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D w '|rw] ?}W S '|m}rwQO}y `=aW R 'xy=Q@; |Q@ R ?} Q[ cr 'u; QO xm|xar=]t QO xOWxO=iDU= sQi Q@ |Q=}U@ u=} QH Ctw=kt \@=wQ "CU= u=} QH jta[13]'nv}v=t %\@=wQ x@ u=wD|t =yv; |xrtH R= xm 'OvDUy j@]vt �2 |xrO=at� Q[=LQwmPt\@=wQ R= |O=OaD '1 pwOH QO "OQm xQ=W= """w [16]'|rvwQ@ [15]'|RW [14]'|U}r

"CU= xOW x�=Q=C=QP \UwDt Q]k d u; QO xm 'CU=Na = 0 0253d1=8 '|U}r |x]@=Q QO

"CU= QDt|r}t ?ULQ@ |@ wUQ"u=} QH Ctw=kt \@=wQ "1 pwOH

x; y; � |=yu=wD u=} QH Ctw=kt |x]@=Q `HQtx = 2

3 ; y = 12 ; � = 0 V = (1=n)R2=3S1=2 [13]nv}v=t

x = 12 ; y = 1

2 ; � = 14 V = (1=Na)

pRSD1=4 [14]|U}r

x = 12 ; y = 1

2 ; � = 0 V = cR1=2S1=2 [15]|RW

"O=R=t |WQ@ VvD sQi x@ xO=iDU= OQwt QDU@ Q=@ CqO=at "2 pwOH=yu=wD w ?}=Q[ QDU@ Q=@ |x]@=Q |xDi=} Q}}eD sQi QDU@ Q=@ |x]@=Q `HQt

cs = 8qg(Gs�1)d350

( d50(Gs�1))1 5qs = cs(�0 � �c)1 5 qs = 8

qg(Gs � 1)d350(� �0

d50(Gs�1) � �c d50(Gs�1) )1 5

[19]Qrwt w QD}B Q}}tm = 0; j = 1 5 �c=( d50(Gs � 1)) = 0 047

cs = cdqs = cs�0(�0 � �c)

qs = cd�0(�0 � �c) [20]|w@ wOm = 1; j = 1 �c = 0 061 + 0 093d; cd = 0 17d�3=4; �0 = �RS

cs = 11 2qg(Gs�1)d350

( d50(Gs�1))1 5qs = cs(�0 � �c)4 5=�30

q�s = 11 2(��0 � 0 03)4 5=��30[21]QmQ=B

m = �3; j = 4 5q�s = qsq

g(Gs�1)d350; ��0 = �0

d50(Gs�1)

�c=( d50(Gs � 1)) = 0 03

51

"""\@=wQ

T=@Dk=

QO|OL

|=y|Qw

�DVkv'12 w 10 CqO=at R= ?}DQD x@ S w � x@ C@Uv �0 w Pbed R= |�RH |Q}ojDWt =@

%OwW|t xH}Dv ?14 w hr=14 \@=wQ

@Pbed@�

=[ (1 + x+ �)� + 2(1 + �)p1 + z2

(� + 2p1 + z2)(� + z)(2 + x+ �)

+(2 + 2x+ �)z + (4+2x+2�)z

p1+z2

�

(� + 2p1 + z2)(� + z)(2 + x+ �)]Pbed (hr=14)

@Pbed@S

= �y(2 + x+ �)S

Pbed

@�0@�

= �� + 2(1 + �)p1 + z2 � (2 + �)z

(� + 2p1 + z2)(� + z)(2 + x+ �)�0

@�0@S

= (2 + x+ �� y)(2 + x+ �)S

�0 (?14)

jta '(Pbed) p=v=m hm ZQa C=Q}}eD |ovwoJ lQO Qw_vtx@ xm CU= K[=wS = S(�) |x]@=Q u}}aD x@ R=}v '(�) pmW pt=a =@ (�0) |WQ@ VvD w (D)

xHwD =@ w QDU@ Q=@ p=kDv= |x]@=Q l} ?=NDv= =@ xm CU= u; |=} wo VN@ u}= "CU=S = S(�) |x]@=Q uOQw;CUOx@ 'u=} QH Q@ sm =L |i=[= |x]@=Q l} u=Oki x@x@ u=wD|t QDU@ Q=@ p=kDv= |x]@=Q ltm x@ |rw 'CU= umttQ}e K} QY CQwYx@

"OQ@ |B � =@ dS=d� = 0 C=Q}}eD |xwLvQs = PbedqsCQwYx@ p=v=mZQapm QO QDU@ Q=@ u=wvax@Qs uDiQoQ_vQO =@'`]kt |Q=O}=B s=ovy QO OyO|t u=Wv xm OOaDt |y=oW}=tR; Oy=wW x@ xHwD =@�[22]'OQ=O xt=O= u=vJty QDU@ O=wt p=kDv= =t= 'CU= xOW hkwDt xQ=vm C=QP CmQL|Q=O}=B Q@ Q=O}=B |xUOvy |v}@V}B |r}rLD |=ypOt R= |O=} R O=OaD O}m -=D R}v w�CU= xOW xDiQo Q_v QO p=ai QDU@ ZQa u=wvax@ [1]Pbed 'pO=aD u=tR QO =yxQ=vmR}v Qs '�0 = �0(�; S(�)) w Pbed = Pbed(�; S(�)) \@=wQ =@ u; ?}mQD w|xv}W}@ u}}aD Qw_vtx@ lv}= "OwW|t h} QaD Qs = Qs(�; S(�)) CQwYx@

%�15 |x]@=Q� OQm |Q}ojDWt � x@ C@Uv Qs R= O}=@ 'QDU@ Q=@ p=kDv= C}iQ_

dQsd�

= @Qs@�

+ @Qs@S

dSd�

(15)

p=v=m ZQa OL=w QO QDU@ Q=@ p=kDv= |x]@=Q wQs = Pbedqs |x]@=Q uDiQoQ_vQO =@w hr=16 \@=wQ CQwYx@ 15 |xrO=at QO @Qs=@S w @Qs=@� '�3 |xrO=at�

%OvwW|t x@U=Lt ?16@Qs@�

= @Qs@�0

@�0@�

+ @Qs@Pbed

@Pbed@�

= Qs[(m�0

+ j(�0 � �c) )@�0

@�+ 1Pbed

@Pbed@�

] (hr=16)

@Qs@S

= @Qs@�0

@�0@S

+ @Qs@Pbed

@Pbed@S

= Qs[(m�0

+ j(�0 � �c) )@�0

@S+ 1Pbed

@Pbed@S

] (?16)

hr=14 CqO=at R= @�0@S w @�0

@� ' @Pbed@S ' @Pbed@� C=Q=@a |Q=Po}=H =@ 'xt=O= QOw hr=17 \@=wQ 'QwmPt \@=wQ |R=UxO=U w ?16 w hr=16 CqO=at QO ?14 w

|r}rLD pOt |xaUwD "3"2|x]@=Q CQwYx@ `]kt |UOvy |=yQDt=Q=B T=U=Q@ (�) Oa@|@ pmW pt=a =OD@=

%CU= xOW h} QaD 4� = Pbed=D (4)

OvDUy p=v=m jta w hm ZQa |xOvyOu=Wv ?}DQDx@ D w Pbed 'u; QO xmjta '(�) pmW pt=a?UL Q@ `]kt |UOvy |=yQDt=Q=B Q}=U 'xt=O= QO "�1 pmW�CQwYx@ G}=Dv '|}xkvRwP `]kt |=Q@ "Ov=xOW u=}@ (z) p=v=m |@v=H ?}W w '(D)

%CU= 8 |r= 5 \@=wQ

A = D2(� + z) (5)

P = Pbed + Pbank = D(� + 2p1 + z2) (6)

R = A=P = (� + z)(� + 2p1 + z2)

D (7)

W = Pbed + 2zD = D(� + 2z) (8)

|xOWQD \}Lt '|[Qa `]kt CL=Ut ?}DQD x@ W w R 'P 'A '=yu; QO xm|x]@=Q u}}aD Qw_vtx@ "CU= QB `]kt K]U ZQa w '|m}rwQO}y `=aW 'p=v=mp=v=m |@v=H ?}W '(Q) |@O '(s) xy=Q@; |rw] ?}W '(cr) |Q@ R ?} Q[ =@ jta|xrO=at� |oDUw}B |x]@=Q R= (Q=A) C@Uv CU= |i=m (�) pmW pt=a w '(z)?}mQD 7 w 5 CqO=at =@ TBU w OwW 2 |xrO=at QO (V ) CaQU u} Ro}=H �1|UOvy |=yQDt=Q=B x@ \w@ Qt \@=wQ w �9 |xrO=at� pY=L |x]@=Q ?}mQD R= "OwW=@ R}v |WQ@ VvD w '\UwDt CaQU 'QDU@ ZQa |x]@=Q '�7 |r= 4 CqO=at�

%�12 |r= 9 \@=wQ� Ov};|t CUOx@ QwmPt |=yQ}eDt

D = (� + 2p1 + z2)x=(x+2+�)

(� + z)(x+1)=(x+2+�)(Q=cr)1=(x+2+�)

Sy=(x+2+�) (9)

Pbed = (�D) = �(� + 2p1 + z2)x=(x+2+�)

(� + z)(x+1)=(x+2+�)(Q=cr)1=(x+2+�)

Sy=(x+2+�)

(10)

V = (Q=A) = (S2yQ(x+�)c2r)1=(x+2+�)(� + z)(x��=(x+2+�)

(� + 2p1 + z2)(2x)=(x+2+�)

(11)

�0 = ( RS)

= S1�(y=(x+2+�))( Qcr )1=(x+2+�)(� + z)(1+�)=(x+2+�)

(� + 2p1 + z2)(2+�)=(x+2+�)(12)

pt=a R= |a@=D CQwYx@ p=v=m |rw] ?}W =OD@= QO '|R=UpOt R= CtUk u}= QOw 10 '9 CqO=at x@ xHwD =@ =Pr w xOW xDiQo Q_v QO S = S(�) p=v=m pmWwD = D(�; S(�)) 'Pbed = Pbed(�; S(�)) CQwYx@ R}v =yQ}eDt Q}=U '12s}y=wN =Q ?13 w hr=13 |=yx]@=Q wQu}= R= "Ov=xOW h} QaD �0 = �0(�; S(�))

%CW=OdPbedd�

= @Pbed@�

+ @Pbed@S

dSd�

(hr=13)

d�0d�

= @�0@�

+ @�0@S

dSd�

(?13)

52

2/1|

xQ=tW'

33�2

|xQwO'

�1396

u=DU@=D�

h}QW

u=Qta|

UOvyt |OL |Qw�D wO uOw@ RQ=sy Qou=}@ Qt= u}= "OwW|t pY=L (Qsmax) ?wUQ

?wUQ p=kDv= C}iQ_ |xv}W}@ w �u=} QH u=wD |xv}tm =}� (Smin)?}W |xv}tmOvDUy u; Qou=}@ 18 w ?17 'hr=17 CqO=at 'Qo}O CQ=@ax@ "CU= (Qsmax)

|@ wUQ Q=@ xm Ovm|t s}_vD |}xvwox@ =Q OwN ?}W 'uOWQ=O}=B CyH QO p=v=m l} xm?}W xm |Dr=L QO w OwW Q@=Q@ ?wUQ p=kDv= C}iQ_ |xv}W}@ =@ p=v=m x@ |OwQw(D) jta w (Pbed) QDU@ ZQa R= ?}mQD l} \ki 'OUQ@ OwN Q=Okt |xv}tm x@p=kDv= C}iQ_ |xv}W}@ pO=at 'Cr=L u}= w Ow@ Oy=wN Q=oR=U QwmPt CqO=at =@

"CU= ?wUQxm |]}=QW QO 'OwQ|t Q=_Dv= xm Qw]u=ty 2 pmW x@ xHwD =@ u}vJtyO=a@= 'u; =@ Q_=vDt w� ?}W |xv}tm '2 R= QDnQR@ |=y� QO 'CU= �0=�c > 1

"(dS=d� = 0) OwW|t pY=L �p=v=m |xv}y@(Qs = Qsmax) |[Qa `]kt |xv}y@ O=a@= "1"3"2

�m xv}y@ pmW pt=a 'dS=d� = 0 w 18 w ?17 'hr=17 CqO=at ?}mQD R=QO �m |Q=Po}=H =@ "�20 w 19 CqO=at� O};|t CUOx@ 'C@=F ?}W Cr=L QOCaQU w (Dm) jta '((Pbed)m) hm ZQa |x]@=Q '11 |r= 9 CqO=atT}Ov= "�C@=F ?}W Cr=L� OwW|t pY=L xv}y@ p=v=m QO (Vm) u=} QH \UwDt

"CU= p=v=m QO xv}y@ \}=QW |xOvyOu=Wv 'm�0�c

= K2K1

(19)

�0 � �c�c

= j(�m + (2 + �)z � 2(1 + �)p1 + z2)

K1(20)

%=yu; QO xmK1 =(�m+ x+ 1 + �� j)�m + (2m(1 + �) + 2(1 + �)

+ 2j(1 + �))p1 + z2 + (�m(2 + �)2 + 2x+ �

� j(2 + �))z + (4 + 2x+ 2�)zp1 + z2

�m

%w

K2 =(�m+ x+ 1 + �)�m+(2m(1 + �)+2(1 + �))p1 + z2

+ (�m(2 + �) + 2 + 2x+ �)z+ (4+2x+2�)zp1 + z2

�m

x@U=Lt p@=k 19 |xrO=at R= K2 w K1 ?}=Q[ 'xt=O= QO xm CU= QmP x@ sRq"OvDUy

pY=L(Sm) p=v=m |xv}y@ ?}W '12 |xrO=at QO 19 |xrO=at |Q=Po}=H =@%�21 |x]@=Q� OwW|t

Sm =((cr=Q)(�c= )(x+2+�))1=(x+2+��y)

� (K2K1

)(x+2+�)

(x+2+��y)(�m + 2p1 + z2)(2+�)=(x+2+��y)

(�m + z)(1+�)=(x+2+��y)

(21)

hm ZQa |x]@=Q '11 |r= 9 CqO=at QO 21 |xrO=at R= Sm uOQm u} Ro}=H =@pY=L xv}y@ p=v=m QO (Vm) u=} QH \UwDt CaQU w '(Dm) jta '((Pbed)m)

%CW=O s}y=wN =Q ?17@QS@�

= [ j(�� � (2 + �)z + 2(1 + �)p1 + z2)�0

(� + 2p1 + z2)(� + z)(2 + x+ �)(�0 � �c)+ ((�m+ x+ 1 + �)� + (2m(1 + �) + 2(1 + �))

p1 + z2

(� + 2p1 + z2)(� + z)(2 + x+ �)(�0 � �c)

+(�m(2+�)+2+2x+�)z+ (4+2x+2�)z

p1+z2

� )(�0��c)(� + 2p1 + z2)(� + z)(2 + x+ �)(�0 � �c) ]QS

(hr=17)

@QS@S

=

[(x+2+��y)(m+ j)�y]�0 + [y �m(x+ 2 + �� y)]�cS(2 + x+ �)(�0 � �c) QS

(?17)

Qs R= O}=@ 'xv}y@ ?wUQ p=kDv= `]kt |UOvy C=YNWt u}}aD Qw_vtx@ uwvm ="(dQs=d� = 0) OwW xO=O Q=Qk QiY Q@=Q@ u; jDWt w xDiQo jDWt � x@ C@Uv

%OwW|t xH}Dv 18 |x]@=Q '15 |xrO=at R= CQwYu}= QOdSd�

= � @Qs=@�@Qs=@S

(18)

C=Q}}eD x@ ds=d� C=Q}}eD xm CU= K[=w 18 w ?17 'hr=17 CqO=at R==@ w 18 w ?17 'hr=17 CqO=at R= xO=iDU= =@ 2 pmW "OQ=O |oDU@ � w �0=�c|x]@=Q w nv}v=t u=} QH Ctw=kt |x]@=Q '�|r}]DUt p=v=m � z = 0 uDiQoQ_vQOV}=Ri= =@ 'swrat �0=�c <=R=x@ 'QwmPt pmW j@=]t "CU= xOW s}UQD |w@ wO QDU@ Q=@QO xm CU=vat u=O@ u}= "Ovm|t Q}}eD C@Ft Q}O=kt =D |ivt Q}O=kt R= dS=d� '�QO CUQO w OQ=O OwHw q=@ x@ wQ |QakD '� p@=kt QO S |x]@=Q x@ \w@ Qt |vLvt(S) ?}W '(dS=d� = 0) Ovm|t =O}B |kqD |ki= QwLt =@ dS=d� xm |}x]kvCmQL |xv=DU; \}=QW QO '2 pmW x@ xHwD =@ "OUQ|t OwN Q=Okt |xv}tm u=R}t x@

:(Smin) OwW|t pY=L dS=d� = 0 '�m = 2 <=R= x@ (�0=�c = 1)

<=R= x@ w (�0 = �c) CmQL |xv=DU; \}=QW QO R}v 'hr=17 |xrO=at x@ xHwD =@uDiQoQ_vQO =@ x@=Wt \}=QW QO xm � � = �(2 + �)z + 2(1 + �)

p1 + z2

R= � = 2 '|w@ wO QDU@ Q=@ |x]@=Q w nv}v=t u=} QH Ctw=kt |x]@=Q 'z = 0p=kDv= C}iQ_ |xv}W}@ 'Qo}O CQ=@ax@ =} @QS=@� = 0 '�OwW|t pY=L u;

|x]@=Q w nv}v=t u=} QH Ctw=kt |x]@=Q <=R= x@ '� =@ dS=d� C=Q}}eD "2 pmW"|w@wO QDU@ Q=@

53

"""\@=wQ

T=@Dk=

QO|OL

|=y|Qw

�DVkv%u; QO xm

� =((�2m(1 + �)� 2(1 + �)� 2j(1 + �))p1 + z2

+ (m(2 + �)� 2� 2x� �+ j(2 + �))z)2

� 4(�m+ x+ 1 + �� j)(4 + 2x+ 2�)zp1 + z2

|xOvyOu=Wv u; |}q=@ OL w CmQL |xv=DU; \}=QW |xOvyOu=Wv '�m |v}}=B OL<=R= x@ xm |QDU@ Q=@ p=kDv= w u=} QHCtw=kt CqO=at?}mQD QO "CU=|@q}U\}=QWm+ j < x+ 1 + � |x]@=Q '�m > �(2 + �)z + 2(1 + �)

p1 + z2

u}rw= [23]'OQ=owmU= w Ovrov= xmv}= x@ Q_v w pOt C=}[Qi x@ xHwD =@ 'CU= Q=QkQ@QO 50 R= QDnQR@ (W=D) jta x@ ZQa C@Uv xv=NOwQ uOW|v=} QW pQDvm Q=}at"OwW|t u}}aD �m = 50� 2z |x]@=Q R= '�m |}q=@ OL 'CU= xOW xDiQo Q_v

�m w |}q=@ w |v}}=B OL u}@ |m}rwQO}y |xUOvy \@=wQ "2"3"2p=v=m l} O=a@= |xOvyOu=Wv '�26 |r= 21 CqO=at� xv}y@ p=v=m |xUOvy \@=wQ|xUOvy \@=wQ w s} SQ \@=wQ =@ j=@]v= R= O}=@ QwmPt \@=wQ 'u}=Q@=v@ "CU= Q=O}=B|=yxv=NOwQ w Q=O}=B |=yp=v=m R= |@ QHD CQwYx@ xm 'QB `]kt QO |m}rwQO}y`@=wD R= |@ QHD \@=wQ QO xmv}= x@ xHwD =@ "OvW=@ Q=OQwNQ@ 'Ov=xOt; CUOx@ |a}@]|xv}y@ p=v=m |xUOvy \@=wQ 'CU= xOW xO=iDU= (Q) u=} QH |@O ?ULQ@ |v=wD

%OQm p}O@D 32 |r= 28 \@=wQ CQwYx@ |v=wD `@=wD sQi x@ O}=@ R}v =Q pY=LSm / Q�1=(x+2+��y)��1

m (28)

(Pbed)m / Q 1(x+2+��y) ��2

m (29)

Qs / Q 1(x+2+��y) ��3

m (30)

Vm / Q (x+��y)(x+2+��y) ��4

m (31)

Wm / Q 1(x+2+��y) ��5

m (32)

%R= Ov=CQ=@a ��5m w ��4

m '��3m '��2

m '��1m '=yu; QO xm

��1m / (K2

K1)

(x+2+�)(x+2+��y)

(�m + 2p1 + z2)(2+�)=(x+2+��y)

(�m + z)(1+�)=(x+2+��y)

��2m / (K1

K2)

y(x+2+��y)

�m(�m + 2p1 + z2)(x�y)=(x+2+��y)

(�m + z)(x�y+1)=(x+2+��y)

��3m / �m(�m + 2p1 + z2)(x�y)=(x+2+��y)

(�m + z)(x�y+1)=(x+2+��y)

� (�m+(2+�)z�2(1 + �)p1 + z2)j(K2)m�

y(x+2+�y)

(K1)m+j� y(x+2+��y)

��4m / (K2

K1)

2y(x+2+��y)

(�m + z)(x�y��)=(x+2+��y)

(�m + 2p1 + z2)2(x�y)=(x+2+��y)

��5m / (K1

K2)

y(x+2+��y)

(�m+2p1+z2)(x�y)=(x+2+��y)(�m + 2z)(�m + z)(x�y+1)=(x+2+��y)

p@=kt QO 32 |r= 28 CqO=at QO �m ?ULQ@ C=Q=@a 'pUm = Q=Ri=sQv QO TBU'pY=L \=kv Q@ |v=wD |x]@=Q VR=Q@ R= w sUQ u; |xOWh} QaD |xOwOLt QO �m|UQQ@ x@ xHwD =@ xmCU= QmP x@ sRq "OvwW|t u}}aD �5 w �4 '�3 '�2 '�1 |=yu=wD|=yVwQ R}v w [25w24]'Ca}@] QO OwHwt OQ=wt QO p=v=m |xQ=vm |x} w=R C=Q}}eDC=Q}}eD <=R= x@ |m}rwQO}y |xUOvy \@=wQ 'Q[=L VywSB QO [26]'OOaDt |r}rLDC=Q=@a |Q=Po}=H R= "CU= xOW u}}aD xHQO 90 =D 30 R= p=v=m |xQ=vm |x} w=R

%�Q}eDt ?}W Cr=L� �24 |r= 22 \@=wQ� OwW|t

(Pbed)m =(Q y

cr�yc)

1(x+2+��y) � (K1

K2)

y(x+2+��y)

�m(�m + 2p1 + z2)(x�y)=(x+2+��y)

(�m + z)(x�y+1)=(x+2+��y) (22)

Dm =(Q y

cr�yc)

1(x+2+��y) � (K1

K2)

y(x+2+��y)

(�m + 2p1 + z2)(x�y)=(x+2+��y)

(�m + z)(x�y+1)=(x+2+��y) (23)

Vm =(Q(x+��y)c2r(�c

)2y)1

(x+2+��y) � (K2K1

)2y

(x+2+��y)

(�m + z)(x�y��)=(x+2+��y)

(�m + 2p1 + z2)2(x�y)=(x+2+��y)(24)

xv}y@ K]U ZQa |x]@=Q '8 |xrO=at QO 23 |xrO=at |Q=Po}=H R= u}vJty%OwW|t pY=L 25 |x]@=Q CQwYx@

Wm =(Q y

cr�yc)

1(x+2+��y) (K1

K2)

y(x+2+��y)

� (�m + 2p1 + z2)(x�y)=(x+2+��y)(�m + 2z)(�m + z)(x�y+1)=(x+2+��y) (25)

%CQwYx@ p=v=m ZQa pm QO QDU@ Q=@ |x]@=Q uDiQoQ_vQO =@Qs = Pbedcs(�0=�c)m((�0 � �c)=�c)j�m+j

c

CQwY QO xm OwW|t pY=L |}x]@=Q '22 w 20 '19 CqO=at =@ u; ?}mQD wQs = Qsmax ZQi =@ w �Q}eDt?}WCr=L QO |va}�Qs wQ Q}O=kt uOw@swratCUOx@ u; R= u=wD|t =Q (�m) p=v=m |xv}y@ pmW pt=a 'p=v=m QO pO=aD \}=QW QO

%�26 |xrO=at� OQw;

Qs=(Qsmax)=jjcs y

(x+2+��y)�m+j� y

(x+2+��y)c (Q=cr)

1(x+2+��y)

� (�m+(2 + �)z�2(1 + �)p1 + z2)j(K2)m�

y(x+2+��y)

(K1)m+j� y(x+2+��y)

� �m(�m + 2p1 + z2)(x�y)=(x+2+��y)

(�m + z)(x�y+1)=(x+2+��y) (26)

'�m = �(2 + �)z + 2(1 + �)p1 + z2 <=R= x@ '26 |xrO=at j@=]t

Qw_vtx@ u}vJty "CU= CmQL |xv=DU; \}=QW |xOvyOu=Wv xm OwW|t Qs = 0p@=k |xOwOLt w Ctqa u}}aD O}=@ 26 |xrO=at 'Qs � 0 Q}O=kt uOWpY=L'Qs � 0 Q}O=kt uDW=O |=Q@ '26|xrO=at x@ xHwD =@ "OwW h} QaD �m |=Q@ |rw@kp=kDv= w u=} QH Ctw=kt \@=wQ u}@ m+ j > x+ 1 +� |x]@=Q xm |DQwY QO%OwW|t OwOLt 27 |x]@=Q QO xOWh} QaD |xR=@ x@ �m C=Q}}eD 'OW=@ Q=QkQ@ ?wUQ

�(2 + �)z + 2(1 + �)p1 + z2 < �m <

[ (2m(1 + �) + 2(1 + �) + 2j(1 + �))p1 + z2

2(m� x� 1� �+ j)

+(�m(2 + �) + 2 + 2x+ �� j(2 + �))z +p

�2(m� x� 1� �+ j)

]

(27)

54

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�1396

u=DU@=D�

h}QW

u=Qta|

UOvyt

(cr) u=} QH Ctw=kt |x]@=Q QO |Q@ R ?} Q[ uOw@pkDUt ZQi =@ u}vJtyS 'Q |=yQDt=Q=B ?ULQ@� xQ}eDtxU |m}rwQO}y |xUOvy \@=wQ 'S w Q Q@ xwqa

%�45 |r= 42 \@=wQ� OwW|t pY=L � crw

(Pbed)m / S�2=�1(c�1r Q)(1+(�2=�1))=(x+2+��y) (42)

Dm / S(�2�1)=�1(c�1r Q)(1+((�2�1)=�1))=(x+2+��y) (43)

Vm/S�4=�1Q(x+��y+(�4=�1))=(x+2+��y)c(2�(�4=�1))=(x+2+��y)r

(44)

Wm / S�5=�1(c�1r Q)(1+(�5=�1))=(x+2+��y) (45)

�m |v}}=B OL QO |m}rwQO}y |xUOvy \@=wQ "3"3"2|va}Qs = 0 x@ �m = �(2+�)z+2(1+�)

p1 + z2 <=R= x@ 26|xrO=at|m}rwQO}y |xUOvy \@=wQ u}}aD Qw_vtx@ "OwW|t QHvt CmQL |xv=DU; \}=QWC@=F Q=Okt '25 |r= 21 CqO=at QO CU= |i=m 'CmQL |xv=DU; \}=QW QO'S \@=wQ w OQm u} Ro}=H =Q �m = �(2 + �)z + 2(1 + �)

p1 + z2

CUOx@ 49 |r= 46 \@=wQ CQwYx@ Q R= |v=wD ?UL Q@ =Q W w V 'D%OQw;

Str = ttrQ�1=(x+2+��y) (46)

Dtr = ctrQ1=(x+2+��y) (47)

Vtr = ktrQ(x+��y)=(x+2+��y) (48)

Wtr = atrQ1=(x+2+��y) (49)

%=@ OvDUy Q@=Q@ ttr w ktr 'ctr 'atr ?}=Q[ '=yu; QO xmatr =( y=(cr�yc ))1=(x+2+��y)(2(1 + �)

p1 + z2 � �z)� (2(1 + �)

p1 + z2 � z(1 + �))(y�x�1)=(x+2+��y)

((4 + 2�)p1 + z2 � z(2 + �))(y�x)=(x+2+��y)

ctr =( y

cr�yc)

1(x+2+��y)

(2(1 + �)p1 + z2 � z(1 + �))(y�x�1)=(x+2+��y)

((4 + 2�)p1 + z2 � z(2 + �))(y�x)=(x+2+��y)

ktr =(c2r(�c= )2y)

1(x+2+��y)

(2(1 + �)p1 + z2 � (1 + �)z)

(�y+x��)(x+2+��y)

((4 + 2�)p1 + z2 � (2 + �)z)

2(x�y)(x+2+��y)

ttr =(cr(�c

)(x+2+�))1

(x+2+��y)

((4 + 2�)p1 + z2 � z(2 + �))

(2+�)(x+2+��y)

(2(1 + �)p1 + z2 � z(1 + �))

(1+�)(x+2+��y)

=@ xOWQmP \@=wQ QO "CU= CmQL |xv=DU; \}=QW |xOvyOu=Wv 'tr T}Ov=XwYNt uRw u; QO xm � �c = ��c d50(Gs � 1) |x]@=Q uDiQoQ_vQO\UwDt uOw@pkDUt ZQi =@ R}v w �CU= C=@ wUQ XwYNt |r=oJ Gs w ?;|m}rwQO}y |xUOvy \@=wQ '(Q) |@O Q@ xwqa (d50) QDU@ C=@ wUQ |xR=Ov=Ovy=wN pY=L 53 |r= 50 \@=wQ j@=]t �Q w d50� ?UL Q@ xQ}eDtwO

C@=F R}v w 29 |xrO=at =@ 4 |xrO=at ?}mQD '32 |r= 28 CqO=at QO pY=L |v=wDQO OOaDt u=QoWywSB |xOWx�=Q= h} Q=aD x@ xHwD =@� (Qs=Q) C@Uv uDiQoQ_vQOOQ=O Q=Qk s} SQ =} pO=aD QO|v=tR|DiQ@; p=v=ml} xmv}= Q@|v@t 'pO=aD swyit =@ \=@DQ=|=yQDt=Q=B w [27]�Ovm Q}}eD |OwOLt |xR=@ QO =} CU= C@=FQs=QC@Uv u; QO xm|Q@ R ?} Q[ w '(d50) QDU@ C=@ wUQ |xR=Ov= \UwDt '(�c) |v=QL@ |WQ@ VvD?ULQ@ xQ}eDtlD |m}rwQO}y |xUOvy \@=wQ '(cr) u=} QH Ctw=kt |x]@=Q QO

%�37 |r= 33 CqO=at� OW Oy=wN pY=L |@O(Pbed)m = eQ((1��2=�3)=(x+2+��y))+(�2=�3) (33)

Dm = cQ((1+ (1��2)�3 )=(x+2+��y))+ (�2�1)

�3 (34)

Sm = tQ�((1+ �1�3 )=(x+2+��y))+ �1

�3 (35)

Vm = kQ((x+��y� �4�3 )=(x+2+��y))+ �4

�3 (36)

Wm = aQ((1� �5�3 )=(x+2+��y))+(�5=�3) (37)

%=@ OvDUy Q@=Q@ e w t 'k 'c 'a ?}=Q[ 'QwmPt \@=wQ QO xma =[ (Qs=Q)

jjcs�m+j� y

(x+2+��y)c ( ycr )

1(x+2+��y)

]�5�3

( y

cr�yc)

1(x+2+��y)

c =[ (Qs=Q)

jjcs�m+j� y

(x+2+��y)c ( ycr )

1(x+2+��y)

]�2�1�3

( y

cr�yc)

1(x+2+��y)

k =[ (Qs=Q)

jjcs�m+j� y

(x+2+��y)c ( ycr )

1(x+2+��y)

]�4�3

(c2r(�c

)2y)1

(x+2+��y)

t =[ (Qs=Q)

jjcs�m+j� y

(x+2+��y)c ( ycr )

1(x+2+��y)

]�1�3

(cr(�c

)x+�+2)1

(x+2+��y)

e = [ (Qs=Q)

jjcs�m+j� y

(x+2+��y)c ( ycr )

1(x+2+��y)

]�2�3 ( y

cr�yc)

1(x+2+��y)

w |@O |=yQDt=Q=B ?UL Q@ xQ}eDtwO |m}rwQO}y |xUOvy \@=wQ u}}aD Qw_vtx@CUOx@ 35 |xrO=at R= =Q S w Q ?ULQ@ Qs |x]@=Q CU= |i=m |rw] ?}W|r= 38 \@=wQ ?}DQD u}O@ 'OQm u} Ro}=H 37 w 36 '34 '33 CqO=at QO w OQw;

%CW=O s}y=wN =Q 41(Pbed)m = e0S�2=�1Q(1+�2=�1)=(x+2+��y) (38)

Dm = c0S�2�1�1 Q(1+ �2�1

�1 )=(x+2+��y) (39)

Vm = k0S�4=�1Q(x+��y+(�4=�1))=(x+2+��y) (40)

Wm = a0S�5�1 Q(1+ �5

�1 )=(x+2+��y) (41)

?}DQDx@ e w t 'k 'c 'a ?}=Q[ x@ xHwD =@ e0 w k0; 'c0 'a0 ?}=Q[ '=yu; QO xmw k0 = kt��4=�1 'c0 = ct(��2+1)=�1 'a0 = at��5=�1 %R= OvCQ=@a

"e0 = et��2=�1

55

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%OWStr / d(x+2+�)=(x+2+��y)

50 Q�1=(x+2+��y) (50)

Dtr / d�y=(x+2+��y)50 Q1=(x+2+��y) (51)

Vtr / d2y=(x+2+��y)50 Q(x+��y)=(x+2+��y) (52)

Wtr / d�y=(x+2+��y)50 Q1=(x+2+��y) (53)

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%CU= Q=QkQ@ 54 |x]@=Q(�0)tr = max �0(�) = �c (54)

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"OyO|tv�m |}q=@ OL QO |m}rwQO}y |xUOvy \@=wQ "4"3"2

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"(Qs > 0) CmQL|m}rwQO}y |xUOvy \@=wQ x]@=Q wv `HQtW / Q0 458�0 503

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3. extrenal hypotheses4. maximum sediment transport capacity5. maximum ow e�ciency6. minimum stream power7. minimum variance

59

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60

London, Discussion on paper No. 6274, 11(2), pp. 145-