glasses or eyes of the travellers: "Later they passed through the magnificent scenery of the Taurus. As they looked down towards the Cilician Gates, standing in the corridor side by side, a sigh came suddenly from the girl" (p. 12). The time frame is equally illdefined; while the weather is constantly mentioned, notably the type, variation and intensity of the cold, there are few chronological pointers. Indeed, no date (year, month, general external information) is given or hinted at, while details of the day of the week, the time -down to the minute -constantly set the pace of the plot; a detailed account given with pointillist precision against a blurred background.

Somewhere that is nowhere

This was not Agatha Christie's first railway crime novel, nor the last, but it is the only one in which the train is the actual murder setting. The drama occurs fairly early on, somewhere and/or sometime after Belgrade, in a nebulous area on the way to Europe. The train stops, trapped in the snow; and then the action unfolds. We know that time never stops; but we are less familiar with the fact that neither does space. However, the stopping of the Orient Express -sudden, unexpected and at night -makes it possible to simulate these joint impossibilities. First, the train stops at some ill-defined place in Croatia -"somewhere between Vincovci and Brod" -at a vague geographical point about 200 km west of Belgrade14 . Standing irresolutely between two places that serve only as landmarks, the train is temporarily in what is clearly an incongruous no-man's-land, in a landscape that is also a negation, that no longer exists because it is nothing but a shapeless mass of snow, white on white, matter on matter. The space, previously drawn in, stretched out and narrowed by the track, is now resorbed, reduced to a shapeless and monochrome mass. Is not white the absence of colour, and cold a suspension of climate and the neutralization of the outside world? The combined effect is to dilute space and absorb time; all that is left is an over-abundance of snow. The snowdrifts are thus both an abstraction and a totality: the absence and neutralization of time and space. Ultimately, the snow becomes time and landscape. It folds itself around the train, like a wrap. The combination of this effect and the unforeseen events makes the Orient Express a complete socio-spatial-temporal object. The train stops suddenly, an event over which the protagonists have no control, but the action carries on and culminates in the murder act, which is a continuation of the movement and thus resembles "a suspension", to use dance terminology (Roux, 2007, 42). An unlikely plot, it is true, but above all, the actions and setting are in phase with the train, which is in a state of equilibrium and disequilibrium between two suspended worlds and two poles, one the impulse driving it from Istanbul station, and the other inexorably pulling it towards its final destination, even if the force of attraction is temporarily hampered by the proliferation of snowflakes.

It should be remembered that this murder is unusual in two ways: it is collective, each of the twelve assailants stabbing the victim, who turns out to have been a murderer himself, and the murder will remain unpunished because Hercule Poirot will not reveal the culprits. The Orient Express is also a determining factor of the unusual clemency and directions of the detective/upholder-of-the-law, breaking with his usual pursuit of reason, he too perhaps under the spell of the train. Can a deed carried out in a noman's-land located in an area under oriental influence really be subject to a criminal court? In this case, can the law of retaliation, first mentioned in an engraving on the Hammurabi stele in Babylon, not prevail as the norm, and vendetta not be justified as a form of just vengeance?

Conclusion: There (easy) and back (difficult)

A straightforward outward journey, a complicated return; it is perhaps the train that invented, or at least formalized, a binary rhythm of travel, breaking it down into two separate journeys, with different directions and periods of time. Two starting points, polar opposites in a scalar sequence of events; two sequences materialized by the elastic movement and direction of the rebound: there and back. The return journey is essential, and any delays or disruptions are feared and dramatized. With great subtlety, Agatha Christie set her story explicitly in the context of the return journey. This defused and forestalled any excitement associated with travelling, preventing it from disturbing or interfering with her railway plot. The return is not a journey of discovery, and thus the meaning of the journey loses its interest and becomes dull, blunting the vividness of the landscape. Taking a journey means going somewhere, but what about coming back? The return is the finalisation of the journey.

"Snow?" "But yes, Monsieur. Monsieur has not noticed? The train has stopped. We have run into a snowdrift. Heaven knows how long we shall be here. I remember once being snowed up for seven days." "Where are we?" "Between Vincovci and Brod." "Là là", said Poirot vexedly. (p. 31)

The murder -the ultimate crime, an expression of total social dysfunction, a symbol of cessation, of the end -is committed in the train, an object-movement whose purpose is to move ceaselessly, at a constant speed, following a predetermined route along an indelible and unalterable track. The train in question is the supreme Orient Express, an additional vector of disorder, bearing the enchantment and the curse of the mysterious and threatening world conjured up by its name. Agatha Christie thus made the Orient Express a literary friction motor, as well as a composite object, a "mobile-stabile" like the works of Alexander Calder, both supported in space and suspended from another of its possible dimensions.

In a somewhere that looks like nowhere, in the Far East, at the limits of Europe, in a blurred zone, in the confinement of the train, the crime took place. That thriller promised an event (the murder) and a setting (The Orient Express), and the relation between the two. And the promise is kept: Murder on the Orient Express is indeed a geographical narrative and the literary illustration of particular situations of spatiality generated by the train, a mechanism that links fixity and movement.

  13 A comment made at the very beginning of the novel (p.8), when Hercule Poirot boards the Taurus Express in Aleppo bound for Constantinople (which makes up the first part of the novel). The Taurus Express, inaugurated in 1930, crossed Anatolia and linked Constantinople to Aleppo, where it branched into two, one line going to Damascus, the other to Baghdad.

Pierre Loti, Fantôme d'Orient, p. 57.

Agatha Christie (1890-1976) travelled to Baghdad in 1928; returning from Nineveh in 1931, she took the Simplon Orient Express in the other direction and was delayed by problems caused by bad weather. She used this experience and an episode in January 1929 when the train was snowed up in eastern Thrace for four days. The plot was based on elements of the sensational Lindbergh affair (the kidnapping and murder of the son of the famous American aviator in 1932), as the victim of the Murder on the Orient Express, Samuel Edward Ratchett alias Cassetti, had murdered a child.

In the novel, the city is called either Stamboul (name of the old town of the historic peninsula) or Constantinople, a name that was no longer used after 1926, when the name Istanbul was imposed by the Turkish authorities but never in fact adopted by Agatha Christie.

Agatha Christie married the archeaologist Max Mallowan in 1930, and subsequently, several of her most famous novels were set in the East, where she spent long periods. After Murder on the Orient Express, Hercule Poirot figured in other oriental mysteries: Murder in Mesopotamia (1936) set in Tell Yarimjah (Iraq), Death on the Nile (1937) in Upper Egypt, and Appointment with Death

Vinkovci and Brod are towns in Croatia, 63 km apart, located respectively in the county of Vukovar-Srijem and in Slavonia. In relation to the route of the train, Vinkovci is

km from Belgrade going towards Zagreb.

Summary of the places and times in Murder on the Orient Express

-After dinner, the travellers assemble in the restaurant car where Hercule Poirot draws his conclusions and where the novel ends.