539.217
-
. . ¡à £¨¬®¢, . . ¥¤¥¥¢, . . ॡ®¢
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®¯à¥-¤¥«¥¨ïª®«¨ç¥á⢥ë寮ª § ⥫¥© ¯®à®¢®© ¦¨¤ª®áâ¨. ⨠§ ¢¨á¨¬®á⨠¯®§¢®«ïîâ
ãâ®ç¨â쨧¢¥áâ륪« áá¨ä¨ª 樮륯®ª § ⥫¨ ¬®£®ä §ëå ¯®à¨áâëåá।.
áᬮâਬ ¯®à¨áâãî ¬®£®ª®¬¯®¥âãî á।ã, ¢ ®¡é¥¬ á«ãç ¥
á®áâ®ï-éã⢥à¤ëå ç áâ¨æ,¦¨¤ª®á⨨£ § . ª ç¥á⢥¯à¨¬¥à ¬®¦®¯à¨¢¥áâ¨
¯®ç¢ã, £àãâ, ¯®à®¤ã ¨ â. ¤. «ï à áᬠâਢ ¥¬®£® ®¡ê¥¬ áà¥¤ë ¬®¦®
¢ë-¤¥«¨â쮡ꥬ, § ïâë© â¢¥à¤ë¬¨ç áâ¨æ ¬¨
V
s¨ ®¡ê¥¬¯®à
V
. ®£¤ ¯®«ë©
®¡ê¥¬á।ë
V
t=
V
s+
V
,£¤¥
V
=
V
w+
V
g,
V
g¨
V
w|ᮮ⢥âá⢥® ®¡ê¥¬ë
£ § ¨ ¦¨¤ª®á⨠(à¨á. 1).
§¢¥áâ® [3], ç⮤«ï ¥¤¨¨æë ®¡ê¥¬ áà¥¤ë ¢ë¯®«ï¥âáï ãá«®¢¨¥
m
+n
=1;
(1)£¤¥
n
=V
=V
t| ®¡ê¥¬¯®à á।ë,
m
=V
s=V
t| ®¡ê¥¬â¢¥à¤ëå ç áâ¨æ á।ë.
।áâ ¢¨¬ ¢¥«¨ç¨ã ¯®à¨áâ®áâ¨
n
à áᬠâਢ ¥¬®© áà¥¤ë¢ ¢¨¤¥á㬬ën
=n
w+
n
g:
(2)
«ï ®¡ê¥¬ ¦¨¤ª®á⨠¯à¨¢¥¤¥¬ á«¥¤ãî饥 ¨§¢¥á⮥ ®â®è¥¨¥
n
w =V
w
=V
t =nS
w
;
(3)
£¤¥
S
w| á⥯¥ì ¢« ¦®á⨨«¨ ª®íää¨æ¨¥â¢®¤® áë饮á⨠[3]:
S
w =n
w
=n
=
W=W
sw:
(4)
¤¥áì
W
sw|¯®«  £®¥¬ª®áâì,ᮮ⢥âáâ¢ãîé ﯮ«®¬ã§ ¯®«¥¨î¯®à
áà¥¤ë ¦¨¤ª®áâìî:
W
sw =n
w
=m
s;
£¤¥
w | ¯«®â®áâì ¦¨¤ª®á⨠(¢®¤ë), s | ¯«®â®áâì⢥à¤ëå ç áâ¨æ á।ë. ®¤¥à¦ ¨¥ £ § ¢ ¥¤¨¨æ¥ ®¡ê¥¬ áà¥¤ë ¡ã¤¥¬®æ¥¨¢ âì ¯® ¢¥«¨ç¨¥n
g =V
g
=V
t =V
V
=V
t ,V
w
=V
t=
n
,n
w;
(6)
¨«¨ á ãç¥â®¬ § ¢¨á¨¬®á⨠(3)
n
g=
n
,n
w=
n
,nS
w=
n
(1,S
w)
:
(7)â®è¥¨¥ ®¡ê¥¬ £ § ª ®¡ê¥¬ã ¯®à §®¢¥¬ á⥯¥ìî £ §®á®¤¥à¦ ¨ï ¨ ®¯à¥¤¥«¨¬ á«¥¤ãî騬 ®¡à §®¬
S
g =V
g=V
=V
=V
,V
w=V
=1,
V
w=V
:
(8)
ç¨âë¢ ï á®®â®è¥¨ï (3) ¨ (4), § ¢¨á¨¬®áâì (8) ¬®¦® ¯à¥¤áâ ¢¨âì ¢
¢¨¤¥
S
g=1,
S
w;
(9) ¨«¨S
w +S
g=1
:
(10) ¢¥á⢮ (7)á ãç¥â®¬ (9) ¯à¨¨¬ ¥â á«¥¤ãî騩 ¢¨¤
n
g =nS
g:
(11) «ï ®¡ê¥¬ £ § ¢ ¥¤¨¨æ¥ ®¡ê¥¬ à áᬠâਢ ¥¬®© áà¥¤ë ¨¬¥¥¬n
g =m
g
=
g =(m
a +
m
c )=
g;
(12) £¤¥m
a| ¬ áá áã宣® ¢®§¤ãå ,
m
c| ¬ áá ¢®¤ï®£® ¯ à . ç¨âë¢ ï, çâ®
m
a =a
n
a ¨m
c =
c
n
c, ¯®«ã稬
n
g =(n
a
a +n
c
c )=
g =
n
a
ao +n
a
co;
(13)
£¤¥
n
a¨
a| ®¡ê¥¬¨ ¯«®â®áâìáã宣®¢®§¤ãå ,
n
c¨
c| ®¡ê¥¬¨ ¯«®â®áâì
¢®¤ï®£® ¯ à .
।áâ ¢¨¬ § ¢¨á¨¬®áâì ¤«ï®¡ê¥¬ £ § (13) ¢¢¨¤¥ áã¬¬ë ¯à¨¢¥¤¥ëå
®¡ê¥¬®¢ áã宣® ¢®§¤ãå ¨ ¢®¤ï®£® ¯ à
n
g =n
a +n
c:
(14) ¯à ¢¥¤«¨¢® á«¥¤ãî饥 á®®â®è¥¨¥ ¤«ï ¯à¨¢¥¤¥ëå ®¡ê¥¬®¢n
a=n
c =ao
=
co;
£¤¥ ao |®â®á¨â¥«ì ï ¯«®â®áâìáã宣® ¢®§¤ãå : ao = a = g ; (16) co | ®â®á¨â¥«ì ï ¯«®â®áâì ¢®¤ï®£® ¯ à : co = c = g : (17)
§¢¥áâ® [4], çâ® á㬬 ®â®á¨â¥«ìëå ¯«®â®á⥩ ¡¨ ன á¬¥á¨ à ¢
¥¤¨¨æ¥:
ao +
co
=1: (18)
¢¨á¨¬®áâì(14) ¬®¦® ¯à¥¤áâ ¢¨âì ¢ ¤à㣮¬ ¢¨¤¥ á ãç¥â®¬ (9) ¨ (11)
n a +n c
=n(1,S
w
); (19)
⮣¤ ¯à¨¨¬ ï ¢® ¢¨¬ ¨¥ á®®â®è¥¨¥ (15) ¨¬¥¥¬
n a ( co = ao
+1)=n(1,S
w
); (20)
¨«¨ á ãç¥â®¬ (16)
n
a
=n(1,S
w ) ao =nS g ao : (21) «®£¨ç® ¤«ï ¢®¤ï®£® ¯ à ¬®¦® ©â¨ n c
= n(1,S
w ) co =nS g co : (22)
¦® ®â¬¥â¨âì, çâ® á«¥¤ãï ª« áá¨ç¥áª¨¬ âà ¤¨æ¨ï¬ [2, 3] ¬®¦®
¯®«ã-ç¨âì á«¥¤ãî饥:
n
a
=n=S
a
=(1,S
w ) ao =S g ao : (23)
«®£¨ç® ¨§ á®®â®è¥¨ï (22) ¯®«ãç ¥âáï:
n
c
=n=S
c
=(1,S
w ) co =S g co : (24)
ª« ¤ë¢ ï (23) ¨ (24) ¯®«ãç ¥¬ á«¥¤ãîéãî § ¢¨á¨¬®áâì
S a +S c = S g ( ao + co )=S
g
; (25)
ãç¨âë¢ ï (9) ¨¬¥¥¬
S a +S c +S w
=1: (26)
§¢¥áâ® [1], çâ® ¯® ¬¥å ¨§¬ã ®¡à §®¢ ¨ï ¨ ¯® í¥à£¨¨ á¢ï§¨ á ⢥म©
áà¥-¨¬¥îâ á«¥¤ãî騥 áâàãªâãàë¥ à §®¢¨¤®á⨠¯®à¨á⮩ ¦¨¤ª®áâ¨: ¦¨¤ª®áâì
¬®®¬®«¥ªã«ïன ¤á®à¡æ¨¨, ¦¨¤ª®áâì ¯®«¨¬®«¥ªã«ïன ¤á®à¡æ¨¨,
¤¨ä-ä㧨® ï, ª ¯¨««ïà ï ¨ £à ¢¨â 樮 ï ¦¨¤ª®áâ¨.
¡é¥¥ ᮤ¥à¦ ¨¥ ¦¨¤ª®© ª®¬¯®¥âë ¢ á।¥ á®áâ ¢«ï¥â
W
=W
mw
+W
pw
+W
dw
+W
cw
+W
sw
;
(27)£¤¥
W
mw
| ¢« ¦®áâ쬮®á«®©®© ¤á®à¡æ¨¨:W
mw
=n
mw
mw
=m
s
;
(28)W
pw
| ¢« ¦®áâì ¯®«¨á«®©®© ¤á®à¡æ¨¨:W
pw
=n
pw
pw
=m
s
;
(29)W
dw
| ¤¨ää㧨® ï ¢« ¦®áâì:W
dw
=n
dw
dw
=m
s
;
(30)W
cw
| ª ¯¨««ïà ï ¢« ¦®áâì:W
cw
=n
cw
cw
=m
s
;
(31)W
sw
| £à ¢¨â 樮 ï ¢« ¦®áâì;W
sw
=n
sw
sw
=m
s
:
(32)।áâ ¢¨¬ ®¡ê¥¬ ¦¨¤ª®á⨠¢ ¥¤¨¨æ¥ ®¡ê¥¬ à áᬠâਢ ¥¬®© áà¥¤ë ¢
á«¥¤ãî饬 ¢¨¤¥
n
w
=n
mw
+n
pw
+n
dw
+n
cw
+n
sw
;
(33)¨«¨ á ãç¥â®¬ § ¢¨á¨¬®á⨠(4)
n
mw
+n
pw
+n
dw
+n
cw
+n
sw
=nS
w
:
(34)â®è¥¨¥ ®¡ê¥¬®¢ ¢á¥å á®áâ ¢«ïîé¨å à §®¢¨¤®á⥩ ¦¨¤ª®£®
ª®¬¯®-¥â ¬®¦® ¯®«ãç¨âì ¨§ á®®â®è¥¨©(28) |(32). ª®ç ⥫ì®â ª¨¥
á®®â-®è¥¨ï ¨¬¥îâ á«¥¤ãî騩 ¢¨¤:
n
iw
n
jw
=W
iw
®¤áâ ¢«ïï § ¢¨á¨¬®áâì (35) ¢ à ¢¥á⢮(34) ¬®¦® ¢ëà §¨âì ®¡ê¥¬ ⮩
¨«¨ ¨®© áâàãªâãà®©à §®¢¨¤®á⨠¯®à¨á⮩ ¢®¤ë:
n
iw
=nS
w
M
iw
;
(i
=m;p;d;c;s
);
(36)£¤¥ ¤«ïª ¦¤®© áâàãªâãà®©à §®¢¨¤®á⨠¢¥«¨ç¨
M
iw
¨¬¥¥â¢¨¤M
iw
Ps
W
iw
iw
i
=m
W
iw
=
iw
;
(
i
=m;p;d;c;s
):
(37) ¬¥â¨¬, çâ® á㬬 ¢¥«¨ç¨
M
iw
¤«ï ¢á¥å áâàãªâãàëå à §®¢¨¤®á⥩ ¢®¤ë à ¢ ¥¤¨¨æ¥:s
X
i
=m
M
iw
=1;
(i
=m;p;d;c;s
):
(38)«ï ®¯à¥¤¥«¥¨ï á⥯¥¨ ᮤ¥à¦ ¨ï ⮩ ¨«¨ ¨®© áâàãªâãன
à §®-¢¨¤®á⨠¯à¥¤áâ ¢¨¬ § ¢¨á¨¬®áâì (36) ¢ á«¥¤ãî饬 ¢¨¤¥
n
iw
=n
=S
iw
=S
w
M
iw
;
(i
=m;p;d;c;s
):
(39)㬬¨à®¢ ¨¥¬ § ¢¨á¨¬®á⥩ ¢¨¤ (39) ¬®¦®¯®«ãç¨âì
s
X
i
=m
S
iw
=S
w
s
X
i
=m
M
iw
;
(40)¨«¨ á ãç¥â®¬ á®®â®è¥¨ï (38)
s
X
i
=m
S
iw
=S
w
;
(i
=m;p;d;c;s
):
(41)®¤áâ ¢«ïï§ ¢¨á¨¬®áâì (41)¢à §¢¥àã⮬¢¨¤¥¢à ¢¥á⢮(26)¤«ï¢á¥å
á®áâ ¢«ïîé¨å ª®¬¯®¥â®¢ áà¥¤ë ¨¬¥¥¬:
S
a
+S
c
+S
mw
+S
pw
+S
dw
+S
cw
+S
sw
=1:
(42)ª®ç â¥«ì® ¨á室ãî § ¢¨á¨¬®áâì (1) ¤«ï¥¤¨¨æë ®¡ê¥¬ á।ë
¬®¦-® ¯à¥¤áâ ¢¨âì ¢¢¨¤¥:
¨«¨ á ãç¥â®¬ (39)
m
+nS
a
+nS
c
+nS
mw
+nS
pw
+nD
dw
+nS
cw
+nS
sw
=1;
(44)¢ë®áï ®¡é¨© ¬®¦¨â¥«ì| ¢¥«¨ç¨ã ¯®à¨áâ®áâ¨
n
§ ᪮¡ª¨, ¨¬¥¥¬:m
+n
(S
a
+S
c
+S
mw
+S
pw
+S
dw
+S
cw
+S
sw
)=1;
(45)®âªã¤ á ãç¥â®¬ (42) ®¡à â® ¯®«ãç ¥¬ ¨á室ãî § ¢¨á¨¬®áâì (1).
¨á. 1. ¡ê¥¬ ¨ ¬ áá á®áâ ¢«ïîé¨å ª®¬¯®¥â®¢
¢ í«¥¬¥â¥ ¯®à¨á⮩á।ë
§ ª«î票¥ ®â¬¥â¨¬, çâ® ¤«ï¯à ªâ¨ç¥áª¨å à áç¥â®¢ ¢¥«¨ç¨ã á⥯¥¨
¢« ¦®á⨠¨§ § ¢¨á¨¬®á⨠(4) á«¥¤ã¥â ®¯à¥¤¥«ïâì á ãç¥â®¬ ¢¥á®¢®£®
ᮤ¥à¦ -¨ï áâàãªâãàëå à §®¢¨¤®á⥩ ¯®à®¢®© ¦¨¤ª®áâ¨.
¨â¥à âãà
1. ¥®à¥â¨ç¥áª¨¥ ®á®¢ë ¨¦¥¥à®© £¥®«®£¨¨. ¨§¨ª®-娬¨ç¥áª¨¥
®á®-¢ë. / ®¤ । ªæ¨¥© ª ¤¥¬¨ª ¥à£¥¥¢ ..|M.: ¥¤à , 1985.|354á.
2. íà., á« ¢áª¨., ଥ© . ¨§¨ª®-¬ ⥬ â¨ç¥áª¨¥ ®á®¢ë
䨫ìâà -樨 ¢®¤ë.|M.: ¨à, 1971.|451c.
3. «¬ ⮢ . . ¥å ¨ª £àã⮢. ᮢ ¨ï ¨ ä㤠¬¥âë.|.:
âன¨§¤ â, 1988.|415 c.