Tactile Display Using Elastic Waves
Takaaki NARA, Taro MAEDA, Yasuyuki YANAGIDA and Susumu TACHI
Faculty of Engineering
University of Tokyo
Bunkyo-ku, Tokyo
nara@star.t.u-tokyo.ac.jp

Abstract

As a basis for tactile display, it is important to understand the mechanism of tactile sensation. So, we
analyze elastic waves inside a nger in tactile exploration with a two-layered, half-in nite elastic ngermodel, and derive a mapping from shapes of objects todisplacements of mechanoreceptors. It is shown thatthe mapping from displacements of the nger surface
to displacements of mechanoreceptors is a one-to-one
mapping. Therefore, it is necessary to create the same
displacements of the nger surface as in actual exploration for tactile virtual reality(VR). However, themapping from shapes of objects to displacements of thenger surface is shown to be a many-to-one mapping
because of the mechanical property of skin. So, we use
the envelope of the amplitude-modulated Lamb wave
on an elastic plate as an alternative shape to the real
object, because its shape is easily controlled. The generation of the wave is con rmed with an experimental
device of silicon rubber vibrated by voice coils.
1

Introduction

The purpose of this study is the development ofa tactile display, which provides the feel of roughness,ruggedness, or small surface textures of objects in tactile exploration. Many tactile displays using variable
actuators have been proposed [1]-[4]. However, physical phenomena of tactile sensation have not been adequately analyzed, therefore it is dicult to predictphysical states inside the nger, and tactile sensation
as a result, which are evoked by some stimulus generated by the display. The study of the mechanismof tactile sensation is important because it delineates
what should be displayed for tactile VR.
Static skin deformations are analyzed in detail withthe elastic nger model.[5] [6] However, dynamic phenomena should also be investigated because tactile
sensation is obtained by active exploration. Indeed,a tactile exploration gives a spatio-temporal displacement, or stress distribution on a nger surface, which

generates elastic waves inside the nger. Mechanoreceptors embedded in the upper layer of the nger detect the quantity of the wave, and as a result, they
are activated and transmit signals to the brain. Thisis a sketch of how tactile sensation occurs, thus elastic
waves in the nger are essential for tactile sensation.
In this paper we analyze the elastic waves inside thenger using a two-layered, elastic half space model ofthe nger.
Now, let us consider that tactile sensation is obtained because a physical state of the outside world is'mapped' on a physical state of the mechanoreceptors.
Concretely, the state of the outside world is a geometric shape of an object, and the state of mechanoreceptors is a set of their displacements. Then, the expression of the mapping from the shapes of the objects
to the displacements of the mechanoreceptors can bederived by the analysis with the elastic nger model.
Here, it is a problem for VR whether the mapping
is one-to-one or many-to-one, because a many-to-one
mapping can produce the same sensation from two
di erent shapes of the objects. (Fig.1) Although generating an arbitrary shape instantly is very dicult,the possibility exists that the real sensation is evokedby the substitute shape which is created more easily
than the real shape, if the mapping is many-to-one.
A brief overview of this paper is as follows. First,
in the analysis with the two-layered elastic half spacemodel, the mapping F from the displacements of theskin surface to displacements of mechanoreceptor isderived. It is shown that F is one-to-one because of
the characteristic structure of the nger in section 2.
Second, the mapping C from shapes of objects to thedisplacements of the skin surface is considered, andshown to be many-to-one because of the mechanicalresponse character of the skin (section 3). Then, wesuggest a tactile display using the many-to-one property of the mapping. The essense of the method isthat deformation of the nger surface in touching theamplitude-modulated elastic wave is the envelope ofTC
Shapes of objects

F
Finger surface Mechanoreceptors
Displacements Displacements

Many-to-one

One-to-one

So, in this paper, a two-dimensional problem is discussed with the two-layered, elastic half-space ngermodel shown in Fig.2. We set x0axis parallel to theskin surface, and z 0axis perpendicular to the skin surface. The plane, z = 0, is the boundary between the
rst(upper) and the second(lower) layers. The rst
layer represents epidermis and dermis with a depthof H , and lays in the region 0 <z <H . The second
layer represents the subcutaneous tissue and lays in
the region z < 0. Both layers are assumed to be homogeneous, isotropic, linearly elastic media, and havedi erent elastic constants respectively.

The real shape
The shape
which is generated in the display

Figure 1: VR by Many-to-one mapping. C is determined by contact between the object and the skin.
F is determined by the elastic media in the nger.
T , which is a composite mapping of C and F , represents the total tactile operation.
the wave. Because a wave length and a travelling velocity of the envelope are controllable, the arbitrary
displacement of the nger surface, and the arbitrary
displacements of the mechanoreceptors as a result, canbe formed. The experimental device with which thebasic phenomena of the above principle can be conrmed is described in section 4.
2

Analysis of tactile sensation

2.1 Two-layered elastic half-space modelof the nger
Mechanics of tactile sensation have been analyzed
with models of an elastic media of a nger to predice
the neural responses evoked by the stimuli applied toa skin surface. [5] used a homogeneous, isotropic, incompressible, linearly elastic half-space model of a nger. This simple and idealized model predicted SAI response pro les well, but it treated only static stimuli.
[6] also analyzed static problems with more detailed
models composed of linear elastic media of variable
shapes by nite element calculations.
Dynamic mechanical properties of the skin has beenstudied primalily for medical modives.[7]-[11] However, the most detailed analysis with a layered elastic model of the nger in [11] has focused only on SH
waves which vibrate parallel to the skin surface. The
more general analysis dealing with deformations parallel and perpendicular to the skin is required for thephenomenon of tactile sensation.

The first layer
Epidermis
and DermiszwH (x,t)z=Hu H(x,t)

z=z

F

x

0
Mechanoreceptor

wz (x,t)uz (x,t)The second layer
Subcutaneous tissue

Figure 2: Two-layered half-in nite elastic modelof the nger.

When tactile exploration is conducted, twodirectional (x0 directional and z0 directional) displacements, or stresses, are given on the nger surface as a boundary condition. Because stresses anddisplacements at z = H can be transformed to eachother, we take boundary conditions of displacements
here.
The following notation is used:01

u i ; wi(uH ; wH )(uz ; wz )i ; i
i
i
ii: Displacement in the i-th layer(i = 1; 2)= (u1 ; w1 )jz=H= (u1 ; w1 )jz=z: Lame's Constants: Rigidity: Density: P-wave velocity: S-wave velocity

The problems dealt with in section 2 are as follows:First, a dirivation of the mapping F from (uH ; wH )to (uz ; wz ). In other words, we express the displacements of the mechanoreceptors in terms of the displacements of the skin surface. The result is (26) insection 2.2. By using this result, an amplitude distribution inside the nger during the exploration of asinusoidal shape is shown in section 2.3.
Second, a discussion of whether F is one-to-one or
many-to-one. (section 2.4) The result is one-to-one
because of the characteristic structure of the nger|
the upper layer being sti er than the lower layer; 1 >
2 .
2.2 Derivation of the mapping F
2.2.1 Basic equations

Displacement potentials in the i-th layer: (i ; i ),which have a relation with (ui ; wi ) as
@i @ i
0 ;
@x @z

ui =

wi =

@i @ i+ @x ;@z(1)

satisfy the wave equations
1 @ i = r2  ;
2

i @t
2

2

i

1@
2

i

2

@t2

i

= r i:
2

Next, the displacement boundary conditions at z =H in tactile exploration.

 (u1 )z=H = uH = u3H ej (kx+!t) at z = H (10)
 (w1 )z=H = wH = wH3 ej (kx+!t) at z = H (11)(6) and (8) reduce to the expression of P; B in termsof P 0; A0 as(7), (9), as


Q
A



i

i

(5)

If unknown coecients P; Q; A; B; P 0; A0 in (3),(4)can be expressed by (uH ; wH ), we substitute theminto (3),(4) and (1), so that we obtain the expression(uz ; wz ) in terms of (uH ; wH ).
2.2.2 Boundary conditons

The solution assumed in (3),(4) should be subject tothe following six boundary conditions.
First, the continuity conditions of stress and displacement at z = 0.






(Tzz )1 = (Tzz )2 at z = 0(Tzx )1 = (Tzx )2 at z = 0
u1 = u2 at z = 0
w1 = w2 at z = 0

(6)
(7)
(8)
(9)

1

0

r2
k


k 2 B r1 Y r1 Z C P 0= k 2 @ r2A
A0 ; (13)W X
1kwhere ; X; Y; Z; W are de ned as
2
;
1
k2
 22 0 2( 0 1);
k2( 0 1);k2 k2
 22 0 21 0 2( 0 1);
kk
k2 1+ 2( 0 1):
k2

 =

(2)

2 = P 0 er2 z
(3)
0
s2 z
(4)
2 = A e
!
!
k i=
ki=

X
2
= kk2 B
@ k
Z
1
s1



X =
W =

The solution of (2) is assumed as follows:[12] (Here,ej(kx+!t) is excluded.)1 = (P cosh r1 z + Q sinh r1 z )1 = (A cosh s1 z + B sinh s1 z )qq
ri= k 2 0 k 2 i si= k2 0 k2i

0s2 1 

W
P
P0
k C
A
s2
A0 : (12)BY
s1
In the same way, Q; A are expressed by P 0; A0 from


Z =
Y =

(14)
(15)
(16)
(17)
(18)

On the other hand, (10), (11) are reduced to






0k cosh r1 H s1 cosh s1 HP
r1 sinh r1 H 0k sinh s1HBsinh r1 H s1 sinh s1 H
Q
+ 0r kcosh
r1 H 0k cosh s1H
A
1
=



u3H
wH3



(19)

;

by using (1),(3),(4). Substituting (12) and (13) into
(19), we obtainGz jz=H 1



P0
A0

where
Gz =



k 2  u3H 
wH3 ;

= k31



g1 g2
g3 g4



;

(20)

(21)

P0
A0



k21 01  u3H = k3 GH w3
H :

(23)




P 0 = k 21 u3z
A0
k3 wz3

:

(24)

u3z
wz3



= Gz 1 G0H1 1




u3H
wH3 :

(25)

Consequently, when det G0H1 is not equal to 0, themapping F from (u3H ; wH3 ) to (u3z ; wz3) is obtained as
F = Gz 1 G0H1 ;(26)
where Gz is given in (21),(22).
2.3 Example; Application of the mapping
F to a sinusoidal boundary condition
Let us consider tactile exploration at the constantvelocity v0 . As a basic fourier component of thedisplacement of the nger surface at that time, wetake a sinusoidal z0 directional displacement with thespatio-temporal frequency (k0 ; !0 ), where !0 = k0 v0 .
That is to say, the inverse image of F is given as




uH
0
=
wH
A cos K (x + v0 t)
 3 


u
0
H
() w3
=
A(k 0 k0; ! 0 k0 v0) (27)H:
Then, applying the mapping F in (26) to (27), we
can obtain the basic amplitude distribution inside thenger in tactile exploration.(Fig.3)

Therefore, substituting (23) into (24), we obtain

Gz 1



Now, (19) holds not only at z = H but also atarbitrary z :(22)Hence, if det GH 6= 0 (This is shown in 2.4), (20) is
solved for (P 0; A0 ) in the form

g1 = 0X cosh r1z + Z cosh s1 z
0 rr2 Y sinh r1 z + sk1 r22 W sinh s1 z;
1s2sg2 = 0 W cosh r1 z + 2 Y cosh s1 z
kkk0 r Z sinh r1 z + sk1 X sinh s1 z;
1r2
rY cosh r1 z 0 2 W cosh r1 z
g3 =
k
k
r1
+ k X sinh r1 z 0 sk Z sinh s1 z;
1
g4 = Z cosh r1 z 0 X cosh s1 z
+ rk1 s22 W sinh r1 z 0 ss2 Y sinh s1 z:
1In[6]:=

(z=0)

0.5

1 (z=H)wz

2.4 The mapping F is one-to-one
Let us consider the boundary condition of a traveling wave with velocity v0 , where v0 < i ; i. Then,the mapping F in (26) holds for the next theorem.

Theorem If  = 2 < 1 in the two-layered elastic
1
half space, F is a one-to-one mapping.
This theorem shows that the displacement inside a
two-layered elastic medium is uniquely determined by
the displacement of the surface if  < 1, that is, theupper layer is sti er than the lower layer. Accordingto [11], the human nger has
11 = 6 2 105 [Pa]; 2 = 2 2105 [Pa]; hence  = < 1:
3
Thus, the nger satisfys the assumption of the theorem, and the mapping from the displacements of theskin surface to the displacements of the mechanoreceptor is one-to-one. Therefore, it is necessary to repoduce the real nger surface displacement in orderto generate the same mechanoreceptors displacements
as in real tactile exploration.
In the proof of this theorem (Appendix), the condition,  < 1, is essential. When  > 1, the example that F is many-to-one can be constructed[15]. It
should be noted that human skin has  < 1 structure
which does not lose the quality of tactile information.

0A
H

Figure 3: Amplitude distribution inside the nger in exploring a sinusoidal shape. (A=0.1[mm],H=1[mm], 0 = 2k =1[mm])<<2layamp.txt

uz

k z
H

0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04

3

The principle of Many-to-one VR

According to the theorem above, the displacementof the nger surface should be identical with the displacement in real exploration for tactile VR. Then,before the mapping F , the mapping C from shapes ofobjects to displacements of the nger surface is considered next. C is determined by physics of contact
between the skin and the object. So in this section,
we discuss the many-to-one property of C , and investigate a possibility to form the same displacement of
the nger surface as in real exploration without generating the real shape of the object. (Fig. 1)
3.1 Amplitude modulated, almost sinusoidal traveling wave
Let us consider the superposition of traveling waves
with spatio-temporal frequencies (k1 ; !1 ) and (k2 ; !2 ).
The shape of this wave isCSpace ofFinger surface
Displacements

2 ω0ω (time frequency)

Space of
Object Shapes(k2 ,ω2)ω(k1 ,ω1)

2k0

The envelope wave

(k0 ,ω0)(k0 ,ω0)kk(space frequency)

Figure 4: The many-to-one property of C

3.2 The non-lineality of the skinLet us consider the skin model composed of a springcos (!1 t 0 k1x) + cos (!2 t 0 k2 x)
(
k),
a mass (m), and a damper ( ). The essence of the= 2cos !1 02 !2 0 k1 02 k2 cos !1 +2 !2 + k1 +2 k2 : contact is a one-directional transmission of the force;that is, the object can only push the skin inside. Thus,(28) the movement of the skin is the damped oscillation
when the vibrating object is removed from the skin.
The rst factor of (28) represents the envelope wave
Therefore, the skin detects the envelope of the disof the amplitude-modulated, almost sinusoidal travelplacement of the object vibrating with an adequately
ing wave whosehigh frequency. Indeed, according to [7],
 = 103(when the frequency is 100[Hz]) spatial frequency is kmod = k1 02 k2 ;52103 (when 1[kHz]) [dyne sec/cm2 ]!1 0 !2 temporal frequency is !mod = 2 ; m = 1[g/cm3 ]!2k modulated velocity is vg = !k mod = !k1 0
0 k (29)  =2230  2240 [1/s].
mod

1

2

Now we consider touching the amplitude modulated
traveling wave with the nger. If the skin detects
the envelope of the wave, an arbitrary displacementon the nger surface can be produced because thefrequencies (kmod ; !mod ) is arbitrarily controllable by(k1 ; !1 );(k2 ; !2 ) as shown in (29). The advantage ofthis method is unnecessity to prepare all the objects
with arbitrary shapes.
The following two points must be con rmed for thismethod.

 The skin detects the envelope of the amplitude-

modulated traveling wave. In other words, C has
the many-to-one property. ! Discussed in 3.2.

 Amplitude modulated waves whose frequencies(kmod ; !mod ) are controllable can indeed beformed in some medium. ! Discussed in 3.3.

:m=) 0 +

p

0 4mk ' 50[1/s]  10[Hz]:2m
Hence, the skin can detect the vibration with a frequency of more than 10[Hz].
3.3 The method to form amplitudemodulated traveling waves { using theelastic wave called 'Lamb wave'
We use Lamb waves to form amplitude-modulated
traveling waves. The Lamb wave is a traveling wave in
an in nite elastic plate. There exists in nite symmetric(longitudinal) and anti-symmetric( exural) modes,
whose dispersion curves are shown in Fig.5.
To form the envelope wave with modulated velocity
is v0 , a line is drawn with the gradient v0 = !k 0 on the0frequency plane. Let (k1 ; !1 ); (k2; !2) be the frequencies determined by the intersecting points of the line2and the dispersion curves. Then the superposition of
the two Lamb waves with frequencies (k1 ; !1 ); (k2 ; !2 )provides the envelope wave with modulated velocity
v0 .
Especially for v0 = 0, the immovable envelope isobtained by the two Lamb waves which have the same
temporal frequencies.
An arbitrary wave number k0 is obtained by theparallel movement of the line. For small k0 , we choose
the intersecting points on the dispersion curves of twomodes which approach the same asymptote at highfrequency. For large k0 , the rst and the higher modes
are used.
From the above discussions, we obtain the principleof the tactile display using elastic waves as follows:
The principle of the tactile display

To provide tactile sensations of exploring an object with the spatial frequency k0 at the velocity v0 ,we superpose two Lamb waves whose envelope wave
has the spatial frequency k0, and the modulated velocity v0 . This condition is ful lled by generating two
Lamb waves with spatio-temporal frequencies (k1 ; !1 ), (k2 ; !2 ) which satisfy
k1 0 k2 = 2k0 ; !1 0 !2 = 2!0 :Lamb waves
to be superposedω.......
V1(k1 ,ω1) Gradient v0
2ω0
2k0

R
ω0

Experiments

4.1 Construction
Studies on the ultrasonic linear motor using elastic
traveling waves provide clues to forming Lamb waves
on the elastic plate.[14] Although the Lamb wave witha single frequency is vibrated on the metal plate (orstick) in the motor, many Lamb waves of di erent
modes should be vibrated at the same time in thetactile display. Here, the rst symmetric and the rst
anti-symmetric Lamb wave are used. The construction of the display is as follows:
Voice coilfor vibrating
symmetric
Lamb wave

1[cm]

8[cm]

Voice coilfor vibrating
anti-symmetric
Lamb wave

Silicon rubber Voice coils( width 2[cm]) for absorbing
vibrations

Figure 6: The construction of the tactile display.

Modes of Lamb waves

V2(k2 ,ω2)4

ω = kv0
The mode oftactile exploration

The real shape
to be formedkk0

Figure 5: The method to display the sensationin tactile exploration by amplitude-modulated
Lamb waves. The same displacements of the nger
surface are created by using either R or V1 + V2 .

The voice coil perpendicular to the plate vibrates
the rst anti-symmetric Lamb wave ( exural wave),and the voice coil parallel to the plate vibrates the
rst symmetric Lamb wave (longitudinal wave). We
vibrate the plate by the voice coil on one side of theplate, and absorb the vibration by the coil on the otherside. When a impedance of the voice coil for absorbing
vibrations is matched to a impedance of the plate, thetraveling wave is created. Thus, the traveling wave
toward any direction can be generated by changing therole of the voice coils in both sides. We use 8[ ],50[W]woofers as voice coils.
4.2 Material
The main aim of the experimental device here isto con rm the generation of an amplitude-modulated
Lamb wave. We use the low temporal frequency so as
to obtain a large amplitude. However, the spatial frequency of the Lamb wave decreases with a lowering of
the frequency of the vibration, which results in the decrease of spatial resolution of the tactile display. Thus,
we use silicon rubber, which has a low Young's ratio,
to keep the temporal frequency low, and the spatialfrequency high. The dispersion curves of this siliconrubber shows the possibility of creating a Lamb wave
with the wave length in the order of a few centimeters,by the audio frequency.
4.3 Result
By vibrating longitudinal and exural modes with
30[Hz] at the same time, we can obtain the amplitudemodulated Lamb wave, which is shown in Fig. 7. The
phenomenon that the envelope wave progresses, thatis, the modulated velocity is not equal to zero, is also
con rmed.

two-layered, elastic half space model of the nger. We
derived the expression of the mapping from the displacement of the nger surface to the displacement of
the mechanoreceptors, and showed that it is one-toone owing to its characteristic structure | the upper
layer of the nger being sti er than the lower. Next,
the mapping from the object shape to the displacement of the nger surface was considered, and shown
to be many-to-one because of the mechanical characterof the skin. Then we proposed the tactile display using
the many-to-one property as follows: The amplitudemodulated traveling wave by the superposition of the
rst longtitudinal and the rst exural Lamb wave on
the elastic plate produces its envelope shape on thenger surface, whose spatial frequency and moving
velocity can be arbitrarily controlled. With the experimental device of silicon rubber vibrated by voicecoils, we have con rmed the generation of an envelope
wave. To control the wave number of the envelope
wave arbitrarily, the addititonal study on vibrating
the traveling wave of a single mode to a large degreeof certainty is necessary.
References

Figure 7: Experiment{ touching the envelopeof the amplitude-modulated Lamb wave. The
curve touched by the nger is the envelope wave. By
controlling two frequencies of the voice coils, we cangenerate an envelope wave with an arbitrary wave
length and modulated velocity.
4.4 Future work
The amplitude-modulated Lamb wave whose modulated velocity is controllable can be generated in our
display. Now the frequencies are held at 30[Hz] to obtain a large amplitude, but next, we vibrate the plate
at higher frequencies in order that human does not
feel the vibration of the carrier wave. Higher temporal frequencies also provide higher spatial frequencies,
which result in the higher spatial resolution.
To produce traveling waves at high frequencies, itis necessary to match the impedance of the voice coils
and the plate more precisely.
5

Conclusion

In this paper, the phenomena inside a human nger in tactile exploration have been analyzed with a[1] R.D.Howe, et al., "Remote palpation Technology,"IEEE engineering in Medicine and Biology, Vol.14,
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A necessary and sucient condition of F being oneto-one is det Gz 6= 0. To prove det Gz 6= 0, we showi) det Gz=0 > 0;ii) det G is a monotone increasing function of z;
as follows:si) det Gz=0 =

s

!

k 41
>0
k4
2
2
(because of the assumption; v < i ; i)v2v2

10 10 2 10 2

[11] H.E.Pereira et al., "The E ects of Layer Properd
det Gzii) dzties on Shear Disturbance Propagation in Skin,"Transactions of the ASME, Vol. 113, pp. 30-35,= 12 (r1 + s1 ) sinh(r1 + s1 )z
1991.


 


2
r2 s 2 22[12] W.M.Ewing, Elastic Waves in Layered Media, 1 1 0 r1 s21 X 2 0 r2 s22 W 2 + 1 0 k
Z 0 2 Ykk
r1 s 1kFrank press, 19571
r1 0 s1 )z[13] A.E.H.Love, Some problems of Geodynamics, +
2 (r1 0 s1 ) sinh(

 


Dover publications, 1911r2 s 2 2
k2
r2 s 2 2
r1 s 12
2
1 1 + k2 X 0 k2 W + 1 + r s Z 0 k 2 Y
1 1[14] M.Kuribayashi, S.Uehara, E.Mori, "Excitation



1
r
s
r1 s 12
2
conditions of exural traveling waves for a re- + (r1 + s1 ) cosh(r1 + s1 )z 1+
1 0 k2 (XY 0 ZW )2
r1 s 1
versible ultrasonic linear motor," Journal of the
Acoustical Society of America, 77(4), April, 1985
1
r2 s 2
r1 s 1+ 2 (r1 0 s1 ) cosh(r1 0 s1 )z 1 r 0 s 1 + k2 (XY 0 ZW ):11[15] T.NARA, "Tactile Display using Elastic Waves,"(30)
University of Tokyo Master Thesis, 1997 (inJapanese)
Here, from the assumption; 1 > 1 > v,ss!
Appendix { Proof of the theorem in 2.4v2v2
r 1 0 s1 = k
1 0 2 0 1 0 2 > 0:
First, we represent det Gz as a function of z. The
1
1
expression is obtained from (21),(22) as
det Gz = g11 g4 0 g2 1 g3

= 2 0 XZ + rk2 s22 Y W+ 12 cosh(r1 + s1 )z 1 1 0 rk1 s21 X 2 0 rk2 s22 W 2



2
+ 1 0 rks Z 2 0 rk2 s22 Y 2
1 1



+ 12 cosh(r1 0 s1 )z 1 1 + rk1 s21 X 2 0 rk2 s22 W 2
+ 1 + rks





rs
Z 2 0 2 22 Y 2
k
1 1



+ 12 sinh(r1 + s1 )z 1 rr2 + ss2 1 0 rk1 s21 (XY 0 ZW )
1
1



+ 21 sinh(r1 0 s1 )z 1 rr2 0 ss2 1 + rk1 s21 (XY 0 ZW ):112dTherefore, dz
det Gz > 0, if the coecients of the fourterms in (30) are positive. As for the second term in(30), for example,

1X 2 0 rk2 s22 W 2 >



s

=

v2

s

v2

!

1 0 1 0 2 1 0 2 W 2 > 0;=

1Z 2 0 rk2s22 Y 2 >

1 0 rk2 s22 W 2

2


1 0 rk2 s22 Y 2
s
s
v2

2

v2

!

1 0 1 0 2 1 0 2 Y 2 > 0;2

2

because  < 1 =) jX j > jW j; jZ j > jY j. The otherthree coecients are shown to be positive in the sameway. (Q.E.D)