On transversely isotropic, orthotropic and relative isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Part III: The irreducibility of the representations for three dimensional transversely isotropic functions

0020-7225193$6.00+ 0.00 Copyright @ 1993PergamonPressLtd

1~. 1. EngngSci. Vol. 31. No. 10, pp. 1425-1433,1993 Printedin GreatBritain.All rightsreserved

ON TRANSVERSELY ISOTROPIC, ORTHOTROPIC AND RELATIVE ISOTROPIC FUNCTIONS OF SYMMETRIC TENSORS, SKEW-SYMMETRIC TENSORS AND VECTORS. PART III: THE IRREDUCIBILITY OF THE REPRESENTATIONS FOR THREE DIMENSIONAL TRANSVERSELY ISOTROPIC FUNCTIONS Q.-S. ZHENG Department of Civil Engineering, Jiangxi Polytechnic University, Nanchang, Jiangxi 330029, P.R. China (Communicated

by A. J. M. SPENCER)

Abstract-In this part we prove the irreducibility of the representations derived in Part II for three dimensional transversely isotropic scalar-valued, vector-valued, symmetric tensor-valued and skewsymmetric tensor-valued functions of symmetric tensors, skew-symmetric tensors and vectors.

1. INTRODUCTIONt

The complete and irreducible representations for transversely isotropic tensor functions constitute a rational basis for a consistent mathematical modelling of the complex behavior of tranversely isotropic materials. The completeness of the representations derived in Part II for tranversely isotropic tensor functions is actually also proved in Part II due to the derivation procedure of the representations. In the present part we prove the irreducibility of the representations obtained in Part II by employing the procedure developed by Pennisi and Trovato [l], who prove+ in [l] the irreducibility of the representations for isotropic tensor functions derived by Smith [2] and sharpened by Boehler [3]. Let {e,, e2, e3} denote a three dimensional orthonormal basis. The symmetric tensors , , E6 and the skew-symmetric tensors a,, Q2, Q3, which are defined in (11.2.1)s and El, E2, . . (11.2.2) respectively, are employed. The following identities are very useful in order to simplify the proof of the irreducibility.

(&EI%,

Q&z&,

(Q2E1Q2,

Q2E2Q2,.

W3E1Q3,

Q332Q3,

. . . 9 QJVW

=

(‘4

-E3,

. . , Q2&Q2)

=

t--E39

0,

. . . 9 fl3bJ23)

=

C-E21

--El,

-E2,

JL

-El,

0,

to,

(Q2E,P3 + Q3E1P2, . . . 7Q2Ec5523

+

fi3bQ2)

=

(EJ,

(Q&IQ, + QtE&

...

+

WeQ3)

=

(0,

(Q,E,Q2 + &E,Qi,

...,

+

Q2W-V

=

((40,

, Q3W-h

Q&Q2

(4%

Es,

0,

0)s

Et,),

0, 0, 2E,, J%,

0,

&,

-Es,

-Ee,

2E2,

-Es,

-Ed,

--Ed,

--J&j,

2E3):

(1.1)

t&e Parts I and II, preceding in this issue, for references. *By using Line 36 in Table 1 of [l], Pennisi and Trovato fail to prove the irreducibility of the invariant tr AW’BW in the representation for isotropic scalar-valued functions because tr AW’BW - 0 for any W if A = B. #The prefixed “II” means Part II. 1425

1426

Q.-S. ZHENG

and

EiEl = El,

E2E2

E4E4 = I - El,

E5E5=I-E2,

E&3 = Es E6E6=I-E3,

S2,Ql=El-I,

Q2Q2=E2-I,

S&&,=E,--I;

&Es f EJEz = O/O,

E3E1fE1E3=0/0,

E,E2fE2E,=0/0,

E5E6 f E6E5 = E4/-Q1,

E6E4fE4E6=E5/--Q2,

E4E5fEsE4=E6/-P3,

RzQ3f R&

&&fSZ&=E5/-R2,

Q,Q2~.2P,=E6/-R3;

JGE~fE~E2=Ed-L E2EsfEsE2=0/0,

E3E‘,fE4E3=E4/-S&,

ElEsfESE1=E5/-QZ, E~E~fE&=Ed-L

EzEcjfE6E2=E6/-S&,

E&fE5E3=EJQ2, E,E,c,fE,E,=O/O;

ElS21TPlEl=0/0,

E2&

E,P1TS2,E3= -E4/Ql,

Elt22TQ2E1=-E5/P2,

E2S22TQ2E2=0/0,

W2

EJ&~Q~E~=I%/Qs

E4223TQ3E2= -E6/Q3,

E3P3TQ3E3=0/0;

E4QlTP1E4=2E3-2EJ0,

E&T@Es=

E681TS21E6=ES/-P2,

E&TPzE4=E6/-Q3,

E5P2FQ2E5=2E1-2EJ0, E5Q~5:Q3E5=E4/-LZ1,

= Ed/--Q,,

E,E4fE.,E1=0/0,

E4R3TQ3E4= -Es/--Q,,

=

r

E2,

Q&2

=

Ed-b,

-Ed-Q3,

‘F

Q2E3

=

WQ2,

E,P2TQ2E6=-E4/-S'&, E&TFP3E6=2E2-2E,/O. 0.2)

In (1.2), the EIEs f E5E1 = Es/-Q2 means EIEs+ EsEl= E5 and E1E5- EsE,= -P2, E,& T Q3E6 = 2E2- 2EJO means E6Q3- &Ed = 2E2 - 2E1 and E,& + R3E6= 0,and so on. In order to prove the irreducibility of the representations derived in Part II, we use Tables l-15 in the same sense as we used Tables 1.2-1.8 or Tables l-4 of [l].

2. THE IRREDUCIBILITY

OF THE

REPRESENTATIONS

UNDER

T,

Let S be the skew-symmetric tensor associated with the privileged direction t of the symmetry group T1. Denoting e, = z, so that Q = Q1, we only need to prove the irreducibility of the functions bases given in (11.3.1)-(11.3.3). From Tables 1-3, there follows the irreducibility of (11.3.1)-(11.3.3), or equivalently, the irreducibility of the representations (11.3.4)-(11.3.7) for transversely isotropic functions under c.

3. THE IRREDUCIBILITY

OF THE

REPRESENTATIONS

Let t = e,. We construct Tables 4-7 in order to prove the irreducibility (11.4.1)-(11.4.4) for transversely isotropic functions under T2.

4. THE IRREDUCIBILITY Let

R = 8,.

representations 5. THE

We construct (11.5.1)-(115.4) IRREDUCIBILITY

OF THE

REPRESENTATIONS

UNDER

T2

of the representations

UNDER

T3

Tables 8-11 in order to prove the irreducibility for transversely isotropic functions under T3. OF THE

REPRESENTATIONS

Let t = e,. We can construct Tables 12-15 so that the irreducibility (11.6.1)-(11.6.4) for transversely isotropic functions under T4 is proved.

UNDER

of the

T4

of the representations

Tensors and vectors--Part

6. THE

IRREDUCIBILITY

OF THE

1427

III

REPRESENTATIONS

UNDER

T5

Comparing the lists (11.7.1)-(11.7.3) of the invariants under G with the lists (11.4.1). (11.4.3) and (11.4.4) or lists (11.6.1), (11.6.3) and (11.6.4) of the invariants under T2 or T4, we see that by the use of the values of the variables as the same as that given in Tables 4, 6 and 7 or Tables 12, 14 and 15, the irreducibility of (11.7.1)-(11.7.3), so that of (11.7.4)-(11.7.7), is proved. Acknowledgements-This paper was written during a visit to the Department of Theoretical Mechanics at the University of Nottingham under a Royal Society Fellowship. The author is grateful to Professor A. J. M. Spencer for his helpful comments and his hospitality.

REFERENCES [l] S. PENNISI and M. TROVATO, ht. I. Engng Sci. 25, 1059-1065 (1987). [2] G. F. SMITH, ht. J. Engng Sci. 9, 899-916 (1971). [3] J. P. BOEHLER, ZAMM 57,323-327 (1977). (Received 6 August 1992; accepted 23 November 1992)

APPENDIX Tables l-15 Table 1. (Q = Q,) V

V’

1 A = E, + (Es + E,)/V6 - 2E, 2 A=E,/d2 3 A=E,-E,+E, 4 A=E,-E, 5 6

7 8

-A = E, + (E, + E,)/V6 - 2E, A=E,

tr tr tr tr tr tr tr tr

A A2 A3 AQ2 A2Q2 A2P2AP W2 WP

-A=E,, B=E, -A=E,, B=E,+E, A=E,+E,, -B=E, -A=E,, B=E, -A=E,-E,, B=E,+E, A=E,+E,, -B=E,-E, A=E,-E,+E,-2E,, B=I-E,+E,-E, -A=E,-E,, W=51, A=E,, -W=Q, A=E,, -W=sL, -A=E,, W=R, W=R,, -v=n, w=Il,, -v=n,

tr tr tr tr tr tr tr

AB

tr tr tr tr tr tr

AW2 AWR AWQ’ AW’Q WV WVP

-A=E,-E3+Eg -A=E,-E, A=& -A=&-E,+E,+E, w=n, -w=n

A=E,

A=E,-E,+E,+E, w=o w=n

9 A=E,, B=E, 10 A=E,, B=E,+E, 11 A=E,+E,, B=E, 12 A=E,, B=E, 13 A=E,-E,, B=E,+E, 14 A=E,+E,, B=E,-E, 1.5 A=E,-E2+E3-2Es, B=I+E,+E,+E, 16 A=E,-E3, W=Q, 17 A=E,, W=8, 18 A=E,, W=0, 19 A=E,, W=P, 20 w=n,,v=n, 21 W=SL,,V=R,

Table 2. (Q = Q,) 1 A=E, 2 A=E, 3 4 5 6 7 8 9 10 11 12

A=E, A=E, A=E, A=E, A=E,+E, A=E,+E, w=n, W=R, W=R, w=n,

I 512 A A2 AIL-RA A% - RA2 RAQ QAR’ - Q2AQ W2 RW-kWQ Q2W - WR2 QW2 - w%I

AB2

A2B

ABQ AB’Q A2BSI AQ2BQ

1428

Q.-S. ZHENG Table 3. (Q = Q,) 1 A=E,

R

2 3 4 5

AR+PA AQ= - Q=A w RW-WR

A=E, A=E, w=n, w=n,

Table 4. (z=e,,&=rc@x-xx%) V

V’

A=E,+E,+E,-(7/6)E, A=E.,+E, A=E,-E,+E, A=E,-E, A=E., W=P,+Q, W=P, x=0 x=e,

-A = E, + E, + E6 - (7/6)E, A=E, -A=E,-E3+Eg -A=E,-E, A=E, w=sL, w=n, x = e2 x = e,

tr A tr A= tr A'

-A=E,-E,+E,,B=E,-E,+E, -A=E,-E3+E4,B=Es+E6 A=E,+E,, -B=E,-E,+E, -A= E, - E, - (E5 - E,)/v2, B = E, - E, + (E, + E,)/v2 A=E,-E,, W=Q, A=E,+E,,W=Q2 A=E,+E,, -W=S& A=E,+E,, -W=IR,+Q, A=E,+E,, -W=51, A=E,-E,+v2E.,, -W=Q,+Q, W=Q,, -v=n, w=n,-n,, -v=n,+n, -w=Q,-~‘,v=P,+Q,+P, w=n,+n,+n,, -v=n2-Q’

tr tr tr t.

14 15 16 17 18 19 20 21 22 23

A=E,-E3+E4,B=E2-E3+E4 A=&-E3+Eq,B=E5+E6 A=Es+&,B=E2-Ej+Eq A=&--E,-(Es--&)/d2 B=E,-E,+(E,-Ed/v2 A=&-Es, W=Q2 A=E,+E,, W=9, A=E,+E,,W=Q, A=E,+E,,W=Q,+a, A=E,+E,,W=a’ A=E,-E,+q2E,, W=Q,+Q, w=n,.v=n, w=n,-n,,v=n,+n, W=n2-n3,v=n,+n2+n3, W=Q,+Sl,+Q,,V=Q,-Q,

24 25 26 27 28 29

A=E,,x=e, A=E,, x=e2 A=E,+E5,x=q W=P3,x=e2 W=P,+S&,,x=e, x=e,, y=e,

A=E2,x=e3 A= E6, -x=e2 A= E,+ E,, -x=e2 W=Q3, -x=e2 W=Q,+Q,, -x=e2 x = e,, -y=e,

x-Ax t-Ax t.A% t.Wx t-W% x-Y

30 31 32 33 34f 35 36 37 38 39 40 41 42 43 44

A=E,,B=Eh,C=2E2+E3+d2E, A=E,, B-E,,, W=a, A=&, B-Es+&, W=IL, A=E,+&,B=&, W=n, A=t&+E,,B=E,+E,+&,W=Q, A=E,,W=R,,V=U, A=E,,W=Q,,V=R, A=E,,W=Q,,V=Q, w=n,,v=n,,u=n, A=E,,B=E,,x=e, A=E,, W=SZ,,x=e, A=E,. W=Q2,x=e2 W=P,, V=S&x=e, A=E,,x=e2, y=e, W=Q,,x=e,, y=e,

A=E,,B=E,,-C=2E,+E,+d2E, A=E,,B=E,, -W=Q, A=E,,B=E,+E,,-W=Sl, A=E,+E,,B=&, -W=P, A=E2+Es,B=E,+Es+Egr A=E,, -W=Q,,V=Q, A= E,, -w=n,,v=n, A=E.,, W=P,, -V=S& W=Q2, V=Q,, -u=n, A=E,,B=E,, -x=e, A= E,, W=P,, x=e3 A=E,, W=SZ,, -x=e, w=n,,v=n,, -x=q A= E,, -x=%, y=e, W=Q,, -x=q, y=e,

tr ABC tr ABW tr AB’W tr A’BW trAW’BW tr AWV tr AWV’ tr AW’V tr WVU t * (AB - BA)x x*AWx t . (AW + WA)x t.(WV-VW)x x*Ay x.Wy

1 2 3 4 5 6 7 8 9 10 11 12 13

t-At

t-At tr w=

t-wt x*x t-x

AB AB2 A2B ABt

tr AW2 tr A2W2 tr A2W2AW t. AWt t . A’Wt t . WAWt tr WV t*WVt tWV% t . w2vt

-W=Q,

tsubstituting for the incorrect variable sets in Line 36 of Table 1 in [l], we may use the two sets {A= E, + Es, in order to prove the irreducibility B=E,+E5+&,W=P,}and{A=E2+E,,B=E,+E,+E,,W=-a,) of tr AW’bW.

Tensors and vectors-Part

1429

III

Table 5. (z = e,) 1

A=E,+E,+E,+E,

2 3 4 5 6

A=E,+E,+E,+E, A=E,+E,+E,+E, w=n,+n,+ra, w=n,+n,+n, x=e2

7 8 9 10 11

A=E.,, B=ES A=E,, W=Q, w=n,,v=n, A=Eq,x=eZ W=Q,,x=e,

Table 6. (x=e,,

x

(AB - BA)r (AW + WA)r (WV - VW)% Ax wx

Z,,=r@x-x85) I r@r

1 2

A=E,+E, A=E,+E,

3 4 5 6 7 8 9 10 11 12

A=E,+E, A=E,+E, A=E,+E, A=E,+E, w=n,+n, w=n,+sr, w=SL,+n, w=n,+n, x=e2 x=e,

13 14 15 16 17 18 19 20 21 22 23

A=E,-Es, B=E,+E, A=E,+E,-E,+Eg, A=-E,+E,-E,+2E,, A=E,, W=S& W-R,, v=n, w=n,, V=R, W=sk2,V=P, A=Es,x=e2 W=S&, x=e2 W=Q,,x=e, I.= e,, y=e,

A

A2

r@Az+ArBr r@At+A*r@r W2 r@Ws+Wx@r WrQWz w58w%+w*T@w~ X9X r@x+xQz

AB+BA AW-WA WAW’ - W’AW A*W - WA2 wv+vW WV2 - VW w*v - VW2

W=Q, W=sZ,

A&#-w x@Wx+WxBx W&+&W x@y+y@x

Table 7. (t=e,,&=r@x-x8x) 1 2 3 4 5 6

A=E,+E, A=E,+E, A=E,+E, W=R,+Q, W=R,+P, W=R,+R,

rBAr-AxQs %@A%-AtBr Ar@At-A2~@Ax W rBWs-Wr@r r@W%-W%@r

7

x=e2

&

8 9 10 11 12 13 14 15

A=E,-E3,B=Eq A=E,-E3,B=Es+Eg A=E,+E6,B=&-Es A=&-E,, B=E,+E,+E, A=E,, W=Q, A=E.,, W=S-& w=n,,v=n, A=Eqrx=e2

AB-BA AB2 - B2A A*B - BA2 (ABZOZ} AW+WA AW2 - W*A WV-VW x@Ax-AxQDx

16 17 18

A=Eqrx=e2 W=R,,x=e, x=e2, y=e,

A&+&A W&-&W xQy-yBx

Q.-S. ZHENG

1430

Table 8. (Q = Q,) V

V’

1 9 4

tr A trA2 tr A3 tr AR2 tr A2Q2 tr A2R2AQ tr w2 .trQW X’X x*Qk ’

(as in Table 1) (as in Table 1)

5 6 7 8 9 10 x=eI x = e,

x=0 x=e,

11 12 13 14 15 16 17 (as in Table 1) 18 19 20

(as in Table 1)

tr AB tr AB2 tr A2B tr ABR tr AB’CI tr A’BQ tr AR2B5L tr AW2 tr AWIl tr AWQ2 tr AW’Q tr WV I tr WVQ x-Ax x.A% x*AQx x * QAQ% x . A2Qx

;; 23 24 25 26 27 28

A=E.,,x=++e3 A=Es+E,, x=%+e, A=Es,x=e,+e, A=E,-E,+Eg-2Es,x=e,+e2+e3 A=E,,x=e,+e,

-A=E,,x=e,+e, A=E,+E,,x=e,-e, -A=E,,x=e,+e, A=E,-E2+Ej-2Es,x=e,-e2+e3 A=E,,x=c,-e,

29 30 31 32 33 34 35

W=Q,,x=e, W=S&,x=e,+e, W=LL,,x=e,+e, W=Q2,x=e,+e, x=e,, y=e, x=q,y=e, x=e,+%,y=e,-e,

W=Q2,x=e, W = R,, x = e, - e2 W=SL,,x=e,-e, W = 4, x = e, - e3 -x=e,, y=e, -x=%, y=e, -x=e,+e,,,y=e,-e,

x.w% X’QWX x - n2wx x*nw% X’Y x*P x*R %

A=E,, A=E,, W=&, W=Q,,

x.Ay x - (Aa + QA)y x.wy x * (WS2 - QW)y

36 A=Eb,x=ee,,y=e, 37 A=Es,x=e,,y=e, W=&,x=e,,y=e, :: W=&,x=e,,y=e,

-x=e,,y=e, -x=e,, y=e, -x=e,,y=e, -x=e,,y=e,

Table 9. (Q = Q,) 1 x=e,+e, 2 x=e,+c, 3 x=eI +e,

X

4 A=E,,x=e, 5 A=E,,x=e, 6 W=&,x=e, 7 W=S&,x=e,

Ax (Aa + PA)x Wx (WQ - QW)x

zx

Tensors and vectors-Part III

1431

Table 10. (Q=S2,, I:=x@y-y@x) 1 *

‘1

a=

;

4 5 6 7 8 9 10 11 12 13 14 15 16

A (as in Table 2)

x=e,+e, x=e,+e, x=e,+e, x=e,+e,

17 x=e,, 18 x=e,,

AZ AQ-ILA A=Sl- QA= RAR QAQ= - R=AR W= nwtwn n=w - WR2 . nw= - w251 x@x x@Px+Qx@x S-h@Qx QXf8Q%+Q=X~QX

y=e, y=e,

x@yty@x Slz+m

Table 11. (Q=S&, X=xQy-y@x) 1 2 3 (as in Table 3) 4 5 6 x=e, te, 7 x=e,+e,

11 APtQA AQ= - @A W nw-wn x&S-h-Rx@x x@,sL%-sIsr&x

8 x=e,, y=e, 9 x=e,,y=e,

Table 12. (c = e,) V

V’

1 2 3 4 :

(

as in Table 4)

3 9 x=e,

(as in Table 4)

x = e,

10

11 12 13 14 15 16 (as in Table 4) 17 18 19 20 z: 23 24 25 26 27 28 29

A=E,+E,,x=e, A=E,, x=e, A=2E2-2E,+E,-E,,x=e,+e, W=Q,,x=e, W=26L,+Q2+Q3,x=e,+e2 W=Q,tR,-Q,,x=e,te,

‘trA tr A2 tr A3 r*Ar z. A=% tr W2 r-W% .x-x (c * x)2

tr A%

(as in Table 4)

A=E,+E,,x=e, A=Es,x=e, A=2E,-2E3+E,-E,,x=e,+e, W=Q,,x=e, W=2Q,+P2tQ3,x=e,te3 W=Q,+Q,-Q,,x=e,-e,

tr AB2 tr A2B r.ABr tr AW2 tr A2W2 tr A2W2AW z. AWz t - A=Wt t . WAWt tr WV

t-wvt t*wv=t .t*w=vt X.AX

x-A% (x.t)(t.Ax) x*w% (x * t)(t * Wx)

(x * Wt)(t . Wsr)

1432

Q.-S. ZHENG

Table 12-Continued V

V’

30 x=e2,y=e2 31 x = e, + e,, y = e, - e, 32 33 34 35 36 37 38 39 40 41 41 43 44 45 46 47 48 49 50 51 52 53 54 55 56

-

-x = e,, y = e2 -x = e, + e,, y = e, - e2

X’Y (r*x)(r*y)

tr ABC

(as in Table 4)

(as in Table 4)

A=E,, B= Es, x=d2e,+e, A=&, W=P,,x=e,+e, A=E,+E,, W=n,,x=e, A=E,+Es, W=R,,x=e,+e,+e, W=Q,,V=Q,,x=e,+e, W=Q,,V=Q,,x=e,+e, W=SZ,, V=Q,, x=e,+e, A=E,,x=q,y=e, A=E,+E,,x=%,y=e, A=E,-E,+E,-E,, x = e, + e, + e3, y = e, - e3 W=Q,, x=e,, y=e, W=Q,+Sl,,x=e,,y=e, w=n,+n,+n,, x = e, + e2 + es, y = e, - e3 A=E,,B=E6,x==,y=e, A=Es,W=Q,,x=e,,y=e, W=P,,V=Q3,x=ee,,y=e,

A=Eg,B=Es,x=d&-e, A=E,, -W=Q,,x=e,+e, A=E,+E,, -W=Q,,x=e, A=E,+E,,-W=Q,,x=e,+e,+e, -W=Q,,V=Irr,,x=e,+e, -W=P,,V=Q,,x=e,+e, W=S&, -V=Q,,x=e,+e, A = E.,, -x = e2, y = e3 A=Es+Eg,-x=e,,y=e, A=&-E,+E,-E,, x=e,+e,+e,, -y=e,-e3 W=Q,, -x=e,,y=e, W=n,+Q,, -x=q,y=e, w=n,+n,+n,, x=e,+e,+e,, -y=e,-e, A= Es, B=E,, -x=e,?, y=e, A = Es, W = as, --I. = e,, y = e3 W = n,, V = Qs, --I. = e,, y = e3

Table 13. (%= e,) 1 x=e,+e, 2 x=el+e2

;%* x)z

3 A=E,+E,,x=e, 4 A=E,+E,,x=e, 5 A=E.,+E6,x=e2

Ax Al (Ax@r-r@rA)x

6 W=Q,+Q,,x=e, 7 w=n,+n,,x=e, 8 W=Q,+Q,,x=e,

zx (Wr@s+s@rW)x

9 A=E,,B=E,,x=e, lo A=E,, W=Q,,x=e, 11 W=R,V=Q,,x=e,

(AB - BA)x (AW + WA)x (WV-VW)x

tr ABW tr AB=W tr A’BW tr AW’BW tr AWV tr AWV’ tr AW’V tr WVU x*ABx x. AWx x . A’Wx x . WAW% x*WVx x*Wvx x * w2vx X-A X-A 5 x.(AzQr-r@zA)y x*w x-w r, x*(WeQe+t@xW)y x.(AB-BA)y x - (AW + WA)y x.(WV-VW)y

Tensors and vectors-Part

III

1433

Table 14. (r=e,,Z=x@y-y@x)

‘I

r@r

:

A A2

3 4 2

T@AT+ATQT r8A%+A2r@r

(as in Table 6)

W2

7

T@WT+WTBT WT8WT

f 10 11 x=e,+e, 12 x=e,+q 13 14 15 16 17 18 19 20 21 22 23

.wT8w%+w2T@wT 181 (T.1)(T691+1@T) ‘AB+BA AW-WA WAW’ - W’AW A2W - WA2 Wv+vW WV2 - VW ,w%-VW2

(as in Table 6)

A=E,+E,,x=e W=R,,x=e,

xGDAl+Al@x

+ez+e,

W=&,x=e;+~+e~ x=e,+e,y=e,

24 x=e, +e2, y=e, 25 A=E,,x=e,,y=e, 26 W=R,,x=e,,y=e,

Table 15. (r=e,,

l@Wl+WlQl Wx@W%+W1@W1 xQy+yO1 %82kr+ZZ@Z AT-?3 m+zw

Z=xQy-y@x)

‘s@AT-

: 3 4 (as in Table 7) 5 6 7 x=e,+e, 8 8 10 11 12 13 14 15 16 17

AT@S

s@At-A%@T AT’BA~-A=T’BAT

W T@wT-wT@T ,T@wt-w%@T (T-i)(T@i-SCOT) AB-BA AB2 - B2A A2B - BA2

(as in Table 7)

A=E,-E3,x=e2+es A= Es+ E6, x=e2 A=&-Ej,x=e,+e2+e, W = n,, x = e2 + e, :t W=&, x=e,+e, 20 x=e,+e,,y=e, 21 x=e,+q, y=e, 22 A=E,,x=%.y=e, 23 W=S&x=e,,y=e,

{~T@T}

AW+WA AW2 - W2A WV-VW xQAx-AxBx x@Af-A%bx (A(x @ x)(x @ 1)) l@Wl-WXBX l@W%-w'zxox I: T@rT-rT@T Az+zw wz-LW