SOME METHODS FOR ASYMPTOTIC DISTRIBUTIONS
IN THE MULTIVARIATE ANALYSISl
by
Minoru Siotani 2
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 595
December 1968
" -.- .
.
lThis research was supported in part by the National Institutes of Health of
General Medical Sciences, Grant No. GM 12868-05 •
2University of North Carolina and Nihon University
TABLE OF CONTENTS
Po.ge
A~ltcl~ _., ---- ------------------ - - ---- --- - -- -- - - 'rii
S~ am.d ~c1um - - - - - - - - - - - - - - - - -'- - -- I
MetltMl ~QM'U1~~AtaLAti~------------3
I. I.
CTemuz.o1 clL<JClLif1wn ~ tht m~thB-d - - - - - - - - - - - - - - - - 3
1.2.
Hotill~ ~ To --<1ta1M1Lc - - - - - - - - - - - - -- q
I. Z.I. Non-nuii ~%pt>:lic dii1ii~ ~
up t> 8/l-m n-% _.- - - q
l. Z. 2. P.w.tm;~ poirn1D 1 To:t lin tItt 7W11 ~ - - - - - - - - - - 29
I. 3
01:ft.vL uOlTTt1'lli - - - - - - - - - - - - - - - - - - - - - - 33
1. 3. I. TM JULtw &i two H~~ ~ To"- ALa1iotw>.• - - - - 33
I. 3.2. 'vIold - Andvw071A ciaMi{fco1i.on -6~tic. - - - - - - - - - - 34
I. 3.3. Jfu. maJX-~ ~ ~F dMt~CUJ - - - - - - - - _.. 36
J : _. I
- 0 0.. ~ Bl'S.t/t),
(1'
Bl'S.tu,vW'Ct), OtnuJ Bl'5.tu.UIlt•
t...,
.
~I
App~<M/X.
. TCUJ"U!,()
.t~(t) /J1l. :.5 .1./.-40
AppemcLux 2. ToJ,.W ~ .6U1m6 ~ Ul1M..cJ.
Un § I. 2.1. - - - - 44
Pant T.
2
T:
(3)
FeWmo
Pant 1I. TM ev-a1u.a1Um
Z. I.
1. 2.
2. 3.
~ M1fTYLP1otLC v~ anu1 C~ ~
~etiono ~ IN-iakJit ma1Ju;CUJ - - - - - - - - - - - - - - F~u1M - - - - - - - - - - - - - - - - - - - - - - - - - - - T~
50
5\
plUllJimUJ nMulM - - - - - - - - - - - - - - - - - - - 53
Nan- nun M1f¥L>Uc diafJ7i1ru1ian- ct tfa tfA1 A1a1JA1ic ~ - -- 56
AAJt11L~ j;n tIt.t iJtw.c1mu ~ cmvutvnu ~ - - - - - - - --
II
2.4-.
N01l- null
~1Mic
d101iriJnJum ~ fut 4tilJAluo CM m1d11ph
imckp.QJn,dtma im. tIr1 ALp-dmun
COA.e.. - - - - - - - - - - - - - - - - 58
~ - - - - - - - - - - - - - - - - - - -. - - - - - - - - - - - - - - -
65
iii
ACKNOWLEDGE MENTS
~ ~ to
r:lwurnan c>[
m.g ~ ~ UJ p~ G. E. Nic!wtJan, Jr.
ckpa.4mvnt G8J~ gJ-~ m£ th ~P~ ~ vi.4i1ing
M.f'1U<'L>
tw
~ and th -c acilitito G6/2- 7TltJ ./l.W.IJ.I1.,ek.
01M Af.w.a1 ~ M1 cUu ~ p~ J (Ygtwm, ck.i~ t1£
DLpanlmwnZ ~ Sta1ia1iUJ, Nilwn UriMJeJuUt~ 'G~ f1M ap~ "i rrua leave
.in /Jfill &b :it 1Je~g a ~ pWMl ~ tIt£ rU~nt.
J unru1d JiJu 11> tItMJe p~ N. L. J~{rn w-R.o JlRA.d ~
~
ckatt
J
amd mack ~u1 ~F1imw·
aloo ~X-PIl.eM 17lfJ t/z,~ tJ p~ 1 E. CjU~~. T1¥o.ntm~ e-6
Bio~to1M1ico, ~ ~ -dtq:>ptYit.
1
SOME METHODS FOR ASYMPTOTIC DISTRI BUTI ONS
IN THE MULTIVARIATE ANALYSIS
MINORU SIOTANI
Uli.WeJt6J1j ei N~ Co.Mtma amcl NiiLcm l1mMM.i~
SUJrrvm,(¥~ amcl Ant1iw4etum.
VllILUnw ~taL:oliU) iJn th ~ ~M ClmU ~ ao ~ ti
'vLiMwnt mrilJUao. 0iUD ~ ~ ~ ~L1i.c .LJt1mp4 ~bUWmo
"b ~ AJ~1iCi1
0Jl.e
d..wCWJ.<J..eJ..
~71 Pant I , a mLthcd. ~ ~f"'. Atitio~ anu:1 ito o.,p~ wi111Je
cLwC<U1<J.Q.d. £.it
z ={~"
... ,ZI7I}
~ ~c,{ 10 iJn.d.LPR/ll-~
1>e ().
pxm ~.vwrJimv
dilJ1JuJu1ui ~ 1JJ
().
mo.twx ~ ~
p-1!O.IliAtJ. Mn- ~
1W'1JnUJ cliotulnilion ~ ~ mo:f;U;x, ~ ( > 0 , i.e., ~ ~ ) anu1
lit n SlI k a px p INi.oJw.it rrw1Jwx dio1Mlni1Ld ,irt.rkp~rWntlrJ &£ z ~
loth
W~ ~ Wp(OZ,1l) w11:k
~ E. Jftvn, titt A1:Jiti.a1ico ~
ida
n ~ ~ ~ cmd ~
-G.w. ML ~ T( Z. Sll) &C tIu. Cwm
-VdS;'ZZ'), ~ t1wt
~cWn.
tJu duJ~bitio-M e-C TCZ,L) (l/lt ~ ~~ 1:D
~rling ~ tftt irruu.t irJ..uw dut 1c B. L. INdrJ,. [ /7 , 18 J a;nd G-. S.
7 ] anuJ. t/iti;u ~, tIr.t die1iu1ni1imw wi111>e
.iIrL tILt
J Mn.W [
.expo.mcllcL
rem
~
R{T(Z,Sll) ~ ten>} = Ao + ~
+
~: + O( ~3)'
This research was supported in part by the National Institutes of Health of General
Medical Sciences, Grant No. GM 12868-05.
2
~
A. , A, am.cJ. Az. ant jlluhf.il1U1..emt ~ n.
~ d.£oClLibtm.a tAt ~ ptlbClrlwu ~ tk mJ1Jwd. (§ J. I ), tftt HoWliIna
2
~ged To /Jta1ioUc in twJ.uJ .i.tn eMail 10 etUainIt tILt 1iMn. &C ~ n-2 iJn
t1L£. .vx.pa;ruJ.tci ~ ~ tItt di.a~fm amJ ahJo 10 p;uwirl.t. a CDnfI1i1t ~
~ tL ~ (§ I. 2. , ). S(J7nJ. taM.tA wftLclt. 7TWJ Je ~ ~ appltfnra tItJ
1Tl11ilrM. ~ ~ inL Appen.diao J MU1 2 at tItt.RRU1 ~ Pw I. JKe ~
2
~ pLWIrVWr poi'n1o e-c To .int tht null caM, ~ dio~ kfltl Vn § I. Z.Z.
~1t § I. 3 , otIwt M.ClImpiJA ~~ cam k 1Ju.atuJ. .f,y mPL 71tltb1 flAt ~o1uJ..
J~ w. tlttaa1io ~ 1mc Ho1:dl~, T:-~ (§ /.3.1 ), Wo1d-A1Ul.uwo-M
claM+ca1ibn A1:a1iJ.ic (§ 1.3.2) MLd tJu mtllXAmuum ~ ~p1 ~
1M! 07IL daw-a tiuJ1 am ~ pinit di.a1iu:A1~ ~ taro
~F ~ta1MJ1iuJ C/JIlIlJeMailn.edby(JU/l.malwd .
j,n pant ][, Aitmpit ina ~ ~ fn camptilim3 tItJ. ~p1Dtic
( § 1.3. 3 ). JILt
Vil/li,lJIrLCt6
QItt
QIrI11 CC7raniMV<:.tO ~ A~1i.co. ~cJr.
~ (§ 2.1).
lW.
~ ~ W~ mo1n.i.aa,
Ju n.w1 ~ ~ ~ 0Ai<J.L0 1A!'kn we. lJJ7JIYit ]A ~
tItt rum-ruJ1 t1Af"Lpt,o;t,U r1io~ ~ ~l 4taJi.a1A.,'co.in tItt ~
1Jy tIt.t M co.11td d..t11r;. mi1Ju,.d. ,gn § 2.2 tItt ~u1to ~ -<)0 ~
~ ~g 1h. ~,u1M (1ftt ~tu1.
00 dr.bur -Mw-"ttt.L ~ QIli. UMJi. tILt iJAt AJta1iotiCA ~ ~
.iAt t.Itt ~c1wu ~ tILt po-pcda1ibn ClJ1J7JJliiJInC ~ (§ 2.3 ) MU1 ~ 7TUdtipiL
Q.$o.¥iA
ilTuLtp~ (§ 2.4) aM
't!uo.tuJ. .
3
Pant I. M.itfwd b1 (). -<1&4
n1>~ ital.i.47:ico.
I. L ~ _~ %th miIMd.
M z = { ~" ... , Zm} 1te a px m fWITU10m rnituit
~ Zi
~ ~ to p- _ate TWl/fTl11i ~
with
0Jli
-ihuiJ.fRJrL~
CJJrrIIfMn
CD1JaJU.MLa
~ l: ( >0)
amd. lit 51/ 1>e a pxp W1Alw.it ma1iwx ~ ~ ~.Mtc1.Mlt ~ Z
0Aul iA ~~et 1D a, CiJilM1 WiAiwnt ~ WP(-tL:, n) wi1Jz- n ~ ~ ~dWL
anuL ~matuix, -frL:,
Le.,
I Snl(ll-~-I)/Z ex,p{ -ttn[ (Lint' SnJ }
(1.1.1)
znp/1. rrP<P-I)/41L:/nI 11/ 2 fr rUTlt/-i)/2]
L'" ,
lit fLO canold..u tJu MAf¥1AU diotubu11un ~ th AJta1.UJic T(Z, Sn) ~c! fA
(L fpAvcWm ~ z arru1 Sn A».dz, tJuJ
(i ) t/u, d.io1iU1ru1i.on ~ T( Z, L) i,a ~, am.d
(ii) ~(Sn) = pJt{ TeZ, Sn) ~ c I Sn} ~ a ~ Sn fuw tilt
.wx-patMion
J+-
o1m-ut
~. \Aft-
LUJi,
tJu Mto.1:innd;
L = ( O"iJ)
and
4
R{T(Z,Sw)~c} =
=J
f
&{T(Z,SIl)~CISll}dWp(+L,n)
5,,>0
~p{1iL(S,,-L:) a} PIL{T(Z,L:)~ c }dWp'1rL,n)
s.>o
=
J
erx-p{titS"d} dWp(tE,n)· ~p{-Vz,L;a} fA{T(Z,L)~C}
5.>0
r
n/1
exp{-liLL: d} P/t{ T(Z,L) ~ C}
=
I I - {L: d
=
G· Pn.{ T(Z,L:) ~ c }
(1.1.2)
G = ~~{-1itr:a - t.&a/I -tEal}·
.9t.Mwu1d .& niW1 that ,itn tItt a1JUve anu1 o.1AtJ ihL 1k. ~,
AtanuLa ~ [t(l-tSrs)afHT(Z,S,,)~ C IS,,}/~s7S Js -1,;.
(I. I. 3)
drs l1d T( Z, L:)
~ C}
ll
UA4* ~uia
2
-ball-AI =1itA+-ttitA + -t1iz.A3 +...
e
=
~P{ ttt(Edi + ~1it,{Eaj + f,-tn,(Ld)4 + ... }
=
1+ ~ "lit(ral + ~J [t"lit(L:a)3+i{~(Eal}lJ
+
+a[ Z1it(~a)'~ + ttt(L aiud~d)3 + t{vdrol}3]
+ O(1i+)
(1.1.4)
5
(1.1. 5)
lJJitem c iA cmwtamt , the dtaiud ~1hUc ilXpMLOiDn CQln.1Je ~a.iIn.ed ~ (/.1. 2 )
and (/.1.5). Jfwte, (jIlt ~ the CM.M wItw. C i.J rwt cmw!amt )rot ~ tIte ~
C == CCn) = Co + C*(n)
urfurte
CJn)
=
OC n- L),
i= 0,1,2, -...
~n ~
CM.M
we ftaNe nud 10 ~
~.
ur~D=a/dCo.
Jima
exp { C*(1l)D}
==
5Xf [ f (I(m + Cz(n) +
=
I
+
C1(l1JD
T
u_
}
D]
{C2(TI)J}t ~ C~(n) D
2
}
(l.l. 2)
6
+ {C3(1l)D + CJ(n) C,(nJD:& +
+ OCn-4 )
~ C,3(n)D3
}
,
e~ = I + [ C,(TI)D + ~ ~ (Jur(Jst c3ndtll ]
+ [ C~T1)D +
l C~(n)D:t
+"* ~t~ <y';ro;t ars ~u (.(n)D
+ ~2 {t ~
~ CfwTCfsta;,1TaTS~udvllT+
+ [c3(n)D + Cz(n)C,(l1)D'" + t C~(n)D3
+*
+)
~o;;rCfstcr:v(JlIJla"dtlla11llldn}
v
~
nn~
~tJ erllTcrstaTS~ { C2(11)D 1- :l C~(1l)D2.}
+ ~. {t~
~ a;ro;ta.;,~xdrs~.9,.,aru}C,(n>D
mm (j.P"sto;.Va7S~alllJ+ i ffiumt
II
II
+ ~{ 2 ~crvr~~va;rarsquallllTaXS
~
+t ~
(j"Oir<rlt<ful1~x 011 ars 0tu awdrJ
~
+
t
~'"F1 rstU 11'" XI~ Oit ~~ 'YuT<J;t Ojv (J.,r dtll/ d" drs dtu dUVaXl! } ]
II
+ O(~)·
~n~t ~aiJn, tltt ~1dic MpwnaiJm &C tItJ. dMMJi,un ~ T(Z.Sn),
we muJ. to .vrro1.u..o1t tIr.e ~ dJA.iN~ o-C Rd T( Z,I:) ~ Co} uitU. I'l.Uput. 1D
..eivmem1A ctb L 1m (t. I. 2) (9It (J. l:~ ). J~ dJ> ~ we.MuJi U4t tltLui.w. ~
~aUon iIrr- ~~
amd ~
e
7
J - R{ T(Z,(L:+E))
~t }
(1./.10)
~
t -fA C Un ( 1.1. 2) (0/2, Co Un (/. I. '6) anuJ. £ 1<> a px p m.o1 ~ ~
compG<lQ.d ~ .&mali iJnC!UltTLRJ111A t Lj to G:J ~ALclt, tIw1 ~ + E .UJ.i1i11 ~itiNe
~. JMm-,01 J~ ~
+ ir.
~ trs ttll tUliJ Ex~ drs dtll (1Ull7 (1):# + .. -J R{T( Z.I:) ~ t}
~
(1.1.11)
On the, MIua tianLd , AMta
N(J! ~ ,l: ), we cam
J
~i
ilXpJl..e,aA)
=
(i -= I . -_.. m)
J 1m 1Itt.
(2 IT flllPIl l L
.fA
rlia1iti1nJ:uJ. A;ru1LpPlTl-~
~
r f· .. Jexrb{- ""~ tit (Z - M)' E-' ( Z - M) } nd
m 2
/
l.
II
D
~ M= (f-i.
t "',
#111}
~ to
';7 •
6?p
(
I.I.IZ)
o.md
D~:
T(Z,(L'+£)s..t·
(1.1.13)
(1.1./4 )
wftm
~ MJ a p><p ~ ma1JWx., cLi~ {~I' '-', ~I'} wUlt, I ~tl < I
,w p~.()lb&,
~.t
we CML 1A7Ja E UJit.k ;.Iu.-Wc.Wn11~
g~ ~ tk 1iullrw~M
I L ,111
=
I L: 1111/2 1I _ ~ Im/2 ,
b+t olf L. JfuA
MM11 ~. Mna the
8
R: T( LY. L L: ) ~ t .
(1.1.16)
9n B't~ that tIt.e p~ ~ le ~, T( LY, L~) ~ k iAul.epud.em:t ~ L (llA , ~ £XQlmpit ,iA "UL( L +Er'ZZ' = "liz, yy' ), tItt 1Jn1tral tin (1.1./ 5)
Air.au1d. k ~d .£;Xp~, anuJ. J Mr.cuid k i/Xpa;ru1u1 ~ /liJJput to
pow-eM ~ £ Lj ~. .9£"tJ,M fA tItt CiJ,<te we ~ ~ tIt€ d.iAiJu..d. ~ ~
Bt{ T( Z •E) ~ t } ~ CDmparUnuJ tItt ~ wi1ih (J. I.J /), anu1. fLvna tILe aAf'l'pt.ot:.e MpCllM-Um ~ tk. ~ 'i T( Z, Sn ).
1l..tJx;t.1it M· CJnUJ.idiJt tIt.t caa.e wR.e..u we cam mMu tAM ~ tAt CMluulvuilit
~Jion ~ T(Z,E). I.d CP(t) k th. C~Uc ~~ o-t T(Z?Sn)
am.d <P(t;-Sn) k tIti
871 iJJ ~(;xed, t.e.,
~ ~(Lc1vzia~ ~ ~ T(Z,SrI)_
a,nd
'P(t; S,,) "'" Ez [~p{ LtT(Z,Sll)} I SlI],
~ L R.
=0
Jfwn- we fuwe
<pct) -ESn{cp(t:
Sll)} =Es)€4p{1a(S"-L)a}J·~(t;L)
"'" e q,(t;L:)
(1.1.17)
9
.,-
urn.utt
tIu AOIm1
@ io
~1t ~ 10
~
oi
E,
~~ M C1. I. 3)
.RIl!o.l!.u.i1L vwUCJU<J d.vwv-~ ~ <PCt; L) wi1ik Mpm 1.0
we CAJnA.id..m ,/>( t ; L: + E ) wkvte £ JA tk JliJmi10JL nuUrwx. &h
J
,~ ~ L: M Un
~ ~:T.
1M c1..wiwJ.
(I.I. 5).
M
c1I1ul CCVl!UJ out the
(I. I. 10)
~ we MNe
tIt£ MAJm.p1Ai.c
~111f1Mic d.1<iLu1ru1ion ~
A1imU1a/l-
uptJht<1Um
M
we
M-pMWllm ~ eJ>ctJ
tn t/u;o ~,
7ruLy i~ ~d 1Jy tid
T(Z, Sn)
'~b'wuda.
I. 2. HM.dWrt@dt ~a1LJ41 To Ato.1w~.
1. 2. I. N071- 7UA11 C14tJW1'~Uc cLwM-utum ci
2
z
2
tit
TQ tLp Ie (JILd.vz, n- .
/L<)
ChnIJ.icJ..vt,
T( Z, Sll)
==
To2
=
T
lIL
S-I
1n
n ZZ, •
I
=2:~.
J=-I do
J6 ,W th Ata1M1ic pM-PMU11J:1
_I
Sll ~d"
H. H~~ [ 3 ] ~1w4thi ~P~'u'
Ho : M = {#I,···,.urn} = 0 (l~t
tl L-1ZZ' fA
(1.2.1)
HI:
M +- O.
J,t 10 w-e11 ~mAm tIw.t
cJ.iA~ M a non-e.tIritao1 cit.i-iJ~ lAJ1th mp ~ ~ ~om
amc1rum~~
w2.
=
"lit 2:-1MM
I
::0
f Jl r:,-' # .
d-'
J
( I. 2. 2 )
J
t. e., ~ F = mp/2 •
D{T_",-IZZ /t}
rn..
l/T.£-:::"
=
e-iJ/2
~
(ut/2 )d
~ J! 2P+d r(F-tJ)
Jt yP+J-'e-X./2.dx: == X (t·cJ)·
2
o.lt-
!lip
,
(123)
.•
10
Sim.cQ, "tiL (I: +Er'zz' -"tit yy' = L:, ~: ~L' we
~
J
eb DiJUclt1t11J ~.
:=
(27tff'1 1- f
I
m/z
J-. J
E I
-m/2
J ~
uaiIng
&¥
~ A101ni, comp~ we Wwe th ~ ~:
f~¥~t
-{ IfI--:e~}
cam, ~
exP{ -tlidY-H)'(I -~ )(Y-H)} Udlj,
~
eW't1ex,p{ 1:1h(I-~)(I-~Er'(I-s)HH/E}
X:p(t;O),
(1.2.4)
~ E La am ~~ ~ tIuit ETX~(t;W:l.)- X~.t2T(t:ut) ~ r=o.I.2.,···. JJW7Tt,
tilt flda1imw inL
~ ~ .. E-I
(1. l. 4- ),
I
J cam 1>e ilX-pJlt<)AU1ltn 1tJzmu, ~ M,
£, E
anu1 E
M
X = (L +Er JE - I atnd
fl
I I -Xli (m/l=
=
L-1MM·
(1.2.6)
1+ .1f1iz,(XA) + -f-{Z1idXAt + mC!iz.XA)2}
+ f§- { ~ 1it(X.1 i + 6 mtit(XLl) tit (XLlt + m(,lh X~ i
2
}
+ 3f4{481h(X~t + 32 m1iL(XLlrlit(x.1i + /zm("tit(XL1t /'
+ 12m2(tt.X~)"Ctit(XAt) + m3 ('ttXAt} + ... (/.2.7)
(1.2.8)
11
~ L r5 = a1S~' J cam k ~ M a. M.IliuJ in powe.rw
~ th d.uJA.tJ. ~ o-b Pn! "tiLL:-IZZ' ~ t}.
W~
M= O.
L. e., lin tILe
n«11
&i
crs'iJ anul"tIuuJ we, CML
CM.e,
(1.2./0)
Jftl c1.uJtu.lnJimz, ~
T: iJn titt rw1l cao£
lIJlU)
gwem ~
l<. Ito [ 5 ] iJn iJu
M-
pORta-
111 ~ up to ~dVL n- 0Ke wn- nu1i didAdru1i.on WM 1ilta1.ed ina M. SLo"tanzi [13 1
lin th.t AlamtQ uro.y dJACUM.fJ1 a1Juve anu:J. l~ -By K. Ito r~] wKo wwJ. tiLt i¥
Il.lfJlJ..<Wlito.1Um ~ 1JrJ. cJuvw..c1vu;o1i.c FcUon ~ To:! cUu:W P. L. HAll [4 ]. But
~ cev1d. ~aim th tvvm ~ ~ n- 2 Q.IrUJ .£JVe1L, tItR, Wvm ~ ~cluz- n- coniaim2
•
J
.uJ. Al~ iJn,~ 'Wvnw ~ lU1J77tJJUca1 ~.
!)n w-Rd ~ we wi11 ~ tIz.t rwn-rw11 dJ.,o~ ~ To'- tq.> to ~
n- 2 irMui. an (I. I. 17 ), ~.Mww.d , at tItt AfJImL 1.imu, tk rl.itai1a ~ tItt
r~ dMCIli&d. i;n § I. 1. Jo d.n -dO, we nud. to ~
ep(t; L+8) = E[elX-P{ Lt 1n,(l:+ Ef'ZZ'} ]
== (ZIT rll~/ll
L flll/2
r· Jexp{
RlIIp
Lttit( L+ ET'zz'-
t tt I;-'(Z-M)(Z -M)'} Vd~~
( I. 2.11 )
~ Rlllp A1MuLo ~ tk ~ mp- A-fJDa. Wi 1ll1l1u tJr.t rum-/1i;n~ ~
~
z
==
py un;th, P Audv tJwt
twmA-
12
(/.2.12)
~ 1 1.<1
(l
~
7Tl41iWt, ci.iaa {1'"
... , rp}
aruJ. we. 71UlAJ ta..U YJ A.uJ'1L tIw.t
IYJ I < I ~ all. J" I, ... , p. :J~
'P(t: 1:+6) === (27t5J11P~1 I -y /111
4
r·J
R." ~p{ it tit YY/-*1idI -Y)(Y-H)(Y-H)'}1Tdj.'J'
N .
;
0.2.13)
~
we put H- P-'M. SinLcL
Lt
=:
1h YY' - ttd 1-)' )(Y-H )(Y-H)'
-ttdI-1)HH ' -i["OtHI-ZLt)I-1}YY / -2tiz(I-1)YH']
- -t1ir.(I-r)HH' + ttidI-Y){(J-2LtH-'Yf'CI-Y)HH'
- flit {(I-zLt>I -y HY-i' )(y- ~)
_
"'P a {(I-ZLt)I-1f'CI-Y)H,
Wf~~~
CP(!: LHl-(1-1L!)""P {!I 1~:~rYl
II
t" exp{-ttt(I -1 lHH' I '
eJXf[ {2(1-2Lt>r'tdI-Y){I -(f-2itf'y r'U-1)HH']· (1.2.14)
<pet; :Eft )-(l-2Ltf''t'1z e6f/1.1 I -ZitCI-zitr'X ,-"'#"
){ ~~[{2(1-Zit)r'1it{I-2it(l-Zitr'xr'.Q,J7
1JJfu1tt ul, Q. am.J.
Mtatic-rw
Xw th. A1amt.f ~M!A M~.
lJ.mg-1.tt
(1.2./5')
UA U4t tILt
13
~(t)
= (1- Zit )-1,
fj.(t) == .QCt)-1 ==
Jiif 'P7Il~(t:
ZLtC/-2Ltr1 ,
(/.2.16)
iA the ~~1ic ~ ~ CMl1JLo1 W-A1fU1'U rJ.iAlrti1nJi.an
mp ~ ~ g,rc.uclom. JRvn
wilh,
0)
r1l1
~(t ; L +e) = I I - ~(t)X !2 e-e.fh ~ [ 1-.QctJtd I - LiCt) Xr'.Q 1'flllpct ; 0)·
9t i<J
A11Jrv
t!at J iJn (/. 2. 5) C1Ind
<f> ( t;
L + £)
W
QJI.R.
~ tItt ,(jfJlYYl1. ~
(I. 2.17)
urith
th.t~~~~:
~
we -/ia;ve tIt1-d~ ~ ~ .up~ in th kth ~.
jn 1k w11 CM-e, i.e., ~ M= 0,
~
'ret;
~c1v
C&VUA-fJoncLo
to
L:+E) = I I-ll(t)X
I-m~ ~lIlp(t;O)
(I. 2. f 0 ).
w.aft, tht old ~ 1M, £/x-p(l;Mion ~M j;n
iJrL
(/1./7) CMt
(1. 2.7)
k J?/X.Pa.meIuJ. itn th ~ ~:
<pet; I:t£) =[ I + { ~L\(tlu.l<
+ +£(tlt:.(tJu.XQ }
anul
(L 2. ~ ), <P (t ; L +E)
14
f
+ ~ Li3(t)"Vt X3 +
f
2
tNt) (titX )(Lit X)
-t-"
n
3li3
(t)( tit Xi~
+ i..eCt)L¥(trtitXJ.Q. + ltct)li't)"thX .Q)(titX.Q) + ~~.eI(t)l13(t)(,litX!2)3
ll
2
+f.eCt)8(t)("t,X )(1itX.Q) +
-t-
It ll(t)!f(t)(titXfCliLXO)
t.ect)8(t)('/it)()(tiX .Q.) + -TR-.tct)ll3(tHVtX)(til,X.Q.)1 }
+t
1
Q(t)!t(tJ"tit X+ll
+ l (ct>ttCt)C(iLX 3I).)(1itXil) + f ict)S(tJCtitX Q)%
1
+-n-i(t)L1+(t)("titX1nXlitXut +
fs4l (t)f}+(t)(titXQ,)4
+ T9: .e.(t)~4(t)(1it X3 Xtiz,XQ) + -(fQ(t)A+(t) Ctir,Xl)("Ot,X )("tit XU)
2
+ ftJl(t)A+(t) (1itX i CtitXD.)
T
X.Q(tJlNtJ(ttX)(1iz.X S2.) + f
3
.t(t)[l4(t)('tit X)(tLtn Kliz.X.Q.)
+ crt .t(t)lt(t) (tit X) (tit XU?
T
+ .e (t)Li+(t) ('lit X2 )(1iz, )(2Q) T -if- .t(t)Ll+(t)("tiI,X 2)(1il.xnl
+ 1~21l(t)fJ+(t)(tir. XrCfi" XJ1J + ~2 .lct)fj4(t)(tit XlCti Xn)2 }
1
+ ... ] 'flll~(t; Cil)
(J. 2.20)
15
(1.2.2/)
~ck M tk cJu~1i.c 1UffLc1Um ~
w-r;tk
mp dt~
~(t; L +£)
~
tk rum- C1/fl1ruJ clti- ~ diatu1rrJi.M
pu.ulmn
DJn-d lW7U1JlituALJj paJUllrn£1m W2. Uioifltg (J. 2. q ) ,
CQ;n 1Je ~ im tJu 1/XpatrulJ.J. ~ ~ ~p.ut 10 puwe!W ~ Eij ~.
lit ILOAmliwc&a tk rwta1imw,
[TS] =Vz.l2-'L rs,
[rs /tU]
=
(rS) = Ut'L-'Lrs Q , (rs Itu) =
A~~)(t) = -W- LHt)[rsJ +
A~~;uCt) =
t'tL-'LrsL'Ltu , [rs ItUI UltJ] =Utl2-'rrs~-IL:tllL""'L1lIII'''' .
tit l:-'l:rsL-'Ltu'Q,
t ..e(t)~(t)(TS)
(TSltu I Vl1T)=ttI:-JE'rsL:"iLtuL-ILUIlTQ,,· ..
(1.2.23)
,
t{ 2Ll(t) tL12(tJ}[rsltll] T f t!(tHTS] [tU] t ~ l (t)i)2(t)[TS] (tU)
+ t .Q(t){ ~(t) +~2(t)} (TS ItU) + +lct)l:1 (t) (TS)(tU),
2
2
ATS:~.VIll(t)= ~ {3l:1(t)+35(t)tL13(t)}[rsltulvlAJ]
(1.2.24-)
2-
+ W{2S(t)+i)3(t)}[rsJ[tuIUUJ]
i
3
+~
~\tJ[rS][tuJ[UlJJJ
+ t.Q(t){~(t) +2~Z(t)+ 6 (t)} (YSltulvUJ) + tl(t){~Z(t)+ &(t)} (rS)(tuIVW)
3
16
+ 4-'8 .Q3(t)d(tHrsXtU)(lll.lT)
A (+>
Ts.tU.,VIl1.x~
+
~ .Q(t}{.(\2(t) +f1 (t}} [rs](tlLll1UT)
+
-ff Q(t)6
(t)
3
3
(t) [rS] [tuH tTut)
+
W.Q(t){ 2l.\1t) +~l(t)HrSI tU](VUf)
+ I~ l(tJl.\3(t}[TS](tU)( llllT)
,
(I. Z.
25)
=m
g {4f1Ct) + btCt) +4£1 (t} + ttCt}} [TS I ru /l1lATl x. ~ ]
3
+-ff {3l1 (t)+ 3~3(t) +f1~t)Hrs](tlL I trW IX~] + ri {48"(t) +4Li (t} +ttCt) HTSltu][1TI1fIX~]
~
2
2
3
+jf{ Zf1 (t}+ ~4CtJ}[TS)[tU][1IlAJIX1A] + 3W~ Li+ct)[rs] [tU] [l1\lJ"] [X~]
3
+t
.Q(t){.!i(t) + 3~'t) +3t13 (t) + ,64(t) }(rs Itul1furl X~)
+ ~ ictJ{.1%(t)+ 2l.\~t)t.64Ct~ }(rs)(tul1TW'/r~) + tj(t){8(t)+Z~'t)+S"(t)}(TS(tU)(1TW' IX~)
+kl(t){~3Ct)+~'tH(TS)(tU)CVllTIXM) + 3~4 .Q't)6'tJ(TS)(t/.O(tTur)(X~)
+ !.e(t){3L!(tJ+ 3tf(t)+ Li+(t)} [rsltllll1IlTJ(J:~) + f1(t){ 2~2(t}+3f1~t)+l1+Ct)HTSltlLJ(VUTI:X:~)
+ Xi(t){~2.(t) +ZLY(t) +Li (t)}[TSHtul ubil X~) +
4
+
*
2-
.Q(tJ{ Zl1 (t) +Li't(t)} [TSJ[tuIVllTJ(X~)
3
ft ict){ZL\ Ct)+ Li+(t)} [rsltu] (1TllT)(X.~) + ft .Q(t){8ct)+ 8+(t)} [TS][tU] (trW/ X~)
3
T
+- .QltJ{ 63(t) + l.\+(tHCTSHtU){"Ul.lTl X~) + ff Q(t) l14(tHrs] [tu][V'kT](X:~)
'5
( I. 2. Z6)
(I. 2. 27)
17
'~
-eM
cumpOJl1<Jfm t{
~
¢(t;L;).
tIu:o MpMW.iJm witIL
~
(1. Z. 20.)
tke d.talrzuJ ci.£Jz1AratiltJ.e<,
Jndo~M,~. we7nJ.VjttaJua.wru.mt~tkt~~iJn
~ rs. t u.
etc. £.it
f1(J
d..eJim.e
(l)
_
81'S.tu.(t) -
I
(3)
Brstllu..ct) =
• ... ..
-3.I
drs
\ ......
lUJ
L(6'3) Am
rstu uri t ) ,
..
(~
I
IT ~ An .tu(t) ,
Q 'tu l1.rr,,(t) =
Urs.. . d
lZ)
~l
-3•I CA
UlI1.xu(t)
($'... ) rstll
., 01
B~)(t) 'Plnp(t; uf) ,
drs
'P(t: L)
dtU
~(t ; 2:) = 2 B~~ll(t) 'fmp(t; Cif),
= -
I
CI. 2. 28 )
(1.2.30)
--
U. 2.31)
(1.2.3l)
Mna tItt ~c ~ 'P(t) ~ To" -7iz. S;' ZZ' cam 1Je ~ Am. tk
~ £;xpam.d.u1 ~ 1ry inw~ ~ fliAJuJt, Mr1:n (1./. /7 ) :
anuJ.
<Nt)
= [
I + ~ ~<rur~ BT~~~lt(t)
+ ..l..nz {- 8~
~ OkrTa;t~1T QLArs,(3:t~'11,,~(t)
...
w
+ O( ~3) ] 'fmp(t; ul)
+ 12 2
(<Yu/~1 (fu1T~X B~~~lI.n:u(t)}
TStlll111n:~
d
•
•
•
d
(1.2.33)
cut tItt A1lJITn.ma1Lan, Un (I. Z. 33), lit M, ~ ~ oli. A1WtplifJ th 1:vvtM
urhiclL apP1A/Z, ,lin tIr.t B:~:.(t) ~, ua,i;na tilt PMp~ ~ tIu. 1iw.u. Jr~ Vtalmph, 1h
1, CWZ!l1J
50~
fW,
RAIJ~ wc!ud :
18
-3', L:(TS/tUllTW]
.
(~)
= [TS/tU/1JIAJ]
1
, E [TSltU] (1TUT) =
-3
.
(~)
+f
[YS/tU](llUJ) -to [TSIVIJ1](tu)
+ (tU/UIJJJ(TS) J'
L
...L
41. (1)..) (TS ItU)( lJl1tl:X:~) =...L
3 {(TS /tU)(UlII/ r~) + (TSI11IlT)(tUIX~) T (T5/ X~)(tUll1lJT)} .
011 tIte n.e~
7ltctMlVUJ .Vtt ~
cmn,pulatimt O/Ii. 1uJ.Ld i;n A~P.MU1i;X I at th MUJ.
~ PClIIt I.
'fLt,x,t we JIlTo.1u.ll1t tIzl ~ ~ 7IOJlinu..o ~ .J u.J,. M L: (J1l1()st [ rs I tU].
L: CY"ll7r ~ o;u (TS ItU)( vun, .ilc. S.iJn.a
(I. 2.34)
(I. 2.35)
U. 2.37)
tIr.e ~cn ~ ~ tItt 1uvtrw &b ~ 'ttPM
a.a l.iotu1 iffi- (A) &b App~ 2.
ClMt
-& ~ calaJoluJ.,
e
19
aF
du1 ~ lo1owz, ~ romptila1icn i/J nxuJ.u1 ff'L th ~
Cffi'lZ~ 1.RJvmo ~ (7S), (rs I tu.), etc. J7&1.eJX.amz,p1.t 1£1 U<J ClYM~
!.it L:-1 =(Q""C1: ... , a'P»). Le., cr- CT) 1>e 1:ltt r-th c~ ~ E- J and U'=(WJ"'.'~p),
i. e., w~ .,Be tJz.e T- tIL Mw o-f> .a. JRRm Il}t fuwe
= Tn L-1 0.' r: Q
=
"tit E-'M M' r-t E r;-' MM'
=
tz,rt,
20
.9{ we put D.:. I,
[15] [ttt IlJUT],
k Aiuru1d .Be ~
~ ~iNe<l
£L<)
to
PCp-t-T)/2, 1k
a chc1- ~
~
~ (JU/l.
vai.ut ~ Z
()WT<J;to;,V'
cempuZa1i.on. JM ~
tIu AU/~ ~ tRiA fAtp-e QIli. A~y.d Un Ap~ 2, (B).
71bur tIte ~a1itnw &b tk mi/t.ul ~ ()JU .eo.da co1cd.o1uJ. ~ ~
A lMnmL£mg tnd tIte ~ witiz-~ ~cJuM 0Jn.d tIwn lL<Wn.3 tIt.t (B) ~
App~c1ux, 2. :Tin iJXOImph
7
2=pw·
21
.'--..-
J~
(C)
~ A~pMLclix 2 C1Jn1aiM
011 v-~ &b -d ~ "l ~ Jni,wj ~p£.
wf!,idv app.£Q/l; iJn ~ ~.
&w- we
CMt
~
with tItt ai-d
oi
( i ) JM tk tvvm
~
( 1. Z. 33)
=
~e1ion ¢>( t) ~
tftR V£fJJlr1ct.uuAlic
c I. 2. 24- ) - CI. 2.16) Q;ncl A pp~ I
f9;UllJL
n-
l
,
t {2LHt)+ E(t)} ~crurCYst {dl E
TStl.l
.
+ In:2
8 ldt)D<J;.rCfst{
rstLl
+
+
[rs /tU] }
21 L[TS][tUJ}
. (1'2)
2
TStU.
{tr L
(rsltIO}
. (1'2)
itt)Ll (t) ZOUro-st{ III, L (rs)(tu)}
+f
2
l'StLl
2
.Q(t)li (t)
L:crurcrst {iT z=
[rsHtU)}
(6'2)
TStl.l
4-
(Pa)
I..
.
+ -fPK(t)
= fp(ptl){Zlict)+5CtJ}
+
(~)
-l.
t.Q(t) {LHt) + f:1 (t)} 20"uro;t
+
T/ Un
evn.c1 :z..
2
2
(t.J+ I) W .Qet){ LJet) + f:1 (t)} +
t
1it~t £ct) t:Ct)
+ ~ w'" .e(t)~1t).
Sim.a
-€(t)
c1u--/j~
10:
~Ct) + I
= (
1- 2itf'
a;nd
.e.Jct) ~
~ tit;.
dM1Ju1nilian lir11 2j ~ ~ ~, it ~ ~ ~ QJlJWJn,f-
"i .i.C t).
tfu- alrove UpfltM.ion with nMfW to p01Ve!W
to
tft.e cJuvud.vzioUc ~c1Wn.
J~ teu a.&ve .UJ ~
2
fNP+I){Q Ct)-I} + f~{.tCt)-2Q(t)TI} + ~(Ptl)wl{.e(t)-.l(t)}
2
t-
t tit ~l {.tct) - 2Q~t) + .let)} + t w
2
{
.e3(t) -1 tct) + Qct)} ,
22
~
<fur<fst B~.~(t) =.E.
Cmp-cd ).Q(tJ
g pCm-p-l) - -WTStu.
.,..
CI.
2. 38)
tIt£ 1vvm &b ~ n- r 1Jn tIt.f iWpMWimt ttb 1(t).
1
Cii) Somt p~ amd Calli. OJl11W.tW to ~ th 1wm. ~ ~ n- • Jfu!l.t<lU.£td
~~
lU)
aJtt
~()WT~t()llv B~~:u.wct) =
£k- P( r +3P + 4){ 3f1Ct) -t 3[f(tJ + ~3(tJ}
+ ~2 PCp+l){ 2tlCt)-t1l3(tJ} + £f&(t)p +
+
*{
w+ + cp+Z)tn.~:t }.ett){~Z(t)+ lil!>(tJ}
-k( p2+3P+4)W~ .Q(tJ{li(t)+2Lf(t)+L1 (tJ}
3
+ i8tn£t .e3(t)Lf(t)
+ ~ (p+I)(l.Q(t){~Z(t) + &(tH + ~CP+t)W11(t){ 2E(t) + il3(tJ}
+
ftw QCt)LYCtJ
lt
= - ~l
T ~
ttrt.e"(t) Lj3(t)
{m -3mCP+I)+ Cpz -t3p+4)} + I~ (m-p-r){mp-w'")Q(t)
lt
- ~ {mp(m+ P+ I) - (3m+p + I ) w2 + lit,it } .ez(t)
+
4i[mP{m2+ 3m( P+ I ) + ( pI +3P+ 4)} - 3 {3 W+ 5 m(P+ I) +(pz + 3p+ 4) }w
1
+3w+ +3(3m+~+2)'lill.Q,2._tz,U3 ]..e 3(t)
+-k-[ {m1+3m(p+J)+(plt+3~+4)}WZ-ZW+-(3m+ZP+4)"ttSt+ 1it~t ].e 4(t)
23
~
cund.
>
7
istfAl1l1l.l)
B (4)
<1 <I
IlT
St<J;"O"IITX TS,tU,U"'.X~
(t -.1!Lt { 3
2
2.
2.
2.
) 384. m P-Zffl (I' +~-t4-)+m(p+I)(P+p+20)-4(21' +5P+S)}
.:....~ (mp-4-)(m-p - J )(ml'- ul) Q(t)
T ~ [mp{ 3m p-Cp+ J )(1'2.+ P+4)} - Z{4m 2P-mCp2.+p+4-) -pcr+I/"} W2.
2
+ 2mw'" + cmp-p2.- p- 4)1iz£t ]
1
.e (t)
- en [mp{ m2p+ mO)2.+p+4-)+ 4(p+ I)} - {6 ml pT .3m(p1+p+4 )-(1'+1)( p +p-8')} w2.
1
+2(Zm+p+l)w'"
+ (2mp-p2.-p-+4)1itS2
1
-
w21nQ1J let)
+ 3~4- [mp{m 3 p+zm2.cp2.+p+ 4-) + m(P+I)(t+ 1'+20) + 4C2 ti+sp+ 5)}
- 4-{ 4m3p+ 5m1 Cp1+ 1'+4-) + mcp+I)( IT+ P+ 32) + 4( ZI'2.+ 5P+ 5)} w2.
+ 0,'6 [ {m p+ zm (p&-tp+4)+ m(~+1 )(r+p+20)+4{Zp2.+ 5"p+ 5 )} uJ
3
1
-2{Zm%+3mCP+l)+(1'2+ Zp+7)}ut
1"
3(2m+P+I)w'-"titg,1
- {2m p+mCp&-tp+ 28)+ 4( 5P-t~)} tzJt + /2ttQ3 - (t",ll2 t ].e5"ct)
2
+
I~Z [2{1TI + zmCP+l)-tCp2+ZP+7)}u5'· +fm2p+m(p1+p+20)+4(5P+8)ltn~t
1
-2,(4m+3p+3)cdth.Q2_241n,Q3 +
+
*{
(m+ P+ r ) uitLSt
+ 4 tn.Q3
-
3(uzJtf ]..e 6(t)
(fn,rl f" } .Q 7(t)
(1.2.4-0)
24
0ftt d.e,ai;w:l ilX-pam.<1uJ. ~vm ~ <PCt)
<1.2.40) iIn10 (/.2.33). Sina
.Q
J
Ct) 'J;,pCt; oJ)
==
.w ~~ lnJ ~~
(1. 2. 38), (I. 2. 3q) am.d.
. -(lIIp+2J )/1
(1- 2Lt)
&Xp {J
-T W2. 8(t) }
CD
• .s
.. JlIl!'i'2d(t; w ),
(1,2.4-1)
'Pet) == 'PmpCt; ur) + fn- ,1...~ Qlm,p)<Pmp+2j(t; w'")
.01-
0
J
+ q6n2
+ 0(~
(}
j; ba(m,p) <"fmptz/ t , uJ)
8
.
eI. 2.4-2 )
J ) ,
~ Oci(m,p)A1
amc1 bim,p)~ wift Jk gwemiln.-a~.
W fex) 1Je tIte ~ ffnLC1i.cn &C T: MU1 1d 8v(x; uJ),& tIr.e
Uon o-t tItl rum-CfIrituJ
itrumrtlwt
r(V/Z)+rJ.-r
e-f,
anon-nnJ1 ~1btic J.u,~11 ~ tItL H01i1litng ~~
(I.
Z. I) La
1We;n; ~
f<x)
ffhtC-
w- A~ ~ witIt v ~ t6 .~ anuJ. rwn-
~ p~ of. Wi
44
dMw~
4
= 3111p(x; wt.) +
-frt ~ UJ(1n, P) 9mrt-%;C x;
+ q~ Tf
t
d- O
bj(m, p)
U)1)
~lrtpt2/X; Wi)
To~
(I.Z.43)
dtffmd
25
(1.2.44)
~
Uo(m,p)
=
0., (m, P)
= -
o.2(m,p)
=
o.3(m,p) =
mpcm-p-I),
2mCm "- w
2
) ,
mpcm+p+I)-Z(Zm-rp+J)w%+ tnQ'-,
2{ (m+p+ 1) w'- - 1iz,J2.'- } ,
(4(m,p)= 1il,(t,
bo(m,p) =mp{ mOmp-8)Cm-2p-z) + m(p+J)(3p2+ 3P-4) - 4(2p2+ 3P -I)},
b,(m,p)= -12 m2 p(m- p-J )(mp - w1 ) ,
bz(m,p) = bm[mp{ 3mp+8m-( p+I)(P~P-4)} -.2 {(mp+Z)(4m-p-I)-(P-.2)(P+J)(I'+3)} u:l
2
+ zmw4- + {mp-cp1.+ 1'-4) l1itrl'- ],
.~-
b/m,p)=-4[mp{lTl(3mpt/6)(m+p+J) + 8m(p+I)+ 4(P2+3P+4)}
-3{3m(mp+4)(Zm+p+l) -m(p+l)qf+P-16)+ 4(p1.+ 3p + 4 )}w'"
+6{ zm2.+ m(p+l)+Z } w4- + 3 f (mp+ 8)(Zm-p-J)+ 4(3P+4)} tt~t
- 3 mw21it.~t - 4 "tn,Q3
J'
b+(m, p) = 3 [mp{ m(m'~+ 8)( mt 2~+2)+ m( P+J)(pz+P+4).+ 4(lr+ 51'+ 5) }
-4f m(mp+ 6)(4m+5p+5) +mCP+! )(If+ p+ 14) + 4(3tr+ 8p + q)} u)'"
+4{ bffl(mtp+ I) +( 1'2 +21'+ 15)} 0.)4 + 4{ 3m%.p+ 36m + I~ P+ 3Z}tz.St
-4 (4m + P+ I ) w1 tLQ2 - 321it.Q3 + ernst)1 ],
bs( m, p) = IZ[ {m (mp + 8)(m+.2 P+ l) + me p+ 1)( 1'2+ P+ 4) + 4- (Z I'll + 5 P+ 5 ) } w1
- Z{ (Z m+ P+ I ) (m + p+ I ) + 8 } (J.)+ - { (m p+ 16 ) (z m+ pt I ) + 8( ~+ 3) } 1iL rl
+ 3 (Zm+t'+l )cJ"tt&t +
3
16 tnQ
-
(tnQ'J.
t J'
26
- b (4m +3 p + 3) u/tz.,Q,2 - 80 1it.Q3 + q(!JLS2.~)2 ] ,
b"Tcm,p) = Iz{cm+p+ I )ufttQ.'" + 4-tt~t
68 (m, p) = 3 (1it,Q,2
M
nClJ?'Vk
- C1iLQ
t
)2.},
t,
AI,~l, ... ,i\~
oma
~ MM'
1Je 1.k cftaM.~1ic;wot
~ ~ mi.n ( p, m). Silna
Tit QJ
==
tct-, 'A
J
d.,
J=
ri Q =L:-'MM~,
I,
~ ~ jj)
z,,··
tIt£
<I. Z. 4-5)
fl£<Udt (/. 1. 4-4 ) 1:RJt, ILO that , at 1.tMt Uf 10 tIte 11/1161- ~ ~l n-2 am-d ~
Fd p o.nul m, 'tiM puwm ~cUcn ~ tIt£ ,tM! dLp.RJrLdA on ~ 1h iwmo~ ~ powetl--AANm ~UJncliO'nAj &£ A~, ~{}IL mJJAuA tit£, ~
uwvk .iM.AJ. J/tQ TU.l!mRIlical W1Lpa.rLiADn ~ prnueM oi tIt£ tMt am.d tIt.e
/\ - M (1rM.u1 on tItt ~ JW1io CJU1JA.Um ) ~
tIte ~~
Ho: M=O ~t H; M+ 0 ~ 1tbW~~a1Um.
JM idlauMta aM.- ApuuJ CM.u> ~ ( I. 2. 4-4 ) .RJXfJl£A~ int Wvm..o oi tM
~
T:-
T:Xu,4
cmmu1a1iNe diA1izi1m1ian ~ ,
FCr) =
F<X)=
f xf(x)dr ,
o
2
Xmp(X;O)
2
2.
+ J!!1:.
4 n [ (m- P-I ) .l' m~( x, 0) - 2mXm/rl"1
Ct, 0) + (m +p+ I) X".... (x,. 0) ]
1..
27
+ ~h%[ {m(3mp-'8)(m-zp-z)+m(p+I)(3!T+3P-4)-4-(Zp2 t 3P-l)} X~~(x;O)
-12 m2.p(m-p-I) X~P+2(X; 0) + 6m{3 mp + 8m - (P+I )(p2+ P-4)} X;pt"JX; 0)
1
- 4 {m(3mpt 16)(mtp+J)+ ~m(p+ I) + 4-( p2+3P+ 4-)} X~p+6(r; 0)
+ 3{mCmp+ g )(m+ ZP+ Z)+ mCl'tl )(P%t p+4) + 4 (2 p2-t SP+5)} X:pts(X;O)]
(I.Z.46)
which ~ k C1J1TlfQJZ£d with tIti .~ (4.3) 1m [ 5 J ck t K. Ito.
2
(f f) m= 1: 9n ~ C1MR-, To .&wnz.e,a an tYlcI.bntll1fJ T -Atatio.tic ~ 1IX.a.c1
c1lA1iu; bu1ion M -known (Ail e. g. [ I ]), o.nu:l
2
F( X:) = 'X ~ (r ; w2. )
- 4-h[ ~ X;(X ;cl) + 2(P-cJ)X~2(X;Wl) -{ (p-on(p+2-W%) -4w% } X~4(X;W")
+ q~n:l[ P(3 p3_ 8 tl + 8 )X~(X; uf) + IZ P\ p-W2)X~2( x; w
2
)
- 6{ p( p+ Z )(tf-6 )-2( p3+3 p2 -6P-/Z )uf + (p2_ b) CiJ4 } X~+(X; W2 )
- 4{ 7p( P+ 2)(p+4)+ 3(1'2_ 71' -34- )(lc P+2-W") -7 w6 } X~,(X; cJ)
28
+4{p1+2?PtQS)w""-4CP+13)w6 + W 8
+ 12 Wi {(~+Z)(p+4 )(pT6) -
}
X;+S(I;cJ)
(3p+ lS)w1 (pt4-cd) -
w' } X~'O(X: uf)
J
• 1
•
2
+ 12W, (p+6-w2) X~+'''"( X,W)
+ 3w8 XP+J6(X,W)
2
ll
+ O(-W)·
(/.2.47)
jRia cam .& eJtuJud ~ cam.paJLiJn.g wt:th tht M.po.ml1u1 ~7I1rt d.uUAJeJ fwm &
-Mva.et c1JA~ &b T 2.
Oil) P- J: ~n tltw CQA£ To2 ~ a rwn-~ F .a~iie Wrtft, m MU1.
n dLF ~ ~cLam. MU1 7LQn~ p~ CJl=L:.~~,p:/(J2 anuJ. we
fuwe tLQJ == w2J , ci = 1,1, ." .
-tn-[ m{m-2)X~Ct:
W
1
zm(m-w )X~.u(t; w
2
) -
2
)
+ {(m- w1 )(m+2.-W2 ) - ZCm+/)w 2 }X~4(X;uf)
T
Zw 2(rntz-w1 ) X~6(l:;W2) + w"" X:n(r; cd')]
+ q~nl[1l1{m-2)(m-4 )(3m-:Z)X~Cr; WZ) -12 m2(m-z)(m-w%) X~%(:x:; CAl)
2
+61n {lO (mt2)(3mtZ)-4(2m +3ID+2)w2 -t: <31f1+2)W"'} X:++(x; uf)
l
- 4 {m (m+2)(mt4-)(3mt4-) - 6(3m%+Qm+8)ci(mtl-w
)-{
3mt4)CA)'} 'X~(r.;cJ)
+ 3{m(mtz)(Tn+4)(m+b) -~ (m+l)(m+4)(:zm+S)w2
+4-(qm%+ 48m+68)w"" - 8' (Zm+5)w' T w8 } X~+,(x:; uf)
29
+ 2uf{q(m+4)(m+6)-4C6m+ZQ)W2 +qw+} X~tI2Ct;uf)
+ 12W' (1n+6-W2 rX:tI4(:CcJ) + 3 w8 X=tI6(I; <J)J
(I. lA·S)
R67ttwvR. ~81iJ th C1WWJpo-ru1Mu.e. ~
*
(I. 2. 5 )
MU1 ( I. 2./7 ), if> we
pn.o-cc.e.d to .£/xpcvn.d (1. Z. 5) iJrL.2JX(J.,~ .JQrrn1., UTCUJ M we duJ ~ ( I. 2. 17) ~
we wauic1 -MNe ili .RJXpMWi.an ~ Bd To 2 ~ t J = f( t) Un tItt JJaImi. ~ M (I. 2.44)
wUJz. X:(t; w };yw1£o.d ~ ~)X; of).
1
1.2.2.
.~----
1
)
P.L~
~ J;2 in
_
* nuiJ..
CMl.
K. Ito [5 ] anuJ. M.SL6tOJLL
[ II ,12] ~ ~€IYitLa am
2
To lin tILt rwl1
c.M.e..
li1
Cel
cJtL-A~ w11JL mp ~. ~
Mf'Lp1Mic ~ ~ P~ ~ ~
k tht Uff1VL 100 c< % (0 < r:I. < 1 ) poinit ~ tke wrilitcJ.
fteulam. J~n,
"Pn.{ To2_tLS~IZZ' ~ tDt(TI)} = X~p.(C~; 0)
41 M~ that to/cm w~ tIte.<?MYU ~
M
(1.2.50)
eC1Z) Un ( /.1. 6), i.e.,
(1.2.51)
30
~ tLd-(n), i-I, Z,
Mi. ~
.d
on Sn. B~..e;x.~"tIu
hit-Mmd. AuU ~ (I. Z. 50) aWll-d.im-S UJ (I. I. 8) Qlnd tIt.enz, comptl.!WrLg kth ~
~ (I. Z. 50 ), we ~ tILt EcllowVna .tqf-~ ~ 1Je A~ W'itIt- ~et to tLd-(n>:
'0'
(Yl,d..uz. n- L anu1 mtlAJ
(I.Z.52)
*
{ tzct.(n)D + tt~(n)It -
+ ~z ~
~aur~ arsdt. t1in)D
)--:? O"llJrO"st Cfuv
fsimt
ors q.1l d
l1 .,-
onuL A.O 07L.
,gntAtt p~
CCl<l.Q.
t,in)
ClJ.m k~j;n~ ~ L:
b-wm
(1.2.52),
wiU:,c.h ~ ~ .vtcltanzr tILt ~ ~ drsdtu anui t,~(n) Un 0.2.53). Ju ~
JUAdt maAJ k ~d
5 ] (9IL [12. 1. l.d X = X~~(~) ... C~, Le.) tht
Uf~ too % peVnl &6 tItt citL-~ diatu-/nJiffn UJ1.1k mp ~ ~ ~dmn, MU1
rwm [
td-(n) =
2
'X +
1
*
[(P+I)(X... + X;t) + m(X4 -X2)]
+ :~1[3ml'(l- mp~l
){(I'+I)(X++Xl)+m(X+-Xz)}2.
- GmI'{(~+ I )( X+ +Xz) +m(14 - X~) }{( P+ I)( X4 -I ) + m( X+- 2X:& + I )}
31
2
-If>{ (p:t+3P+4-)(X,-tX++Xz)+3mCP+l )(X6- Xl) + m(X,-2,X 4 + Xl.)}
+ 3 {4(Z p1+ 5P+5)(X ,+X/i+ 'XI,. + Xl) + If> mCp+I)(X,-Xl
)
+m (p3-tl/T+ 5 p-t 4- )(XI1 t X, - l".- XZ)
2
+zm
(
,r+P+4)(X,- X,-X++ XZ ) +m3 P(X,-3X,+ 3X4 - Xl) l]
+ O(~3)'
'Jo.Uw
~ tcf,(n) aJl.t
gwem
( I. 2. 55)
~
(X ==
0.05 , 0.01 ;
p= 3,4
;
m==
, (I) 10 (2) ZO;
R=/0(Z)30 (5")50,60.80, 100 ilTL [13].
''----
R.e"mwvk,a tm th. 7UJIrIf1/li etJ. RUe ~ 1Iu. powen- ~ tIte. To'" - M .
i>e tIr.t UfpM [DO ~ % poVnt ~ To1 ~Ho: M == a iJJ 1iuJi" L.e.,
£At
7: ( rJ.)
(1.2.56)
U. 1.57)
wftMt.
p=1, z anul M
cliA1iri bu1ioM 01
1
To
m== I, we fuwe tIu 1/X.oit vo.!M.Lo ~ rc ~) Min 1Ju 1IM.et
lJJ7lf1.vt H (~p= Z, A.U H~ [3 ]). f-Wn.c1. we Cilt7t
caiw1a1L tit;. ~imLa1i.
0
vo.1AUA ~ tItL powelL ~
32
(1.2.58 )
\~ u/m.p);o amd
b/m.,,) ~
~.
M.J. All1l17LL M
wu tdo('Tl) ei (t. Z. 55) ~ 't(ot). ~
JU~ LW te .£IX.POlTL<i Bd To ~ t~(n) I H: M+ O} o..ccuu1&rLS to (1. I. 8). .&nr.e
tCll(n) Ln (I. Z. 55) ~ not dtp.e;nd rm L:. 1Ju ~ .RJXpML01Jm Cllm 1Je ~e.d
fwm (I. 2. 58) uiitJr. tr}.(n).Mta1to.J ~ r(oo 1Jy ~ tItL UipMtailm ~ X~CtJn):<iJ.
WiLun p~ 3 amd m> "
we iiaNe to
2
~ D - d/dX
2
am,d
tLcf,(n>
Atanul.
~1h 1'uvmo ~ ~ n- L i;n (1.1.55).
Sinu:.e.
2
D X)X 2 :
ut) =
n:l X:(X 2 ; w:l)
==
g)/(X 1 ;W1 ) .
D ~v('t; ut)
R{ T: > r(d.) I H: M +O} ~ R{ T
2
D
=
i {~V-2(t; w:l) -
:> tol(n)
==
I - Pn { To~ ~ ttl(Tl) I H: M+ 0 }
==
I -
Sy(x'; w1 )}
•
I H: M+ 0 }
['X:p(x';WI)+{t,olCn> 811lp(X :uf) + 4~ _. alm.~)X~~.tZj(X1;ut)}
1
.-0
+ { tzil1) 5m~( t; w:l) +
4-
+tk/.(n) 4'n
~ t~.,.(n) ( ~m~_2(XI; cd) - ~",p(X&; Wi»)
L
0lm.p) ~lII~+1J(XJ: ",i)
J-o
33
(l.l.60)
1.3.~
J.M n,W ~twp Hotde~ T: 4to1MlUA.
Ju cwrPlDtiv iJXpfJlYLOitm ~ p~ poimto ~
I. 3.1.
T( Z, Sn)
= T(( Z"Z&), 5
11 )
( I. 3. I )
cam,
k koluJ. iJn, 1ih£ A,j;mJaJt ~ a,o iIrL
io px m2 • In tIu:a
C/UJ£ T(
Z,L)
§ I. Z. Z , ~ ZI iA p)( mI
anu1 2/.
1itr.-1Z.Z;lvr,'r.-'z"z; 1i.M t-ht di.4?1iifniticn
=
~ethn( I. 3. 2 )
~ f. =m,~/l, fi
=
mz p/2 (J/YI,d B( P., ~) io t1e M.o. ~.
!It
Col k
tJu.
1f~ Aalio-WVna
(o<~<,)·
Jtlvn
tot.(n) = Col + tlol(n)+ t2~(1l)+'"
Pn. {F ~ tl1-(n)
m~ k ~
} -
( I. 3.3 )
A1ud tkat
Bf:. fa( C~)
(I. 3.4 )
J,y ~~ tfu ~a1iona A1mti1o.JL 1b (/. Z. 5Z ) , (I. Z. 53 ), et:,.
JM J te J,e 1UJuJ. lo.RJvV.iuolt tILL ~ o-f,
34
(l. 3. 5 )
(1.3.6)
E. c1 Br..ra(cCIl) =- Br... J.I\(C,,)
El Br..Ps(C,,) anuJ. X- ( L
+ eJI L - I,
1IX.pam.aio"n, ~ J i.m.
Bt:"~+J(CCll)
.6 L Et ::II
J
(
tIr.e ~~ wlMJ
t -\, 1. ) .
JaO.I.Z. '"
(1.3.7)
H.vne.t we. CMt ~ end tL.
M 1In. § I. 2.' (ct. [12] ).
I. 3. 2. WglrJ. - AruJvu,an:o ~ AtJi<J1.ic . M I~t> (~- I,··· . nt ;
L= , • :z.) 1>e lUlnU1mn At1Im~1.u ~ fwm p- VOJ7.i.IJtt 7lh1I!nIJ1 ~ 7t
. N(j:!<L: L ), l. -1.1 onuJ. ~ i. - 1. 1. ) $e .«JImpiL 1n.l.IJIM. l.a S" 1Je a.
i, :
(i,>, (
wL0k.4 ma1JWx ~ JntclLp~ ~ lei.> Ct~!JJ th WiMuvrt
diA1iu1nJum W~(*E, 11). Su.p~ zve KaNe ~ IlIJIrU1J:nn v~ l: w-Kic:Jr. itJ
:w 1>e ~ h.iaIr.vL 7[, QIL 7[;&. J~ WaUl-And.tMun:<J ~~Ati1­
U1i.c -UJ ~ .ly
35
.9t iA ffmown
l.imtitim8 diJJ1ili.fml:itm ~ IN (U) n,. n~, Mu1 n Unu1
BIL N(-f rt, It') o.ctM.J.Un.g (M) I ~ UJ 7r, ~ 7[&.
[ 1 • 16] tMt tItt
tAl i~ kJ N(t Da, 0 2 )
2
~ D iA th. M~ diotama
<1.3./0 )
anuJ. 1Ju ~g th. JlJA~ U> tItL ~~ M-f>JUM1bn ~ tht. cfJAfl.Uio.twe dMfulnJUm Fetum.
lJt u,o d.JnUJt.t tILt c/w.;w,c1.vUoM ~ &C (W ~ tIt )/0 ~ 4(1), Xca"
anuJ. 5 iJy 1J!(t; I"'. 1.(1', Sn). JRvn, we M,v.e
11
'o/tt) = E{e;x.P[ it ('W - t
D:& )/D) 11I, }
36
c/d.,' - ( a/dj.L~d.', d/;}P~~, .... d/afl~' ), d. l,,, am.d a- (1(1 +~rs) a/aars)
a.o~. now-11, UJ c1taJL that we 7TWJ f>MCWl. wi1JL tJu ~ iIn tIrt
~annL W"O--H a,o iIn § /.1. 1nit tIte co..lcJJ1aliun ~ 7lhW Mt Ao dLMftult eta iJn
tJu CM.t ~ tItt rum - CRJn1JuJ. To".
~
=:
I. 3. 3. Jftt 1MJX.iImwm t€ ~~ d.Uita;n,et<J. til ~i' i == '. '" • m J,e
m p- compcmenti 7Wlimo.1 vu1MA w1tMt
cli<iti btJ:iAn, 1.1; tIt.t mp-lIMi.iJ1i
Jciml
1:
k',~I:
K'M!:
kzlL
~
~L
I 1C,1
=
'"
Krill
K1I I '" KG.
.
.....
~ L (: px~) fA poM1We ~. Kij=- Kji. , IfeU' < I cvn.cJ. ~ ®
~ i L:d
S~ we
mtMLO
-IUwe 0. pJ( P mtilJuit S" dio1iU1mtuJ. 1m,~
~ 1JJ Utt wi<JJLMt rliibiJuliJm Wp(1r'L , n), ~ fu mMimwm ~
tIt.t duu.ct
p~.
(1.3./2)
m ~yA dio~
fwm tIu Wfm 1<> ~ iJy
•
37
T~ = mfIX { T~}
== ~ { g L S;'
~ i. } .
(1.3.13)
(1, ~ M
tIr.t pAtaRnl1 ~ ~, tltw.ia no ~ mi1bd ~ wfti.eIL we
cam ~cUhu 1Jr.l /Ja;m~ di.a1tti1w.Wm ~ T~Mt. But we Ita.« till Wt11 ~
~itm- uxcluA.wn ~
.9i we
fuNve
~u1a m~
rWe
a ~ciem!'Jl~
1Iaha ~ t tIt£ lIMt 1mo ~
Z
,
&C tIuA
900d a.pp/lhXimLtiLUm to 8tf T~ ) t
tveJ1L Mn· tfu:o
C~. tItt .wa1.u.o1i(J1t ~ tILt ~UmUJ. 1tirnn- io
di1{icu1t, M ML. a..ppMIXimuilum
m~ 1Je gwen ~ 7h ~pL1A.c M-pMLaitm eC it ~o.JinuJ. 1Jg OWl- maho-d.
f.U)
Z
a
} •
F
In t.ItW CJJrVn£,c1ion M
IUJ
canoirJ.vz, am ~UJtu 1IXfMLaitm
&-C
D{
2
1
t
1}
lit Ti. > C, 'lj > Cz
0.3.15)
~
e i.o tItt ~Mn.t ~0.1
&p.ma1bt, M
(I. I. 3) 9't (I. I. 5 ).
7li!£t we ~ te co1culo.1L
J t j = Hz. { z~ (1: +E r l ~ L > c~ , i J( k +€ r' i j > C:
= (ZTI f~ I L
r' (1- k~. ff'/'-x
}
38
x
J;J ~p [- 2(1 ~ fey) {Z~ L"I ZL -ZkUI;L-'Zj + ~;I:-IZJ} ]diLdZJ
(1.3./6)
,
D* :
-I
Z~CE+E) iL
2
>C, ,
(L3.17)
':7
=j -
(
.%
./z.
/-1<:1)
LJl
(1.3./B)
L' ( 1: +Er' L - I .
J Lj = (27t f" (/- KU)P/2 I I - ~ I )C
x
J.J ex.p [- i {~'( r- i)Y -21<:tj U'(I -l)Jl + Jl'(1 -~)} ] did dll
D
D:
amd
(1.3.20)
u.' I! > Uo
Uo == c~/(I- Ky),
Uo
-
(1.3.2/)
'
C:
l(t-I<~).
DiJricJJJA ~ im. tILt hVml lU1UJ M
LWinLs ~ ~cJM ~
Mt (/.2.4) ( aloo .au I 14- ] ), we iMte
39
-2.
•
_
Xp(X,O) -
ZP/%
Joo t l-, e-.f dt
I
r(p/z) x
(1.3.23)
( I. 3. 2li-)
~
r=o,l.z, .. ·. .9~ wepit LiCl=Ec'-1 (d--I.2), H-('-I<~E,Ezr' anuJ
X = (I:+~r'L - I , ~ we cam 7AJIli.tL (/.3. .22) M
t
Pili
z
Jij _- ( I-Kij)
r- { X(A,+Az)-XLi.Lh}H
-'--..-
I-liZ.
P/.z.
H
-2 ( .
vz.
.
X"U"O),\p(U•• O).
(I. 3, 25)
So- ib we .RJX,pOI1L d J iJ im. ~ ~ ~ en ~ iJrL tIrt AimU.lM. 71UJIMU/L M im § I. Z. I ,
w-e c01.dcJ ~ tIr.t. dtWud cJ..uwa1w-eo 1 fa { ~: L-' ~i > c.1., ~; ~-' Zj > c:} am.d
~emct QI71 ~m-p~ JlIXp~ ~ Pit{ Tt >c: , Tj > c~ }, urfti,cJr. g-iAJ.e<J lLO am
M-tmp1Aic mpanw.iJm, ~ 1h jciInt di<J1iulnJ.iun, &{, taro H~¥ T .dto1ib~.
,g t iIn pMticalaJl" Ie!,i =0, we KMe tILt Ailmpk ~o.
1
Z
_
40
( I)
vn B~~tIL(t):
..1...
21 L: [rs /ttL] -= [rsltU] )
-:&• L:
[rS][tU] C&:»a)
-iT. L.
(rs/tU) (6l)
ir. L. (TS)(tU) = (rS)(tu) •
•
(~)
(rs ItU) ,
+,
L: [rsJ (ttL) - t {[tSJ (ttL) -+
".
(~)
[rs][tu] •
(~)
[tU] (rS) } .
. gU) t
( n:)..un
rs,tu.•.".,():
L
[rs/tu/VW] 1
3 1· (~)
[rsltu/vur].
-frL. [rs][tUJ(ttUT] -(rs](tU].[lTW] ,
. (6'3)
-fr· L:
[TS][tUI1fw] = -L
3 ([rsJ[tUllTlO'J ;- [iu)[rsl1TWJ + [VW][rsltlA.)} •
(6)
it· L: (rsltU/lillT) ... t {(r~ltlA IlJW) + (r5 Ivllfl tU) ;- (tu/rs \1lUT)} •
(~)
-ir· L
(rs)(tulttUT)
(~)
-3"
·
1
3T
L
= -3' {(TS)(tU/1TlAT) +
(tU)(TS/VW)
+ (lfW)(TSltU)} •
(rs)(tU)(lrur) = (rS)(tU)(1TW) •
(~)
fi>
[T5 ItUJ (VlAT) -
+{
.
[rsltlA](V/lT) + [rsjvw] (tU)
+ [tUluwJ (TS)} ,
-L
L. [TSJ(tUllfW') - +(rrsHtu.lllUT) + [tu](rslvw) + [trurJ(TsltrA.)}.
31
·
(~)
iT L:: [T5J [tlAJ (vuT) == i{ [rSJ [tl!J< VIAl') + [TS][VllJJ(~U) + (tUJ[VlIT] (TS)} ,
· (P..)
-3\ L: [rSJ (tU)(1TllT) · (f3 )
+
.
.
{(rs] (tU)(VtlT) 1" [tlAJ (rS)(lTlAT) -+ [VllTJ (rS)(tlA)} .
41
ir-L: [rsltulvW'IX~J
o
+(rrS/tLtllllATIX~J+ [rslx~ltlLll1lATJ -r(1SlvWIX~ltU)},
0=
<~)
trL:
[15J [tUI1TurIX~] = 4. [[rS][tuIVUJIXUJ + (tUJ[TS IVlU'IX~J + [1TlTJ[rsltUI X~]
.
(~)
..,. [x~J rTS I tu /11I1J } •
-1-+
1
o
L: [rs Itu][vltTl XM]
<1>+)
+{ [rsltu] [tTllTL~ij]
-
-t
[rsl Utlt] [tul X~J + [rslx~ Utul u.rJj,
1
-141 L: [rs](t1L][1JurIX~J =-6 {[rsJ(tU.)[l1urIX~J l' (TS]['l1lAJ][tILIX~] ..,. [rSJLt~J[tLlI1TIA1]
• (P..,)
-t [tU][11l.lT][TSI X14] + [tU](X14J[rs l1TlAT] + [1TIlT]~~][rsltuJ},
o
irL: [rs][tU][VlAJJ[X~J = [rs][tUJ(VIAJ][X~J,
iroW
>0 (rs/tu /vlATl XM) - lk'{ (rs/tullTUTI x~) + (rs Itill x~ IVIAJ) + (TS 11l1A1'ltlll xti)
o
(~)
,
I
+- (rs l11lAT I XM / tu.) + (rs / X YI tUIVUT) ;- (Tsl x~ 11J1AT I tu.)
+ (tu.lrsl1JlArlx~) +(tUITSIXMIUW") +(tul1/'1d'ITSIX~)
1
4T
tir, (TS)(tul Ulffl x~) - ttf (TS)(tUIVllTIX M) + (rs)(VlATltul X~) + (rs)(tU.lx~ 11t1lT)
-+
(tU)(rs/l1'urIX~) +(tu)(rslx~ IVllT) +(tlL)(trkTlrSIX~)
;- (lTUT)(TS Itu.1 X~) + (lT1.lT)(tul rsl X~) ;- (1TlIr)(rsIX~ItLL)
+
1, L: (TS ItlL)(1J1AJ I1M) =
-4
(f?,)
o
t 1(rs /tU)(lJUT IX~) + (rs 11fl!T)(tIlIX~) + (rslx~)(tl.tI1TW")},
o
J
(rs)(tu)(1}'W' I XH) ==
4TL::
(1'.. '
o
(~~)(TsltlAIVlAT) -t (X~)(rslll1lTltlA)·tCx~)(tUITSllTbT)J,
•
+{(TSXtU)(UUTI XM) +- (rs)(Vla)(tu IX~)
1"
(rs)<'X~)(tUIVIAT)
I:)
+ (tU)(X~)(TslvW") + (tU)(V'W')(TSIX~) -t (traJ)(x.~)(rsltLL)}'
42
:ir. L. [rs ItUI'l1lATJ(x~) - -if [rsItl!I ulA1](X10
(~,
'T
[r5 Itul X~J (tTlAT) T (rsl tTlal x~ ](tu.)
'T
[tUltTlATIX:JJ(rS)
.
J,
t L: [rsl tu.)(JroIX10 - ...L
6 i [rsltu)(1!W'IX~) ;- (rs/ V1A1](t/J.IX~) -+ [T51 X~](tu./1JlJ)
4f
. (fl.'
+ [tuIVlA1] (TSlxM ) ..,.
...L
41 E [l'SJ (t/J.llTlATl x~ ) -
. ""'>
[tulx~ ] (TS/ll1JT)
+
[VlTI:X:YJ (TSltuJ}
-k-f [rsJ(tlLIV'k1IX~) -r [TSJ(tu IX~ Ivw) + [TS] (VJltrLIXM )
+ [tUJ(TSI11'lI1lx~O
1" [Vllt] ( rs Itulx~1)
-t
[tU] (,~I xy IlTUT) + [tU] (V1A1/rs IX~)
+ [lTl1T] (15 1x:~ ItlL) -t' LV'1AtJCtl.!1 rslx.~)
+ [x~J (rs/tul V1lT) -t- [X.¥] (TSI1flltltlL) + L(~J (tuITS Ivto')} ,
1
4 , ~ [rs]CtlLll1l.lr] (Xli) - *{(TS)(tU.IV1AJ'](X~) -t (tuJ(rslVlATJ (X1D+ [Vlf][rs ItIlJ(x~)
+
[TSJ CtulXYJ (VUT) T [tU.][TS / X~J( V1l1)
-t
[):~J[TSltl.lJ(Via)
-t [rS][lrlATIXlfJ(tU) -t (l1W)[TSII~J(tu.) T [:x:~][rsIV~J(tLl)
+
1
, 2:: [rs ItUJ(trlAT)(Xlf) -4
.
(~)
[tU] rVUJIX~J(TS) + [VWJrtI.HxMJ(rS) + [X~J[tlll111AJJ(TS)} 1
t
{[rsltu](Vh})(XH) + [rsl V1J1) (tu)(X~)+ [TSlx~](t&tXUW')
+ eta I liIA1J(rs)(X~1) -t- [tu 1:t~J (rs)(ttlAT) 'T [1JW'Ir.~J (TS)(tIO},
-L
4 I L: [TS]rtU] (VliT I Xli) - +{[l'SJ[tU] (lilA) /XH) -t (rs) [lTkJJ (tUIXM)
. (Po.)
-t
[rs](X~)(tuIV1lT)
l)
+ [tu](vwJ(rs IX10 'T [tUJ(X~J(~I1TUT) +
[lllATJLXlf](TSItU.)},
-4'
J L: [rs](tu)(uurIX~) - -t{[rs) (tu)(uwlx~) + [rsJ(lllAt)(tUlx~) + [rs)(X~)(tUllllT)
. (Ii>
-t [tl.lJ(TS)(VUTIX~)+ [tlLJ(Vllt)(TSIX~O
+ (tu] (X~)(TSI tTlA}')
+ [V1lJ'] (rs) (tul Xli) + [llwHtU)(TS IXlf) .... [VlY](xy)(rsltu.)
43
-1L [TSJ[tU)(VWJCqO - 4' {[rS][tu][llW] (X~) + [TSJ[tUJ[X~](1J'll1)
41. (~)
+
-+\L: [rs] [tU] (UW)(XM) == t
. <P..>
(rsJ(llWJ[X~](t1L)
+ [tUJ[1tu.r][X~](rS) },
{[TSJ[UlA1](ttL)(X~) + [rsJ[tU](1Tl(1)(X:~) + [rs][X~J(tu)(VIlJ)
+ [tllJ(1JUTJ (TS)(X ~) + [tllJ[X~J (rS)(llUJ) + [1JlIf](X~J(rs)(tu)} ,
-4\ 2. [rsJ (tl.l)(UW)(X~) ==
·00
l f[rs](tll)(Ullt)(XH) 1- [tU] (rs)(11lAT)(X~)+ (lflAJ) (TS)(t1.l)(X:~)
-t rX~J(TS)(tU)(l1lAl)}
44
Ap~2
;JaM,w
tK vnk.o ~ tJL A~
~ 1iuuL ~.iIfl, § I. Z.1.
( J~ A~!I
M.cw ~ ~ tIu ~~ witl..1WJput, 1;, ~ appwutl
Un. tItt ~ , ~e! 80f ~ /WIn() oven., (1, 2.·· . , /') ~ . )
(A) :
L: crlLT<rst [TSJ [ttL]
-
:L: O'Kro;t [TSltuJ -tpep-t/) •
P•
23 (JtQTo;tCTav (rsltuIVIAJ] --tPCP'"-t3P+4),
L3o;To;t~1r[rs][tuJ[VkJJ= p.
L: cr-l Ir ct cr." [rs][tulllkTJ - L: ~q;to;1r[tu)[rsllTlATJ-
L:Cl;r~t<r"['U~](TSltfl)
--tpepT/)
I
L: ~TO'"sto;vo;X(rs Itul V'lAT 1X V] ,. 2: <t,o;tOjlrctx [rsl 1'1411 xv 1tIL]
-= ipe pZ-t %p+ I) ,
I3 O';,O'"stOj.ctx [r5J [tu I VkTl X~] -= L
- L:. o;"O'"stOj.(jlllX [lTlO][Tsltul XV]
<J.r~OSv<r1tX [tttJ[TS 1VW' X~)
.. ~ o;r~tcrrO-W1~J[rsltul1J1ATJ
t P(I>t 1).
=
L<ruTo;to;~<t~[rs/tu.]['U1ATIX~] -.
I
f
4'P
(PTI ,
Z
L: <YllTcto;v<T.rx [rs 11T1.lT](tul X\1 J - L:3 o;;rCJ;tOiu<tx (rs I X~] [tulllw]
=tPCP+I) •
L:: o;rD;tOj1ro;x [rsJ (tuJ rV'1Ar1 X~]
-
23 (furo;to;u(jut [UlAJJ[:r~J [rs Itll]
== t
p~ ( P+ I) ,
e
45
L.(JvrCJ;t<rs,D;:r [rs] [UW] [tul):'AJ
=
23 <YuTCfstOVU'<f.,;t [TS](X~][tlAl1.TW']
= ~ D;.yo;to;,o;x [tlAJ(1ffAT] [rs p:~] = :L: ~r~ (J."o;x [tll][X.~J [rs IVtAT]
==
P
I
(B) :
b
(J""rs(TS) =
1iLn ... w ,
2
L: <YIlTT<it Q;V(YS I tu I UW")
=
L: CfllTy Q;t<fa1T (rs I tTW I tu.)
-=
t (p2+ 3 P+ 4 ) w2.
L: <fuy<rs t (fp<Ywx.(T5 lttL' V'lU' X.~)
=
c:
L:: ~r~t~l1 (tu/rs ItrW)
,
L: cre.tTo;toaV~X(T5 ItILl :t~ I VtU)
= L: o;r 0;; o;v <J;x (r5 IlTW"' X ~ Ittl) == L: a;T o;t QiuCYw:t (rs I X ~ I trW" 1tu)
=
L. ~T()st ~lIa;X (tILl TS I vw" I X~) = I= ~yo;to;17(jn(tu.1 r5 I x~ IW)
::or.
Z <Jur<Ts-t <11/ o;x (tJt.Ul rs I tlA I x~O = L3 ~r~OS1Ta;,x('Vwltu.1 TS I xy)
=
~ (P+ I )2. Wl. ,
L: (jILT CJ;t o;v ~x (TS lvw I tu I Xlj) = L:: o;r o;'t 0;17 o;x (TS I X~ 1tu IUlAJ')
46
=
E ~To;t~tf CJ;x (tul VkT I rs I X~f) - L: ~To;tq;V(fllrX(tul xM I rs I tTlAT)
= ~ (P"t3)
L:~yo;tOivo;,x(7'S)(tl.ll UWIXM)
e.u;l.
0=:L: o;,.~<rWVo;,x(TS)(tlA
IX~ /l1ltT)
= L:<Yuro;t(Jjv o;x (tU)(TS 11JW'IX~) - L: <r.tT~ o;,,~X(ttt)(TSJ :x:~ /VkT)
- L: ~To;tO"iV<r.rX(lllA})(Tsltl.l' XH) - L: <J;,.<t<rp(ju (VW-)( tLLI TS I x ~ )
=
2:: cr;To;to;~CYvXCry )(78 I tulllW') := L: o;y~o;lI ctx (:t~)(tu I TS I vw)
== i< p.. . J )1iz,ft,
1:: c!;y<Tst OSll<r..x (TS)( VW'I tl.tl x~) = L: CYu.T<J;t o;"CT~x (tU)( tl1IT ITS Ix l#)
= ~ crlfT()stOi,'tx (VW)(rs IX~ ItIL) =- B
<fayo;tOivo;x.(X~)(rs' Vk11 tu.)
- tw+ + ftn.O" ,
E
crILTo;~Oiv()IlIX (rs I VllT)(tul XH) -
L: a;,.o;t~v(j»x(rs I xM)(tv.IVtAT)
- t'w'"
-to
-tCp-t%)1;".Q.2. ,
.L: a;.,<rno;v <Y'1lQ:(rs)(tu.) (VWI X~) - .L: crIl.1<YstOiv(j'1IJX (1JtlT)(X~)(TSl tu.)
==
+<1'+1) W tta
Z
,
:L: ~To;to;~o;X (1$)(1110 >Ctl.t IX~) -= 1: a;To;t~vo;x(1.5 )(X\I) (tlLll1W')
==
L: ~To;to;tro;xCtU)(VW)(TS I X~) .. Z
-=
o;,.~o;va;,z(tU)(X~)(TSll1l.\T)
1a03
•
47
'---
(C):
E <J"llTo;t [rs] (tu,)
=-
L: <:rlLyO';t [tu.H rs) - ul .
2:r crW7O"stcruv [rs I trW] CtU,)
~ (f.".<J;to;V [TS It~] CtrW) -
= L:<r.rr<J;t(Ju.v[tuIVUT] (15)
= t<P+l)""~ .
L: <Yw,Q;ta;17 [rs] CtIlIV'W) .. L3 crll7To;t~" [ttL] (TS IlTur)
= 2: (JllTya;t~,,[1Jur](TS\ttl) ... t< ~+ I)W7.
L:cr1lTT O'"St"IT,;" (rS][tU](VW)
L: (Jarr0;t<J;tr(rsJ (tU)(VlAT)
=
L::<Jwr <J;tCTu,,[TS][1TlAT] (ttl)
=
L:: (JlITTo;t~1T [tu]( 1J1AT] (rs) = w ,
-
Z
'=
L: o;,q-t~u [UW) (TS)(tU)
,
2-
(fv,o;t 0';11 [tU] (1S)( U/AJ)
'-
= 1ir,Q:. '
:L: ~ro;tCill'<tx[rs ItlLI vW' J( X~) ::a L. (JlL.,o;tOiv~x[TSlt/,l1X~J (trW")
=
L. a;T~OHv~x[rSI1TurIX~J(tU)=D~.,o;to;"o;,x[t/,tIVIA1IX~] (rs)
i (p-tl)cJ
=
:L: ~rcrS'r08"O-llTx[rs Itu.] (1JkTI:x.~) = L: (fuT<JstOiuo;x [1]"W" I x~ J(rsl tlA)
=a
t
p( pot I )'- U)2
,
23 07t.,(jst~vcr1lT~[15 l1Jw] <tu 1xy> = L: o;yo;t~1ro;,~[TS 1X~ J (tu l11l.O)
= L:=:o;.,o;tO;ua;x[tulllW] (rs I:qO = L:: <TlLy<J,tOi"o;,x (tul ):14] (1SI VUJ)
=
JM/2 jJ~ of>
~ 1:0
t( P+ I) e.u:L.
fu ~
i (pot I)
",,1 ,
L: ~yo;t(j~a;x[rSJ(tl.lllJ'IA1I:r~)
(lJU.
oJ1
48
:L: crUT O'"st Oiu O;x [TS] [VW I X ~ ) (t U)
-
:L:: 0;T o;t- cr", ~x (t U) [ trW" I x~ J( rs )
= L::. ~l'o;to;" cr.x (1Jll') [rs I ttL ) (X M) = L: ~Ta-st~U~.t[X~ J [15 /tu.] (lllAT)
... t
P(p+l) CAJ~,
L: CJ.rO"$t o;v~x [rsJ[tu IlJlATJ (X ~)
= L:: 'ful'o;tCTp(jarx [tu.)[rslllW] (XlJ)
= 1:: ~To;t o;u (Jwx [rsJ[tU/Xlf J (VW)
= L: crlLTo;to;"o;x [tUJ[TSlx1lJ (trW")
=L: (fu,.CJ;tO",vO"n [1TlAJ] [rs I X~] (tU)
-= D O";Tqt cr",cr-;x [I~J [1S IUlT](tU)
.. L:<r.1'Cfn crVvo;,x [lTU1J [ttLr Xli] (15) .. L::~ro;tap~l[x~J [tur vWJ(rS)
=W
L: O";.rO"sr
2-
.
o;v o;.x [rs I tU] (UlAT)(X'f) = L:: a;l'o;t~v~x [VbJl X~ J(rS)(tLt)
= ~ P(p..,.,)in.st ,
L:3. O""ttr O"st()8V(jlllX [rs /V1A1] (tl.L)(X~) =- L: ~r(jsto;v~x[1$ I X~ ) (tl.{)eVW)
L: (J;yO'"stOWvo;x [tU.llJlAT] (rS)(XM) = L'3 Cful'(fstOjvcrll1X (tlA./ X~ ) (T5) (t111T)
=
= ~n2.
,
i:: <J;TCfstOH"o;;x [rS)[tU] (1TW'1 xy) - L:()tlTo;torl1<Y..,X [V'W']rXMJ (rs/tu)
= tP(PTI)W~
~
38
I
0;.1' crSf <Y,v (J"1IIX [15 J[1TUT 1 (t lL I X ~) = L::3 <JuT ~r 0;11" o;,X [1s1I1: ~ ] (tUI 'I)'W' )
L: ~Ta;t ()p0;rx (tU][VW] (TS I X~O ... L:: o;y~a;1TCfur.l: [tu.][J:~J (rs I tTW")
-~U
&\
W
a
49
L: (J"ILT(J;rcrlfV (jWZ [rs Htu] (vW)(XH)
.. L:: cr.l'<rn(jNtT~z [vwJ [X~] (rs)(tu.)
p1A,D,' ,
=0
L::
~To;to;Vo;,X[rS] [VW] (ttL) (lM) -
L: ~rCisto;vcr.,z[rs][x~J (tl.l)(UUT)
L <rur <J;t ~"()W1 [tlA] [VW"J (TS)(XM) == L: cylLTo;to;vo;,x [tr.t][XMJ (TS) (l1lAT)
=
= w4
23 <Yur(J't <rfV o;,x [TS] (t U)( 1IW' I X ~) - L: ~r<T,t'cr.1I' O'".,x [tu J (rs )(11 W' I X ~ )
=
L: (fuTQ;t o;vCfwx [VlAT] (x~)(rs Itu) - B <r.trCYsto;v(J.x [x~J(lTW')(rs I tu)
L: cr;.To;t~lI'(J"wx[TS] (vur) (tr.t I X~)
'--
=
i (1'+1) w
-
E
4
,
<rur<rst~vo;x LrsJ (x~) (tU.ll1W)
- L:: (JuT~ o;v<r.rx [tuJ ('UllT) (TS JX~O ... L:: ~To;to;Vq;X[ttA] (X~)(TS I trW')
= ~ (fura;t o;1I'~X [VlAfJ (rs) (tlAllJW') c: ~ O';T CJ;t ()po;;x [X~J (TS) (tu/ 'Uw)
=-.L: CYuro;t-<J;71 Cfwx(vw] (tlL)(rs I xy) = L:o;.Ta;tcruV~x [X~J(tLJ.)(rst VtU)
= tr,Q,'L: ~To;to;"~x [rsJ [tU] [lrWJ (xM)
=
= :L: ~T 0;"t ~1T<r.,X [rs] [1TW] [X~](tU) -
,
L a;rCJ;tOjuo;x [rs][tIAJ[X~]
(llW)
L: ~To;t Oivo;.x [tlLJ[1flAT](X~](rs)
= pw2. ,
. ~ ~To;t o;vO;x. [rs] (tU)('lJW)(XM) =
L: ~To;to;trCJn[ttl) (TS)( trW)(X~P
=- 2:. ~T<rst 0;11' Clwx [1JW] (rs)(tlA) (X ~) -- Z ~To;to;1T o;,x [X M] (YS)(tU) (trW)
=
til" lit rt ,
50
Pant II
J~ ~ ~
W/"WMic llWUMl.aO MU1. CW~
£l ~ oi wiAJuvU ma1rli~
~ ~ , 7TlOIrUJ 1uJt ~ cmn.t out aa ~ ~
WiMw.atlfl1J.1iziaa anul tILt dt11a ~ ClfJplwi 1:fJ MJiailn- tIrL nwruJ ~
(U) ~ 'rum -ruJ1 ~p1otic d.i.<J1;Ulru1i.urv.
.§n
S UfPG<l£ we iUvv-e (). p-1!aJliait 7l.8/lIHW1 pepu1~ will C1JlfrJ.J1.iaIn ~ E =( OLJ).
lit S - ( S Lj) .& a A1ann.pit CXJlTr1/lia;na rn.o1iwx. ~ n ~ &i ~wdtrm anu1 Mnu
nS i,(J a W~ ~ ~ i.e Wp(L.,n). JR.vn.1ti1J well "k1wwn [I , p.75]
thtJ; S ~~ iJn ~ W L. M n 1,e;ruj~ W ~ , MI4 tILt ~
~ ~ S Mi. G/JJj~ dui1AiJnJld O£Ch~8to P(PT1)/2-~
rtUlJrM1 dio~ UJilh mea;n. E and CIJ7raJliMzwJ
e
(2.0./ )
J.it UJJ rww- ~ m .tJca1~ va1u.td
0Jl1
not ~~ dLpMU1.Mz1 a;nd
~
f, (S), '" , f S) ~ S , urfu:,cJv
'!uwe -GiMt anuJ. Mcmul ~ Un
WI (
a nu~ ~~. J~."J aA¥ ~ ~ "b Jltw-emt 4-.Z.5'
.inL And.uvJun:o -&nrk [ I J, lli limutiln.g ~ ~
~
nAYVrrw1 with ntlML (0, ... ,OJ.
J~ ~ &C ~ am.d ~ iJn
-rJu:" ~ ri.W1IziJnJ:ic-n 14 not ~ 1.MIJ. .971. ~ 2./ we F
~
wPti-clL fLtdJLw tIti 1~1 ~ cmn.puta1ion cmwJ.,~ am.d ~ tht 1.otvv ~
i1ro ~ .£JXCihnfi.uJ M.l iIu.at.td. .
•
51
'-..
2./.
F~M.
le"t
fLO ua.R.
tJzi nMriliDn
drs =
-t(I + ~ I'S) ~TS
'
~ &rs M tIt.t 1<JumLcIeuz. ~.
fwnvulo. I. lit f( s) k a ~eti<rn, ai s ~ tfu ~t anu1 ~.WrrUJ.
~ im-a ~ ~ L. Jlk,ntltt (~~) lJaIUMtu ~ fCS)
lin itA aMJmp1hUc ~ ,fA ~ ~
Van, ( j(S»)"
~ p
t
{feE)
rtn( c.P L)'"
(2. /. I )
ep'J'" dLJ~f(~).
~ BM tItt tLa.UairrutlLtxlwe ltwe
=
(ep~J)'
~
Van,( f( S») = E{ ttJ
- E{
af( S)
d StJ
I
S-1: (S~- <riJ ) }
2-
t.,-t( I + ~Lj)~ls_l:(s'J-er.J)}l
p
=
>i.j.i•.e-l~ {a'Jt(L:) H dJt2f(~)} E (Stj-(frd )(Su-a;.l)
==
{f(E)
=
~ {f(El!'
=
i tf( E)}1 ~ 1.t(~ cf>tj Ojl) (~ ~t 071 )
*
t f.r.r:z;:t
~ cry cfu (~l Ojl + 6";t(f., )
g
4
t'J <1'" 0;. OJ,
= ~ {f(E)}2"tt,( ~Lt.
F~tJa
II. lit fcs) amd
g(S) $e
1iuo ~ &b S
M A;n F~ I,
hi:
n&t ~. ciLp1llYLknt. ~ ~ ~&W ~ i/J ~ irg
Gnr( fCS), 8C$) == if(E) 3(E)"tn,( q;E~r)
(2.1. 2)
52
{1, am, ~ ~,&,
u
tk c1dvvmit7Wln1 &b ~ 7fUilrWx
R = ( r~J ). ~. e.. I RI • ~cl cam-.he UPALOAJM. MIS 1/ I Ds I . urKvtt Ds iA tILt
~ rrw1JuJx, wi.tIJ. ~ .il1nrwn1o, $", Su,'" . S,.". HilnCL 1h. ~pMic
~ ~ I RI cam.& ca1cu1atuJ. 1:y (2.1. f) cw ~ :
= ~ IPI2tn.(~L)2,
Varz,(IR/)
~ IPI
j,a
uo ~
~ ~ &b
tItJ. p~u1a1i.on CMIll101i.on ~ P- D'p#E D~.
Sim.c.e-
~ (L- l ) tj
JtMUlo
~
tlti.
p
amd
(i.J)
il.tnn.vnt
1
e-£ 1:- J
we fuvve
== I.-I - D
t
fwna
trr. ( ~ L)2 = tid I - D-L L)2 = 1ir. ( I - 2. D-L L
(J"
(1'
=
p- 2 ~ + 1it.( D~ L: D~)( DI# L: Dw)
=
tt p~ - p.
V(lid I RI )
=
2
~ IP1 (tL p2 - ~ ).
+ DoL.IT ~ D.J..IT L )
(2. I. 3 )
53
z. z. pJtWi.gu& Jlf.4M1tg.
1. OlJu,n t1nd M. SiMMti [ q ] ~
A)MJe/l.,ai,
~p~
diitrz.LfnJionA ~
~ ~ CMIli101ilm mo.lJziJX. . JIte ~ 7ffM 1rrMtd rm tItL ~tMic
~ ~ eWJ"lVrzd; 1ivmJ!ni, ~ .JQ/YlLph. CMAda1icn ma1iWx. R. ~
pvnt
/l.W)Fad A1cnm ~ tIuit AlO7n£ ~ tIt.&u ~ cam 1Je ilXp!l.IA.<lul M
~­
~ ~ S, a;nd ~ ~p1Mic ~ anu1 ~ Ci1In 1J.e ~
1n4I1L.£Mi4 ~ UAJinr.g th. ~u1M
amd (2.1.2) JUp~.
jn tJuo AJLi1i.cn , UJ.e, 1Mt t.:It.w.t!lUJJJ.1M M 1JJtf1 M ~ Il.t<w1la ~
~ tIua ~ A1)~. 7flMt ~ 1lwm g-we UI.J ~pUCic prnve!L ~ ~
VW7,(,0tUl
fu,t, ~ tIt.t ~
Uf.t IJM,
:E=
(J;n+iA.
tk ~~ rUJ1iU(JM
L II 2:12
L: 1t
PI
L:u. .. Lzt
. ..
1'1
E" E,z' ; . L:"
1'\
!:21
.
•••
PI 1'1'"
(2././)
:
D~ =
L u 0 '" 0
0 2:'n··· 0
o
0 '"
OJ,
~Z •••
~I
crzz
L II
P,
( P- PI + P2 + ... + p\ )
~
L..(m)
==
0;111
• • • 0;11
0; I 0;.1 . . . 0';..
(m=I,z, ....
'! )
C1nd. tfr.t ~ noW.1iurw ~ P, S, anul R.
(m=J,l, .. ·,P)
54
( I ) -rn (IRI - IPI) M ~1MiCAfta ~ ~1iliinitu1 rir1:1t. ~
vanLQIn.Ct 21 Pll( 1ii p2 - p).
0
o.nuJ
Jftl ~1otU j<MtI dio1Julni1Lon &f, rn {(I R(IlI,I Ra,I •... .1R,p,1 ) -( I RIII... ·.\ R"IH
fA n&1Jm,U wi.tk 71tLQIn, (0. 0, .... 0) amcJ. ~
(II)
= 2IPmIIRJ>1 {ttP(~)-i +tL ~J(I-R~~ )P~} = epJ~'
4>ij
P~
wfwu.
II:
(
Fet.,).
at. = J, .•- ) i ; ~ - Lt I, ... ,
~..
u·
(m)
(i{J)
J. W~ L= j •
Z Ipml:l (1iIJ~~J - L)·
=
J~ ~ ~ CIliIvrion ~ 1u~ th. ~~,W ~ AinrltpvrUJ.VniL.
Ho: L:~J:S 0 ~ ali
IT =
i
+ J • fA
lSI
_
IRI
ISIIII 5#1",1 Sui
IRnll Rnl" -\ R"I
.gc at 1w.at Me ~ L lj :d
id MtvrruJ1 ~
711.la/f1.,
(~ .,. j)
fJIl£
(2.2.1)
rwt pw, ~ th.€ CWf11rf1"M.W d..i.a1n.iJn.iLim ~
A == / L 1/( IL: uf· .. /L 1£'& I )
amd
vaJli.a,tn,ct
~X'{1hCD~Lt.,..P}·
( I V)
Jilt 1uJt ~
if
(2.Z.Z)
~ ~ci1Aa
w
=
lSI
(thS/P)P
(2.1.3)
itJ CUlAjTnf~ cktuJnJ:uJ. , ~ L +cr:lr . a.uc-r~
W = I!: 1/( 1iz. E/ P)p • anu1
1
..z W
l5. - n
2 {
p2
(Vt "£/ p t
i"
UL
~
h }
2
-
t'
.
to
N(w, 6%),
w1Wu.
(Z.2.4)
·e
55
( V) Ray - BM.rn-Q/rInlA) At.ep-dnum mul1ip1.t ~l!l11a1i.bruJ [ 10] aM. tItt lTUJ11iplt
~ ~ ili itlJ. ~ UMio.tt Xi. flt7td (i-I) C071LfXJ71emt ~,
{Xl.· ... XL-.J, i = Z. 3 .... ~.
we c1.Mult tIr.Wt ~ 1Jy Ti,!!i! TH I,··', i-I)
~..aann.piL anu1. Pi, :; fj,(J ..... i-I ) .~ pepu1.Jion t.Ittm- we fuwe tIrJ. mialiun.<J
I
.9i
1
(2. Z. S)
r:,...
JM 1IC1t-ruJi ~ diA1iulnJi.crv ~ -Yn {(
,r; )- (PsI, ... , f;)} iA
71.8JlIrrUl1 U1ith 1fU(JIfl, (0, ... , 0) MUl ~ rvru1 emlQ/W1/!/lUJ
A.
;~j, =
4 -I(
-2)2 ,
fj.. I - Pj.
*j
ti- - Z(1- fi,1)( 1- ~f.)( fj,j T ~~I
==
(2.2.6)
i - Z.,·· P
I
•.,.• i-I)-
~:,..,.• i»)'
(L
< J)
(2.2.7)
(2.2.8)
56
(2.2.10)
i" Z,"', P
+
~i"J.. = th..=
1J1. - ,.. F,~~ f.,~d
t (1- f,~. ),
,-I.
( L < J).
Z. 3. llon-ngll a&f"f1Mic ~ ~;tk ~t A1ta:UAk
(2.2.11)
W~ im
t/t& AtJw4mz& t& CO¥a/?l4tr@ ma1itirt:
M S 1Je th l.JDJmf>iL ~ rrui/;U;t. lJJith n dutlUO &C .~
~~
anu111 S .&
wWp( E, n).
S. S. WiLIrA [ /q ] CUMid.uttd tilt ~~ tIuJ
tilt ~ 7TUilIt.iJt, L - (crtJ ) j;n (l p-1taIliJJ.1t 7WlImJli {X'Pdo.limL 1i.a.o th ~­
Wn.e
crt.
P<r~
Ho : L:==I: o = .
f 0"1 . . .
ra'
<r Z
P(J2
,..
:. .....
.. .
FO":t FlY 1
•..
~ G- ~"t:k px p rna1JWx, ~ ~
am.cL
cJ..uriNed tIt£ ~ JUilio
1
=<r {U-fH+fG},
(2.3.1)
(J1
QIll
a11.
out/um" ~ H
0 ,
(ml
anuJ. -[ l/(P-I)] ~ P < I,
urfu:,ciL j,() ltYli111n
M ().
~
&b S ,
(2.3.2)
applicatimL ~ tid ~ ~ § 2.1, we ~ tJu rwn- rw11 ~­
p:WM ~ &l t. OA ~, it W7lJJ!lIfYW1 Wr1JL m£11In,
UA
etn
'r
=
P~(p-I)P-I ILl
{~L:(PI -G) }p-I
th'LG
(2.3.3)
57
(.Z.3.4- )
.&gt
=
PkgP+(P-J)lbgCp-J)+.&aISI
- (P-l )
10{tn S(pI - G)} - ~ (tit SG-).
HiJrLa
~ i.j =
ay ~ I L I - (p- atJ kg{'tit L (PI - G)} J)
~I -
-I
G-
(1i.J
kg (tit r G)
G
~=i: -(P-J)1itL(~I-G)-1iti:G'
(~Ll'=I-2{(P-l) (~I-G)l:
1itL:(PI-G)
+ G-E }
Ut1:G
+{(p-')fi(~)::~) +
l£G t ,
trd~L:t=-P +tn{(p-J)iP~(~~) + tn,GiG }2.
~~ (2.3.5) itn1.D (2.3.4),
Vardt)
=
tILt t1MJ1rLf'1bUc 1la/1iJ1na ~ t
(2.3.5)
La
rfwn ~
ir~ [Ut{(~-J) ~~(~~~):t thcr~E'G }l_ pJ.
,gt,W .e~ 1.0~. tltat urkz. L: =E
o == ()2{
(I-FH
+ FG},
VCJJ/.(t) ==
(2.3.(;)
o.
UJhm I: =F La. titt ~Uic ~ rt& 1JrJ Ata1io1ic t ~ .kg
(2.3.2 ) .i.d 1l&vmo.1 ~ 1rli.ML (2.3.3) anui VMiMtCt (2.3.6).
58
2.4. Tlon-rudl ~ ~~ e&:1JA1 4a1i41ic4 &n ltwltf424
~ iIn
~.
~n ~ A.tc1Um- we AJuJ1 F a rrtM.L ~ .l.!XAImplt jIlL ~ 1h
~M ~ § 2.1 0Jte M.2d /lJ.~. S tan1iJn.e wiUe- tIt.t ~amu. ~ M
(m ) .Ln § 2.2, 1d l.U) ~ tIt.t A1:.tf-Mum ~ [10] ~1Lo~ thL
tItt*-m
~ ~ ~~.
.
H0
:
L. £j - 0
Ju ~-a111it~
~
~ ihu1tp~
all i. J Ci.,. J )
fA p~ i.Jrl1j; (tt -1) ~P~'
(t-l)
Ho
C~PcnuJJ.m8
.• E It -=L: ZL -= ..... ="
-0
L.t-J.£.'
wtIt.w paililiun, tk ~ IlA1ic CIli1vUcn ~ H ~ ~
o
fA ~ ~ (2. Z. I ),
-
k -ca~d iIrztD
Cain
VI V2, ..•
'U\-,
wfuA.t St'l - S. ,gt iA ftrwum tIw.t UlrLcWv tItt 1ULRi. ~~ H0,
~ ..6ta:liA1.ic~ itntUJ>.MLdurt anu1 4J G. E. P. BS'X r 2. ]
- {n16 appaox.i;m~
~.
(Z.4./ )
i=2.,.·.,'6
t - t( PI + ... + Pt+I ) } lo-g
di.itru1rutuL
QA
Vi,···, Vi-I
(2.4.3)
V"t
tIr.t cIu:-.d~ v.it1l PL.'( p, + .. , + P,) ~ ~
JRMt we ~ tILt ~9 taMt.
r
~
;
59
HypothesLs
HlII . ~
X~
CrLtenon
-t - -r(p,+pz)} ~~
0
~
-{n
H. : LLl+'='" "~i..L+I-O
11'.Lo
-{ n -
H<,-I)
. L: 1\=... = L 'tl.,... 0
o .
Vi-I
O'
11
=
(t)
Ho : L··=O, L+J
~J
d.f.
~~l
t -t(P.+··'+r.+I)~1fL,
R+f ( P. +... + Pi.)
- {n- -t- 1(B +... + P'l )}ics lli-, P, (p.+." +f\-I)
I
p3 - 4 P;
1
- {n- 2' - 3( pz- 't r:) } ~ V- 1(pl- '2; rt )
v-
H 1<> o.cCt, pttcL ii ORul em.ty Ai all H~ D~ 0Jl1 O£aptid Jnw.d m Ute cb.,- J~
~tJ .ilruJfta1u1 J"n tItt alnnte taMt , and. ~ H itJ ~c1ul.
1l.ow lit IUJ CUMid.uL tIr.£ aarm~
d.Lo1nA1nJicn ~ (Vi,···,
~ 1WTt!
~ H ~i.) , L = J. ... , ~ - , i.J 1Iw.e., .9t i~ .1 ~w .dU t.Iwt tJu d..taiJu.d diA1Ai1ru1i.cn i¢
1WlIrn.a1 witJL mtllJn v-e.clAn (,:\ I, ... , .:\,-,), w1Wtt
0
0
F
Vi-I )
(2.4.4 )
Owv mtUm wm ~ WCPmpu1.e- tILe OAAf1YLfJ:Mic ~ rruilJwK.
ot tIcio ~
diotulruJiorv .
JiMt
we ~ tItt lla!ltaIn.a
Van, (vi,)
,r: {
= vo.n
== A:
I Srt'tIJI
IS[\.1 II SL+I.l+1
.I
oi
Vi"
Ci == I, ... , ~-1
).
8M tL. d11to., ~ we MNe
}
(
E[ ~~[~I:I ISntl]l-/ Lu+ul) + ;1![~1 (I Sml-I ~[i] I)
ab9.AL (
_
~2
+ aiL: . I ISL+l.i't,1 I Lit',lt,I)J
ltl.ITI
2
=
AL E[ D:;i.+fll (I Stt+IJ-1 L n+ll l) - ILlml (I Sm I-I Lcul )
I
(
- IE.L'tI.l+1 I I Sttf.lHI-1 LI+1,1+11 )
]z
60
=-
A~[ 11: I 12Vwd IS(t~I)I) + ILl \21!an.(IS m l) + II:
CL+IJ
•
[ll
I
1~I,t~1
11
VQlt(lSL~I.t~,I)
- 11:·[Ll It 1: lLtll I Cw(lSml,IS tii'IlI)- IL:[L+IJ I~L: 1+1,1+1 I COV(ISu+IlI·ISi....ltll)
- IL:.
bJ
It!: .I CouUSml,ISltl,i+,I)J
1:tt.~11
Jo mo.b.oit ..eac1 liMn, iIn (2. ft., 5). we nud tIti ~awin3 ~ :
(i) Cw ( IS ru I, I SrJl /.) , ( L ~ J ).
:T1Wm (2.1. 2 ). we 1LaNe
Cw (ISml·/ StJJ I ) m =
Til.'
ePL 'It >:
=
=
d I
.~ ILmllL:[J]1 tit (~1: '!'L: ).
I~rLJ I I== [~I~~0 0].
.
0 d.~
d.p ~ C?
0
[~;:; ~J[~tu :W:~;, ~] [~'J' :]
[I~+.+~
: J[ I~,.+~ ~]_[I~+." ~]
(2,4-.5)
61
(Z.4-. b)
(2.4.7)
(2.4.8 )
(i~J).
(iii)
Cov ( I SnJ I, IS JJ I ), ( i <. J).
3'ltrJt t{ aiJ.. we
twe
Ccv(lSml·IScii l )
'....-
2.
.
-
= nlL:mIILdil1it(~L"¥L:)'
S
iM/"fJ
11'~
_['E;~0 00]
~L -
-
1
. '1:"
llHo...":
r-I (1:"1+1'
•...••
n- +rL' (n
'2: ,Itl' •. 1:" .
.........::
I
_._-
0:
'*
1t
)
0
I
o
I~
o
p.+",
l'
PL"
I
~'t(I~" ~
Ip, ·+Pt : L:;~]
!
L.1.Lil
. f' )'
y'l
'(Pj~),
FJ'l+(":; p'... ~""1\ )
"'Ii
l
i E;:l:
i ~~l
L.~jtl
"
.
Lt.JtI Lt,
••• -.- ••••• --.--.f .. - ... ---.--.----: 0
:
0
E Lj
1: i .(tl'" LI.i-r:
••••••••• L_._ • • • . • • • • • . • -
rJ+···tp'J-r
r(..
:
;
o
. .__ . 0
0
o
0
lOt
- - --- - --:- - -- -- -- ---.- - - - - --1 - - - - - - - - - '1- - --- - - ----- - - -.
1
__ .... __ J.
-II
II
O!
I,,
_
"
0
:
0
!
0
62
•
O
•
I
I
"
0
I
': 0 "
I
- - - - - - - - - - - ... - - •• - - - - - - - - - - - - -r - - - -
• --_.. _.. _.o. _....
:
0
0
I
T -
- - - - - - - -
; 0:
_--~-_·_------------t--_
.
0
~+···i"P•
- - - - - - -
~----------------
P~i" ...... ~'"
L;;
I:Z(Ljl''''Ldt ) ! EZ(!:J.l+I,·.. I:J·j-i): I~ 1
(r.WI '··· ,Lli ) ~.
---------- - --1------ ---------t .. ---1------------ -- .. _..... 0
i 0 :
0
~~+···i"P\
I
o
L;~]
:
I
(Ell)
.t. "1:;1 (Ej" ...,EJL )
-
J
0
0
0
(iV) G,v( ISitl./SjJI)
=
•
*
*
*
0
0
0
0
0
0
0
0
0
I
tILLi./IL:JJr1id~~:LiJLhILJJ
Vwd ISiLI) = t l i LL I2 P.·
(2.4.10)
(2.4.11)
63
(2.4./2 )
(Z.4./3)
wiu:,ciL m~ k
to
~d
(2.2.2).
J~
we
fwNe
1k
~etuu.on:
~ H;L) iA rwt 1:Iw..t. tie ~ diA~ rl1h ~ ~
11.W1TUl1 with mi11/YL At amd 1>aJlLrvna (Z. 4-. 13 ).
1l1Jxt w-e compu1a- tk eotraJl,iama o-t
Vi anu;{
1UeMNe
Cw( V;
~.
JMm
v..)
J
= Cw{
I Sn+l] I
7
ISC)+I) I
ISmll St+l. L+ll IStJJ II
(Z.4-.6) - CZ.4-.II).
Sj+I,jtll
}
Va
(L <. j ).
By tM d..ill:a mJ1Ju,.J
64
-(h+
... -thrL't1 ) -(h+
... +h)_h
rl
r,
rL
rL+!
(Z.LI-.lll-)
lit
lLO
~f' tJu
alrove ~.,
W-il.£m rume ~ tA.t H~L), i = I, '" , ~-, , 0Jl£ 1iuu, (V" '" , 11,_,) ()/ti, M¥"'P1:Ji!.iCafty ~1rU.1ru1:td ~d.?rvg W Th. ('1>-1) - vOlliak nwm.o1 r1W~ ~
rtLeMl- ve~ (A" ... , AS- r ) udlvz.e At QIlt ~ ~ (Z.Lf..'I-) Mtd COlroJti.,ama
~ CQ1n.p~ ~ ~urrio (2.4./3) amd (Z.4./4).
65
REFERENCES
[ I]
Anc1..mdon, 1. w. (I q58). An ~~cWm W MuUwo.n.La1t
Sto.UoticA£ A~, JaWn W~ anuJ. SaM, NewYtYtk.
(2]
Box. G-.E. P.
C/Q4Q). r'A ~ ~ ~ ~ac1Mo ~
lJuii,Mo-rl orJ.1vriA.." Bumtilirika, 36 . 3'7 - 34-6 .
""
[ 3) Hotel JiITlS' H. ( I q 5/ ).
II
A 8MWUJ,.!3ed T ~t anuJ. ~ ~ mulli11TAlMte
diaptMLun ." p~ && th Sewru1 B~ SytmP~ em
Ma1hemr.a:tuo1 Sta1iA1icA Mu1 Hurbakli1y,~. ~ ~
PIllM. 23 -4-2.
[ 4- ] HML, P. L. (/ q4 5 ). liOn 1k powelL ~ ~ tftt E 1
Tl-~l,"
wt anu1 tIr.t
Awn. Mo.th. Statwt., t§, 278-286.
[5] Ito, K. (I q56 ). A4F'p1D1ic ~u1a.t ~ 1h ~1iuJm1Um o-f,
Ie
Hote/l.unr ~d T; Ata1io1ic," Arm. Math. Statio!., 2J,
\OQI-1/05.
[6] Ito, K. (I q 60). A~p1Mic ~o.t ~ tJrL diJJ1nLbu1Um
It
~oli~
&i
Hote1lUn3~
T: .Jt.a1iJJ1ic - IT:' A1IITl. Moth. Statwt.. 3J , 114-~-1I53.
[7J Jann~, G. S. UQS4). Ir Te<>L1
"b
~ ~theo.wiln
wnilJfllliJJ1L anuJ.
muftiArrvtiatt amaiAa.¢hJ urIWn 1h ~ ~ tItJ, papuMiem ~
ant wnJonow-n," .8Unne1:n.iIw.. 4:!, Iq- 43 .
66
[8] Oko.mtow, M. (I q 63). "A~ ~1TLPtotic -expllhL<licn fr6'L 1JIe cJ..u,1iu1nJi.un, ~
the iitn.w.rL d..ia~ ~." Awn. Mat4t. SWiAt., 3~,
12~6-/301.
[ q] OLkim., 1. a,nd S i.otami, M. (I q64). " Qorpooc cW1niMimz ~ ~
&t a. c~ 'TTUJ1iWx,." TecItm.Lta1
[10]
R~ No.6, Stam~ ILnW.
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