e7K=epF8a;L=Ke 7apO4K 8 Bo:8
B8:Bop #�H, �a;8=a #�HA, Map8O OPHEBA
'acBL 2. -:c?ep8<e=B A2pa<e=:o
1. To:8 2 pea:B82=Kx M;e<e=Bax
po4o;68< =aHe 8cc;e4o2a=8e 8 ?epe94e< Be?epL : M;e:Bp8Gec:8< Fe-
?O<, xapa:Bep=Ke pa7<epK :oBopKx (L) 7=aG8Be;L=o <e=LHe 4;8=K 2o;-
=K (L << ). Cpa2=8< ?o2e4e=8e =e:oBopKx cocpe4oBoGe==Kx M;e<e=Bo2
cxe< 4;O 8=epF8a;L=Kx 8 1e7K=epF8a;L=Kx Bo:o2.
B ;N1o9 Fe?8, O2;OeBcO ;8 o=a pa7o<:=yBo9 8;8 =eB, <o3yB cyIecB2o2aBL
Bo:8, o1pa7o2a==Ke :a: 8=epF8a;L=K<8, Ba: 8 1e7K=epF8a;L=K<8 7apO-
4a<8. a: ?pa28;o, 2 7a<:=yBKx M;e:Bp8Gec:8x Fe?Ox (:upx3oDo2c:ue Fe-
nu) Bo:8 87 M;e:Bpo=o2 ?po2o48<ocB8 7=aG8Be;L=o ?pe2KHaNB Bo:8 87
1e7K=epF8a;L=Kx 7apO4o2. <e==o ?o MBo9 ?p8G8=e <K 8x =e �7a<eGae<
?p8 M:c?ep8<e=Ba;L=Kx 8cc;e4o2a=8Ox M;e:Bp8Gec:8x Fe?e9. O4=a:o co-
oB=oHe=8e <e64y Bo:a<8 pa48:a;L=o <e=OeBcO 2 42yx c;yGaOx.
Bo-?ep2Kx, 4o;O Bo:a, co74a2ae<o3o M;e:Bpo=a<8 ?po2o48<ocB8, y<e=L-
HaeBcO ?p8 y2e;uGe=uu GacmomK :o;e1a=89. -Bo o1yc;o2;e=o :a: 8=ep-
F8a;L=ocBLN M;e:Bpo=o2 ?po2o48<ocB8, Ba: 8 c:8=-MDDe:Bo<, cy6aN-
I8< c;o9, 2 :oBopo< MB8 M;e:Bpo=K 4286yBcO 2 ?po2o4=8:e.
Bo-2BopKx, 4o;O M;e:Bpo=o2 ?po2o48<ocB8 2 cy<<ap=o< Bo:e ?a4aeB 2
pa7o<:=ymKx (=e:upx3oDo2c:ux) Fe?Ox. Ta:8e Fe?8 1y4yB ?o4po1=o pac-
c<oBpe=K =86e.
B :aGecB2e ?p8<epa <o6=o ?p82ecB8 48?o;L �epFa, 87o1pa6e==K9 =a
p8c. 1. Cy<<ap=K9 Bo: I, co74a2ae<K9 8cBoG=8:o< =a?pO6e=8O U, c:;a-
4K2aeBcO 87 42yx Bo:o2:
I = I + I , (1.1)
1
34e: I  Bo:, co74a2ae<K9 1e7K=epF8a;L=K<8 7apO4a<8 (=e:8px3oDo2-
c:89 Bo:); I  Bo:, co74a==K9 M;e:Bpo=a<8 ?po2o48<ocB8 (:8px3oDo2-
c:89 Bo:).

P8c. 1. 8?o;L �epFa
-;e:Bp8Gec:aO :8px3oDo2c:aO Fe?L 7a<K:aeBcO 27a8<=o9 e<:ocBLN C,
cyIecB2yNIe9 <e64y �yca<u 48?o;O �epFa. He:8px3oDo2c:aO Fe?L 7a-
:a=G82aeBcO =a ye48=e==Kx e<:ocBOx, o1pa7o2a==Kx �:o=Gu:a<u yco2
48?o;O �epFa.
po2e4e< =e1o;LHoe cpa2=e=8e MB8x Fe?e9 4;O =87:8x GacBoB.
1. a<:=ymaO (:upx3oDo2c:aO) FenL. Pacc<oBp8< ;8=e9=K9 ?po2o4=8:
1e7 ?oBepL, 4;8=o9 L (L << ). To: :8px3oDo2c:o9 Fe?8, o1pa7o2a==K9
M;e:Bpo=a<8 ?po2o48<ocB8, co74aeB 2o:py3 ?po2o4=8:a ?o;O
"A
E = -gradĆ - 8 H = rotA . a: ?o:a7a=o 2 [1], ypa2=e=8O 4;O
"t
?oBe=F8a;o2 8<eNB 284:
� v

"A = -�j = -�� v ; "Ć = - (1.2)
� 2c
34e: �  ?;oB=ocBL ?pocBpa=cB2e==o3o 7apO4a M;e:Bpo=o2 ?po2o48<ocB8;
v  cpe4=OO c:opocBL M;e:Bpo=o2 ?po2o48<ocB8; A 8 Ć  ?oBe=F8a;K
?o;e9 M;e:Bpo=o2 ?po2o48<ocB8.
B c8;y Bo3o, GBo v << c, 2 84ea;L=o< ?po2o4=8:e 1y4yB cyIecB2o2aBL
?pa:B8Gec:8 Bo;L:o <a3=8B=oe ?o;e 8 8=4y:F8o==oe M;e:Bp8Gec:oe ?o;e,
o1pa7o2a==Ke 2e:Bop=K< ?oBe=F8a;o< A .
2. Pa7o<:=ymaO (=e:upx3oDo2c:aO) FenL. Te?epL ?epe94e< : ypa2=e=8O<
4;O 1e7K=epFua;L=Kx Bo:o2 (=e:8px3oDo2c:8e Fe?8). B c8;y Bo3o, GBo
2
j = c� , <K <o6e< pacc<oBpeBL 2o;=o2oe ypa2=e=8e Bo;L:o 4;O � , ?o-
c:o;L:y ypa2=e=8e 4;O Bo:o2 ?o cyIecB2y Bo 6e ca<oe.
1 " �
"� - = 0 (1.3)

c "t
Ec;8 4;8=a ?po2o4=8:a 2ecL<a <a;a ?o cpa2=e=8N c 4;8=o9 2o;=K
2ĄL
( kL = << 1), Bo34a "� H" 0.

-Bo o7=aGaeB, GBo 3pa48e=B ?pocBpa=cB2e==o3o 7apO4a 1e7K=ep8a;L=Kx Gac-
B8F grad� <a;, B.e. Gepe7 ;N1oe ?o?epeG=oe ceGe=8e 2eB28 M;e:Bp8Gec:o9
Fe?8 ?poBe:aeB ?pa:B8Gec:8 o48= 8 BoB 6e 7apO4 2 e48=8Fy 2pe<e=8. B MBo<
c;yGae <K <o6e< ?pe=e1peGL 7a?a74K2a=8e< 8 pacc<aBp82aBL ypa2=e=8e
"A = -�c� (1.4)
1e7 yGeBa ypa2=e=8O (5.3). o Dop<e MBo ypa2=e=8e co2?a4aeB c ypa2=e=8-
e< (5.2) 8 cBpy:Bypa ?o;e9 o:a7K2aeBcO oGe=L 1;87:o9. CooB2eBcB2e==o,
u=4y:mu2=ocmu u e<:ocmu ?o oB=oHe=8N : 1e7K=epF8a;L=K< 7apO4a< 8
Bo:a< 4o;6=K 2ecB8 ce1O ?p8<ep=o Ba:, :a: o=8 2e4yB ce1O ?o oB=oHe-
=8N : 7apO4a< 8 Bo:a< 87 M;e:Bpo=o2 ?po2o48<ocB8.
O4=a:o pa7;8G8e 2ce 6e cyIecB2yeB. e;o 2 Bo<, GBo ?o;O =e:8px3oDo2-
c:8x Bo:o2 2ce34a 8<eNB nonepeG=K9 xapa:Bep. p8 oBcyBcB288 27a8<o-
4e9cB28O c M;e:Bpo=a<8 ?po2o48<ocB8 o=8 =e 8<eNB ?oBepL 2 ?po2o4=8-
:ax. Bo7<o6=o, 8<e==o o=8 oB2eGaNB 7a O2;e=8e c2epxnpo2o4u<ocmu <e-
Ba;;o2 ?p8 oGe=L =87:8x Be<?epaBypax.
B pa7o<:=yBo9 M;e:Bp8Gec:o9 Fe?8 ?p8 oBcyBcB288 ?apa78B=o9 e<:ocB8
<e64y :o=Fa<8 Fe?8, :oBopaO <o3;a 1K 7a<:=yBL :8px3oDo2c:yN Fe?L,
M;e:Bpo=K ?po2o48<ocB8 ?pa:B8Gec:8 =e co74aNB Bo:a. e7K=epF8a;L-
=Ke 6e Bo:8 o1pa7yNB cBoOGyN 2o;=y c y7;a<u mo:a =a :o=Fax Fe?8. o
MBo9 ?p8G8=e 2 pa7o<:=yBo9 Fe?8 cyIecB2yeB xoBO 1K <a;e=L:89 Bo:,
?po?opF8o=a;L=K9 sin(Ąz/), 34e z  paccBoO=8e oB BoG:8 o1pK2a Fe?8.
Ha?p8<ep, Be;eDo==K9 =ayH=8: (8;8 pe?po4y:Bop), ?o4:;NGe==K9 c
?o<oILN 4;8==o3o ?po2o4=8:a Bo;L:o o4=8< :o=Fo< : ?oBe=F8a;L=o<y
?po2o4y Bpa=c;OF8o==o9 ceB8 1y4eB =e3po<:o �2eIaBL. �po<:ocBL 72y:a
3
?pa:B8Gec:8 =e 7a28c8B oB 4;8=K coe48=8Be;L=o3o ?po2o4a, B.e. oB pac-
cBoO=8O <e64y =ayH=8:o< 8 =y;e2K< ?po2o4o< Bpa=c;OF8o==o9 ceB8.
H86e <K 8cc;e4ye< =e:8px3oDo2c:8e Fe?8 =a =87:8x GacBoBax.
2. #e48=e==aO e<:ocBL
B cBa=4apB=Kx yGe1=8:ax D878:8 pacc<aBp82aNBcO 42a 284a e<:ocBe9:
ye48=e==aO 8;8 co1cB2e==aO e<:ocBL 8 27a8<=aO e<:ocBL 42yx Be; (B.e.
:o=4e=caBop), :oBopKe 87o1pa6e=K =a p8c. 2. #e48=e==aO e<:ocBL ecBL
<eBa;;8Gec:oe Be;o, 87o;8po2a==oe oB 4py38x ?po2o4=8:o2.

P8c. 2. 2a 284a e<:ocBe9
B yGe1=8:ax MBo9 e<:ocB8 o1KG=o =e y4e;OeBcO 2=8<a=8O, 8 o?8ca=8e
e<:ocB8 o3pa=8G82aeBcO Dop<y;o9
q = CĆ (Ć" = 0) (2.1)
34e: C  2e;8G8=a ye48=e==o9 e<:ocB8; Ć  ?oBe=F8a; ?po2o4OIe3o Be;a
?o oB=oHe=8N : 1ec:o=eG=ocB8; q  7apO4 =a ye48=e==o9 e<:ocB8.
CocBa28< Be?epL Fe?L 87 3e=epaBopa, 42yx coe48=8Be;L=Kx ?po2o4o2 8
42yx <eBa;;8Gec:8x Be; (ye48=e==Ke e<:ocB8), 87o1pa6e==yN =a p8c. 3a.
p8 =a;8G88 ?epe<e==o3o =a?pO6e=8O U 2 2eB2Ox pa7o<:=yBo9 Fe?8 1y-
4eB BeGL Bo:. -By Fe?L <o6=o 7a<e=8BL M:282a;e=B=o9 cxe<o9 7a<eIe-
=8O, 87o1pa6e==o9 =a p8c. 31. Ha MBo9 cxe<e C1" 8 C2"  ye48=e==Ke e<-
4
:ocB8 ?ep2o3o 8 2Bopo3o Be;a, C12  27a8<=aO e<:ocBL <e64y MB8<8 Be;a-
<8 (:o=4e=caBop).

P8c. 3. -:282a;e=B=aO cxe<a pa7o<:=yBo9 Fe?8
Ec;8 cG8BaBL 1ec:o=eG=o y4a;e==yN ?o2epx=ocBL S ?po2o4=8:o<, Bo pa-
7o<:=yByN Fe?L <K :a: 1K c2e;8 : 7a<:=yBo9 :8px3oDo2c:o9 Fe?8. Ho
Ba:o9 2K2o4 oH81oGe=. a<:=yBo9 :8px3oDo2c:o9 Fe?LN c;y68B ?oc;e-
4o2aBe;L=oe coe48=e=8e M;e<e=Bo2: Be;o 1  ?po2o4=8:  3e=epaBop U 
?po2o4=8:  Be;o 2  e<:ocBL C12  Be;o 1. Ec;8 e<:ocBL C12 <a;a 8 e<:o-
cBLN <o6=o ?pe=e1peGL, Bo ocBa2HaOcO GacBL cxe<K ?pe4cBa2;OeB co1o9
=e:upx3oDo2c:yN Fe?L, ?o4G8=ONIyNcO c2o8< 2o;=o2K< 7a:o=a<.
B Ba:o9 Fe?8 Bo: 2 pa7;8G=Kx ceGe=8Ox ?po2o4o2 <e64y :a64K< Be;o< 8
3e=epaBopo< 1y4eB pa7;8Ge=. B Fe?8 ycBa=o28BcO cBoOGaO 2o;=a Bo:a 8
=a?pO6e=8O. o4o1=aO Fe?L o1;a4aeB ;N1o?KB=K<8 c2o9cB2a<8.
Ha?p8<ep, ?p8 M:c?ep8<e=Ba;L=Kx 8cc;e4o2a=8Ox <K ?KBa;8cL 87<e-
p8BL Bo: 2 coe48=8Be;L=o< ?po2o4e, :a: ?o:a7a=o =a p8c. 4, c ?o<oILN
?p81opa ABO-5 (BecBep =a ?o;y?po2o4=8:o2Kx 48o4ax). Ha?pO6e=8e 2
;8=88 900 B, GacBoBa 18 :�F. Ha MBo< 6e p8cy=:e ?o:a7a=K ?OBL c?oco-
1o2 2:;NGe=8O MBo3o ?p81opa. O:a7a;ocL, GBo ?p8 ;N1o< c?oco1e 2:;N-
Ge=8O cBpe;:a ?p81opa oB:;o=O;acL ?pa:B8Gec:8 =a o4u= u mom 6e y3o;.
-BoB y3o; =e 7a28ce; oB ?o;o6e=8O pyGe: ?epe:;NGaBe;e9 48a?a7o=o2
87<epe=8O Bo:o2 8 =a?pO6e=89.
5

P8c. 4. o:a7a=8O ABO-5 2 pa7o<:=yBo9 Fe?8
A=a;o38G=oe O2;e=8e 2?ep2Ke o?8ca; 2 c2o8x 8=Bepec=Kx M:c?ep8<e=Bax
Ba;a=B;82K9 87o1peBaBe;L A2pa<e=:o [2], [3].
3. -:c?ep8<e=B A2pa<e=:o
 co6a;e=8N, 2 [2] <K 8<ee< Bo;L:o c;o2ec=oe o?8ca=8e M:c?ep8<e=Ba.
o MBo9 ?p8G8=e <K 2occBa=o28< MBy cxe<y, o?8paOcL =a [2], [3] 8 7apa-
=ee ?poc8< y G8BaBe;e9 8728=e=8O 7a 2o7<o6=Ke =eBoG=ocB8.
Ba:, 5.08.90 2 o4=o9 87 ;a1opaBop89 M- 1K; ?po4e<o=cBp8po2a= M:c-
?ep8<e=B, cxe<a :oBopo3o 87o1pa6e=a =a p8c. 5.

P8c. 5. pe4?o;a3ae<aO cxe<a M:c?ep8<e=Ba A2pa<e=:o
6
B M:c?ep8<e=Ba;L=yN ycBa=o2:y 2xo48; <aH8==K9 3e=epaBop 1 <oI=o-
cBLN 4o 100 :BB, 3e=ep8pyNI89 =a?pO6e=8e c GacBoBo9 8 :�F. -BoB <a-
H8==K9 3e=epaBop ?8Ba; ?ep28G=yN o1<oB:y Bpa=cDop<aBopa Tec;a 2.
O48= :o=eF 2Bop8G=o9 o1<oB:8 1K; c2o1o4e= (=8 : Ge<y =e ?o4:;NGe=,
:a: ?o:a7a=o =a p8c. 5). o 2Bopo<y :o=Fy 1K;8 ?o4coe48=e=K ?oc;e4o-
2aBe;L=o c;e4yNI8e M;e<e=BK: Bep<oM;e:Bp8Gec:89 <8;;8a<?ep<eBp 3,
Bo=:89 2o;LDpa<o2K9 ?po2o4 4 (4;8=a ?po2o4a 2,75 <, 48a<eBp 15 <:<)
8 �28;:a A2pa<e=:o 5.
B8;:a A2pa<e=:o ?pe4cBa2;OeB co1o9 7a<:=yBK9 :o=Byp, co4ep6aI89
42a ?oc;e4o2aBe;L=o coe48=e==Kx 48o4a, y :oBopKx o1IaO BoG:a ?o4coe-
48=e=a : o?8ca==o9 pa=ee Fe?8, 8 =a3py7:y. Ha3py7:o9 c;y68;8 =ec:o;L-
:o ?o4coe48=e==Kx ;a<?oGe: =a:a;82a=8O.
o MBo9 pa7o<:=yBo9 Fe?8 A2pa<e=:o c<o3 ?epe4aBL oB 3e=epaBopa : =a-
3py7:e (;a<?a< =a:a;82a=8O) M;e:Bp8Gec:yN <oI=ocBL ?opO4:a 1300 BB
(!). -;e:Bp8Gec:8e ;a<?oG:8 Op:o c2eB8;8cL.
a7a;ocL 1K, MBo O2;e=8e ;e3:o o1JOc=8<o. Cxe<a co4ep68B ye48=e==yN
e<:ocBL, o1pa7o2a==yN Fe?oG:o9 ;a<? =a:a;82a=8O 8 coe48=ONI8<8 8x
?po2o4a<8. p8 ?o;o68Be;L=o< ?o;y ?ep8o4e =a?pO6e=8O Gepe7 48o4 D1
2 MBy Fe?L BeGeB 7apO4=K9 Bo:.  ?oBe=F8a; ye48=e==o9 e<:ocB8 2o7pac-
BaeB. p8 oBp8FaBe;L=o< ?o;y ?ep8o4e =a?pO6e=8O MBa e<:ocBL pa7pO6a-
eBcO Gepe7 48o4 D2, ?p8o1peBaO =o2yN 2e;8G8=y ?oBe=F8a;a. apO4=o-
pa7pO4=K9 Bo: Io 8<eeB 2ce34a o4=o =a?pa2;e=8e 8 e3o 2e;8G8=K 4ocBa-
BoG=o, GBo1K ?o44ep682aBL =a ;a<?ax =a:a;82a=8O 2 28;:e A2pa<e=:o
cpe4=NN <oI=ocBL 1300 BB.
To: I1 2 Fe?8, coe48=ONIe9 :o=eF o1<oB:8 Bpa=cDop<aBopa Tec;a c 28;-
:o9 A2pa<e=:o, 4o;6e= 1KBL 1;87o: 8;8 ?p81;878Be;L=o pa2e= ?o 2e;8-
G8=e Bo:y Io. Ec;8, =a?p8<ep, =a3py7:a 28;:8 A2pa<e=:o ?pe4cBa2;OeB
co1o9 ?oc;e4o2aBe;L=oe coe48=e=8e 6-B8 42yxcoB2aBB=Kx ;a<? =a:a;8-
2a=8O, Bo pa7pO4=o 7apO4=K9 Bo: Io 1y4eB pa2e= 1 A. CooB2eBcB2e==o Ba-
:o9 6e ?p8<ep=o 2e;8G8=K 4o;6e= 1K; 1K 1KBL 8 Bo: I1.
O4=a:o Ba:oe o1JOc=e=8e ?poB82opeG8B c;e4yNIe<y Da:By. Tep<oM;e:-
Bp8Gec:89 <8;;8a<?ep<eBp 3 7aD8:c8po2a; oGe=L <a;yN 2e;8G8=y Bo:a
I1 (I1 H" 2 <A(!)), a Bo=:89 2o;LDpa<o2K9 ?po2o4 4 4a6e =e =a3pe;cO!
7
<e==o MBo o1cBoOBe;LcB2o ?oc;y68;o 3;a2=o9 ?p8G8=o9 Bpy4=ocB8
o1JOc=e=8O pe7y;LBaBo2 M:c?ep8<e=Ba A2pa<e=:o.
B pa1oBe [3] ?p82e4e=K pe7y;LBaBK a=a;o38G=Kx 87<epe=89 ?p8 1o;ee
2Kco:8x GacBoBax 8 1o;ee =87:8x =a?pO6e=8Ox. O=8 :aGecB2e==o ?o2Bo-
pONB pe7y;LBaBK o?8ca==o3o 2KHe M:c?ep8<e=Ba.
p82e4e< =e:oBopKe 2K2o4K, c4e;a==Ke A2pa<e=:o 8 e3o :o;;e3a<8 =a
oc=o2a=88 MB8x 8cc;e4o2a=89.
1. To: Io 2 28;:e A2pa<e=:o ;u=e9=o y2e;uGu2aemcO c pocmo< GacmomK
(48a?a7o= 87<epe=89 5...100 :�F) 8 ?pa:B8Gec:8 ;8=e9=o 2o7pacBaeB c
pocBo< =a?pO6e=8O 3e=epaBopa ?p8 ?ocBoO==o9 GacBoBe. -Bo c284e-
Be;LcB2yeB o e<:ocm=o< xapa:Bepe M;e:Bp8Gec:o9 Fe?8.
2. Ma3=8B=oe ?o;e 2 ?po2o4=8:e, coe48=ONIe< 28;:y A2pa<e=:o c 3e=e-
paBopo<, =e 1K;o o1=apy6e=o.
3. To: I1 1K; oGe=L <a; ?o cpa2=e=8N c Bo:o< Io 8 npa:muGec:u =e o1=a-
py6u2a;cO =u men;o2K<, =u <a3=umoM;e:mpuGec:u< u7<epume;e< mo-
:a. o MBo9 ?p8G8=e =a;8G8e 2 coe48=8Be;L=o9 Fe?8 (mpa=cDop<amop
Tec;a  2u;:a A2pa<e=:o) ?oc;e4o2aBe;L=o coe48=e==Kx pe78cBopo2
(4o =ec:o;L:8x 4ecOB:o2 MO<), :o=4e=caBopo2 8 8=4y:B82=ocBe9 o:a-
7K2a;o Gpe72KGa9=o <a;oe oc;a1;ONIee 4e9cm2ue =a Bo: Io 2 28;:e
A2pa<e=:o.
4ecL <K =e 1y4e< o1cy64aBL 38?oBe7y a2Bopo2, 87;o6e==yN 2 [3], o
?pe4?o;a3ae<o< <exa=87<e ?epe4aG8 M=ep388. OB<eB8< ;8HL ee oc=o2-
=Ke <o<e=BK.
Co3;ac=o 8x BoG:e 7pe=8O 2 Fe?8 BeGeB npo4o;L=K9 mo: c<eIe=uO, =e
co74aNI89 <a3=8B=o3o ?o;O. Be;8G8=a � <eBa;;a u7<e=OemcO c y42oe=-
=o9 Gacmomo9 oB 1 4o " 8 MBo 87<e=e=8e c2O7a=o c Da7o9 ?o4a2ae<o3o 2
Fe?L =a?pO6e=8O. po4o;L=K9 ?epe<e==K9 Bo: c<eIe=8O I1 o1;a4aeB
�c2epxnpo2o4OIu<u c2o9cm2a<u, B.e. =e 2K4e;OeB Be?;a 2 pe78cBopax 8
?pa:B8Gec:8 =e co74aeB =a =8x ?a4e=8O =a?pO6e=8O.
B 28;:e A2pa<e=:o MBoB Bo: ?peo1pa7yeBcO 2 o1KG=K9 ?y;Lc8pyNI89
Bo: Io. O4=a:o, 2 ?o4o1=o< o1JOc=e=88 =e 2ce Oc=o. Ec;8 87<epOBL =a-
?pO6e=8e U 2 BoG:ax AB M;e:BpocBaB8Gec:8< 2o;LB<eBpo<, a Bo: o1KG-
=K< a<?ep<eBpo<, Bo ?p8 2KG8c;e=88 <oI=ocB8 2 =a3py7:e =a1;N4aeBcO
=ecoom2emcm2ue c :;acc8Gec:8<8 7a:o=a<8. BKG8c;e==Ke <oI=ocB8
8

P = I R , P = I U 8 7a:o= O<a U = I R (34e R  co?poB82;e=8e Fe?8,
co4ep6aIe9 ;a<?oG:8) =e cooB2eBcB2o2a;8 4py3 4py3y. A2BopK ?8HyB,
GBo pe78cBopK �:a: 1K BepONB c2o8 =o<8=a;K [3].
4. o=Bpo;L=Ke M:c?ep8<e=BK
He:oBopKe 2K2o4K 2 [3] =a< ?pe4cBa2;ONBcO co<=8Be;L=K<8. Ha?p8<ep,
oBcyBcB28e <a3=8B=o3o ?o;O y Bo:a I1. -BoB 2K2o4 ?poB82opeG8B M;e:Bpo-
48=a<8:e, ?oc:o;L:y ;N1o9 Bo: c<eIe=8O co74aeB 2o:py3 ce1O <a3=8B=oe
?o;e ?o4o1=o o1KG=o<y Bo:y ?po2o48<ocB8. Co<=8Be;L=K< ?pe4cBa2;O-
eBcO 7a:;NGe=8e o1 87<e=e=88 � <eBa;;a. o4o1=oe 87<e=e=8e 4o;6=o
1K;o 1K 2K7K2aBL =e;8=e9=Ke O2;e=8O 2 o1KG=Kx ?po2o4=8:ax. -B8 =e-
cooB2eBcB28O 2K72a;8 =eo1xo48<ocBL M:c?ep8<e=Ba;L=o9 ?po2ep:8
=e:oBopKx 2K2o4o2 pa1oBK [3].
-;e:Bp8Gec:aO cxe<a :o=Bpo;L=Kx 87<epe=89 ?pe4cBa2;e=a =a p8c. 6.
Cxe<a co4ep68B 3e=epaBop 72y:o2o9 GacBoBK 1 (4o 20 :�F), ?o2KHaNI89
Bpa=cDop<aBop (1:30) c Depp8Bo2K< cep4eG=8:o< 2, 2o74yH=K9 Bpa=c-
Dop<aBop 3 4;O 87<epe=8O Bo:a I1 :oc2e==K< <eBo4o<, 28;:y A2pa<e=:o
c 48o4a<8 4, 2 :oBopyN 2:;NGe= <=o3o?pe4e;L=K9 <8:poa<?ep<eBp 5 8
pe78cBop R 6, a Ba:6e ye48=e==yN e<:ocBL C" 7.

P8c. 6. Cxe<a ?po2e4e=8O :o=Bpo;L=Kx M:c?ep8<e=Bo2
9
o;LFe2o9 Depp8Bo2K9 Bpa=cDop<aBop 2 8<eeB 42e oB4e;L=Ke o1<oB:8
4;O y<e=LHe=8O 27a8<=o9 e<:ocB8. #e48=e==aO e<:ocBL C" 2K?o;=e=a
87 a;N<8=8e2o9 Do;L38 H8p8=o9 20 c<, pa7<eIe==o9 ?o ?ep8<eBpy ;a-
1opaBop88 4x5 <.  c2o1o4=o<y :o=Fy A Bpa=cDop<aBopa 2 <o6=o 1K;o
?o4coe48=OBL ?po2o4 (4o 1 <) 4;O y2e;8Ge=8O ?oBe=F8a;a =a 28;:e A2-
pa<e=:o. ;O 3pa4y8po2:8 87<ep8Be;O Bo:a 3 <e64y BoG:a<8 A 8 B <o3
2:;NGaBLcO MBa;o==K9 <=o3o?pe4e;L=K9 <8:poa<?ep<eBp ?epe<e==o3o
Bo:a 8.
oBe=F8a; ?epe<e==o3o =a?pO6e=8O <o3 :o=Bpo;8po2aBLcO c ?o<oILN
87<ep8Be;O ?oBe=F8a;a 9 (Bo6e 28;:a A2pa<e=:o), Iy? :oBopo3o <o6=o
1K;o ?o4:;NGaBL : ;N1o9 BoG:e cxe<K. B =e< 8c?o;L7o2a;cO <8:poa<-
?ep<eBp =a 50 <:A.
-Ba M:c?ep8<e=Ba;L=aO ycBa=o2:a 8<e;a c;e4yNI8e 4ocmou=cm2a. To:8
I1 8 Io 1K;8 2 =ec:o;L:o pa7 2KHe, Ge< 1e7 ye48=e==o9 e<:ocB8 C". O=8
7=aG8Be;L=o ?pe2KHa;8 Bo: 87<ep8Be;O ?oBe=F8a;a 9.  =e4ocmam:a<
cxe<K <o6=o oB=ecB8 c;e4yNIee. CyIecB2yeB =e1o;LHaO napa7um=aO
e<:ocBL <e64y c2o1o4=K< :o=Fo< A Bpa=cDop<aBopa 2 8 ye48=e==o9
e<:ocBLN, a Ba:6e 2 e<:ocB8 C" 2o7=8:aNB =a2o4:u ceme2o3o =anpO6e-
=uO ?po<KH;e==o9 GacBoBK 50 �F.
-:c?ep8<e=Ba;L=Ke 8cc;e4o2a=8O ?po4o;6aNBcO 8 2 =acBoOIee 2pe<O.
Ho =a oc=o2a=88 y6e ?o;yGe==Kx pe7y;LBaBo2 87<epe=89 <o6=o c4e;aBL
2K2o4K, =e 2o 2ce< co2?a4aNI8e c 2K2o4a<8 pa1oBK [3].
1. K;a no4m2ep64e=a ;u=e9=aO 7a2ucu<ocmL Bo:a Io 2 28;:e A2pa<e=:o
oB GacBoBK ?p8 =e87<e==o9 a<?;8By4e ?epe<e==o3o =a?pO6e=8O.
2. K;a no4m2ep64e=a ;u=e9=aO 7a2ucu<ocmL Bo:o2 Io 8 I1 oB =a?pO6e-
=8O, :oBopaO =apyHa;acL ?p8 oGe=L =87:8x =a?pO6e=8Ox 3e=epaBopa 1.
3. K;o o1=apy6e=o =a;uGue <a3=um=o3o no;O, co74a2ae<o3o Bo:o< I1.
-Bo c2o9cB2o 8c?o;L7o2a;ocL =a<8 4;O 87<epe=8O 2e;8G8=K MBo3o Bo:a
87<ep8Be;e< 3.
4. 7<epe==oe 7=aGe=8e oB=oHe=8O Bo:o2 Io/I1 cocBa2;O;o 2e;8G8=y oB 4 4o
10...12. B GacB=ocB8, o=o 7a28ce;o oB 4;8=K ?po2o4=8:a, ?o4coe48=Oe<o3o
: c2o1o4=o<y 2K2o4y A 2Bop8G=o9 o1<oB:8 Bpa=cDop<aBopa 2.
Ta:8< o1pa7o<, 1K; ycBa=o2;e= 2a6=K9 Da:B. To: I1 o:a7a;cO 2o <=o3o
pa7 <e=LHe Bo:a Io. -Bo pa7;8G8e, 2o7<o6=o, <=o3o 1o;LHe 87<epe==Kx

10
oB=oHe=89, ?oc:o;L:y o=o <o3;o 1KBL �7a<ac:8po2a=o 2;8O=8e< ?apa-
78B=o9 e<:ocB8 (o1pa7o2a==o9 c2o1o4=K< 2K2o4o< A 2mopuG=o9 o1<om-
:u mpa=cDop<amopa u ye4u=e==o9 e<:ocmLN), ?o :oBopo9 4o;6e= 2ce34a
?poBe:aBL :8px3oDo2c:89 Bo:.
5. $878Gec:8e O2;e=8O 2 M:c?ep8<e=Be A2pa<e=:o
O?8paOcL =a 87;o6e==Ke pa=ee BeopeB8Gec:8e pe7y;LBaBK 8 pe7y;LBaBK
M:c?ep8<e=Ba;L=Kx 8cc;e4o2a=89, <K <o6e< ?pe4;o68BL c2oe o1JOc=e-
=8e MDDe:Bo2, o1=apy6e==Kx A2pa<e=:o.
;O ?pocBoBK o1JOc=e=8O 1y4e< cG8BaBL, GBo ?apa78B=aO e<:ocBL <e64y
c2o1o4=K< 2K2o4o< 2Bop8G=o9 o1<oB:8 Bpa=cDop<aBopa Tec;a 8 28;:o9
A2pa<e=:o ?pe=e1pe68<o <a;a. Ta:aO Fe?L =e O2;OeBcO :8px3oDo2c:o9, 8
2 =e9 ?poBe:aNB Bo:8, o1pa7o2a==Ke 1e7K=epF8a;L=K<8 7apO4a<8.
7 Beop88 4;8==Kx ;8=89 872ecB=o, GBo 2 ?po2o4e 4;8=o9 L, ?o4coe48-
=e==o< o4=8< :o=Fo< : 2Kxo4y 3e=epaBopa, ycBa=a2;82aeBcO cBoOGaO
2o;=a. #7e; Bo:a 8 ?yG=ocBL ?oBe=F8a;a ?p8xo4OBcO =a c2o1o4=K9 :o=eF
;8=88 L. Pac?pe4e;e=8e Bo:a 2 ?po2o4e 8<eeB 284:
2U Ąx
I (x) = sin (5.1)
w 
34e: 2U  ?oBe=F8a; c2o1o4=o3o :o=Fa ?po2o4a; w  2o;=o2oe co?poB82-
;e=8e ?po2o4a ( Ą O<); x  paccBoO=8e oB c2o1o4=o3o :o=Fa ?po2o4a; 
 4;8=a 2o;=K.
po2e4e< Be?epL G8c;e==yN oFe=:y 4;O M:c?ep8<e=Ba A2pa<e=:o. o-
c:o;L:y 2ce 4a==Ke =a< =e 872ecB=K, <K ?po2e4e< ?p8:84oG=Ke pacGe-
BK. pe4?o;o68<, GBo ?oBe=F8a; =a :o=Fe coe48=8Be;L=o3o ?po2o4a 8
28;:e A2pa<e=:o cocBa2;OeB 2e;8G8=y ?opO4:a 2 :B. B MBo< c;yGae ?p8
oBcyBcB288 ?apa78B=o9 e<:ocB8 c2o1o4=K9 2K2o4 2Bop8G=o9 o1<oB:8
Bpa=cDop<aBopa Tec;a 1y4eB 8<eBL ?oBe=F8a;, ?o :pa9=e9 <epe, pa7 2 10
 50 2KHe (O2;e=ue nepe:oca nome=Fua;a, o1=apy6e==oe =a<8), Ge< �=a-
3py6e==K9 2K2o4 2Bop8G=o9 o1<oB:8 Bpa=cDop<aBopa. p8<e< 4;8=y
L = 5 <. ;8=a 2o;=K, cooB2eBcB2yNIaO GacBoBe 8 :�F, pa2=a 37500 <.
11
#G8BK2aO, GBo 2o;=o2oe co?poB82;e=8e ?po2o4a cocBa2;OeB 377 O<, 7a-
?8He< 2Kpa6e=8e 4;O I1.
I (x) H" 5,3sin 8,3�"10- x H" 4,4 �"10- x
p8 x = 5 < Bo: pa2e= I1 = 2,2 <A. -Ba 2e;8G8=a cooB2eBcB2yeB Bo:y, 87-
<epe==o<y 2 M:c?ep8<e=Bax A2pa<e=:o. Be;8G8=a Bo:a Io =e <o6eB ?pe-
2KHaBL 5,3 A. ;O HecB8 ;a<? ?o 200 BB Bo: Io cooB2eBcB2yeB 1 A, 4;O Ge-
BKpex ;a<? ?o 300 BB Io = 1,5 A 8 B.4.
4ecL : <ecBy 2Kc:a7aBL =e:oBopKe 38?oBeB8Gec:8e coo1pa6e=8O o 27a8-
<o4e9cB288 8=epF8a;L=Kx 8 1e7K=epF8a;L=Kx 7apO4o2. #c:ope==oe
4286e=8e M;e:Bpo=o2 ?po2o48<ocB8 4o;6=o 2K7K2aBL ?oO2;e=8e 1e7K-
=epF8a;L=o3o Bo:a 87 pa7=o8<e==Kx 1e7K=epF8a;L=Kx 7apO4o2, :oBopKe
1y4yB 4283aBLcO 2 ?poB82o?o;o6=Kx =a?pa2;e=8Ox. -;e:Bp8Gec:8e 8
<a3=8B=Ke ?o;O 1e7K=epF8a;L=Kx 7apO4o2, 2 c2oN oGepe4L, 4o;6=K
�mop<o7umL 4286e=8e M;e:Bpo=o2 ?po2o48<ocB8, B.e. ocyIecB2;OBL o?-
pe4e;e==K9 om1op M=ep3uu. -Ba oBo1pa==aO M=ep38O 4o;6=a ?epe4a2aBLcO
:p8cBa;;8Gec:o9 peHeB:e. e7K=epF8a;L=Ke 7apO4K o3pa=uGu2aNm c:opo-
cB8 4286e=8O M;e:Bpo=o2 ?po2o48<ocB8 8 �2Kpa2=82aNB MB8 c:opocB8.
C 4py3o9 cBopo=K, <a3=8B=Ke 8 M;e:Bp8Gec:8e ?o;O 1e7K=epF8a;L=Kx
7apO4o2 c?oco1=K 2K7K2amL =anpa2;e==oe 42u6e=ue M;e:Bpo=o2 ?po2o-
48<ocB8. -;e:Bpo=K ?po2o48<ocB8, 8<eO 8=epF8N, coxpa=ONB c2oN c:o-
pocBL ?oc;e 2o74e9cB28O.
O1cy48< Be?epL 2o?poc o ?poxo64e=88 1e7K=epF8a;L=Kx Bo:o2 Gepe7 p-n
?epexo4 48o4a. Pa=ee Ba:aO ?po1;e<a eIe =8:e< =e 8cc;e4o2a;acL. o-
MBo<y <K 2Kc:a6e< =e:oBopKe 38?oBeB8Gec:8e coo1pa6e=8O.
ycBL ?oBo: 1e7K=epF8a;L=Kx 7apO4o2 ?poxo48B Gepe7 p-n ?epexo4. o-
Bo: MB8x 7apO4o2 GacB8G=o oBpa6aeBcO oB ?epexo4a, a GacB8G=o ?poxo48B
Gepe7 48o4. Ec;8 ?o;o68Be;L=Ke 1e7K=epF8a;L=Ke 7apO4K 4286yBcO oB
p-c;oO : n-c;oN (oBp8FaBe;L=Ke, cooB2eBcB2e==o, 2 o1paB=o< =a?pa2;e-
=88), Bo o=u 2K7K2aNm noO2;e=ue mo:a 4KpoG=o-M;e:Bpo==o9 ?po2o48-
<ocB8 87 oc=o2=Kx =oc8Be;e9. Ec;8 6e 1e7K=epF8a;L=K9 Bo: 8<eeB o1-
paB=oe =a?pa2;e=8e, Bo Bo: 87 8=epF8a;L=Kx =oc8Be;e9 o1pa7o2aBLcO y6e
=e <o6em. 4ecL 1y4eB cyIecB2o2aBL e<:ocB=o9 Bo:, 1;a3o4apO 1apLep-
12
=o9 e<:ocB8, 8 c;a1K9 Bo:, o1yc;o2;e==K9 <a;o9 o1paB=o9 ?po2o48<o-
cBLN 48o4a.
Ec;8 MBa 38?oBe7a 2ep=a, <K <o6e< ?pe4;o68BL c;e4yNIee o1JOc=e=8e
M:c?ep8<e=Ba A2pa<e=:o. a<eB8<, GBo <K 1y4e< pacc<aBp82aBL ?poFec-
cK, npome:aNIue 2 oGe=L :opom:u9 u=mep2a; 2pe<e=u � = 2L/c H" 3,510 7
ce:y=4K, :oBopK9 7=aG8Be;L=o :opoGe ?ep8o4a ?epe<e==o3o =a?pO6e=8O
T = 1,25�10 4 ce:y=4K.

P8c. 7. To:8 2 �28;:e A2pa<e=:o
a4aNIaO 2o;=a Bo:a 87 1e7K=epF8a;L=Kx 7apO4o2 ?o4xo48B : BoG:e B
28;:8 A2pa<e=:o 8 pa72eB2;OeBcO. Bo;=a I1?poH, ?poxo4OIaO Gepe7 48o4
D1 (:a: ?o:a7a=o =a p8c. 7), 2K7K2aeB ?oO2;e=8e Bo:a Io 87 M;e:Bpo=o2
?po2o48<ocB8. -B8 M;e:Bpo=K 1y4yB 4283aBLcO, 2 :o=eG=o< cGeBe, : 48o-
4y D2 8 ?poxo48BL Gepe7 =e3o. Bo;=a 6e 1y4eB 7apO6aBL ye48=e==yN e<-
:ocBL C". py3aO GacBL 2o;=K, ?o4xo4O : BoG:e B 8 48o4y D2, 1y4eB oBpa-
6aBLcO, 8 ;8HL <a;aO ee GacBL I1?poH 4o94eB 4o ye48=e==o9 e<:ocB8 C".
OBpa6e==aO oB ye48=e==o9 e<:ocB8 2o;=a pa72eB28BcO. O4=a ee GacBL
I2oBp ?po94eB Gepe7 48o4 D2, 2K7K2aO 4286e=8e M;e:Bpo=o2 ?po2o48<ocB8
8 ?o44ep682aO Bo: Io. py3aO, oBpa782H8cL oB 48o4a D1, 2=o2L 2ep=eBcO
o1paB=o : ye48=e==o9 e<:ocB8 8 oB =ee : 48o4y D2.
o;LFe2oe 42u6e=ue M;e:Bpo=o2 ?po2o48<ocB8 2 28;:e A2pa<e=:o, o1pa-
7yNIee Bo: Io, 1y4eB coxpa=OBLcO =e7a2ucu<o oB Bo3o, 7apO6aeBcO ;8 ye-
48=e==aO e<:ocBL 8;8 6e 84eB ?poFecc ee pa7pO4a.
13
To: 2 coe48=8Be;L=o< ?po2o4e I pa2e= pa7=ocB8 ?pO<o3o Bo:a (84yIe3o
: 28;:e A2pa<e=:o) 8 o1paB=o3o (84yIe3o oB 28;:8) I1 = I?a4  IoBp. OBcN-
4a c;e4yeB, GBo Bo: Io <o6eB 1KBL 2o <=o3o pa7 1o;LHe Bo:a I1.
Ec;8 ?oc;e4o2aBe;L=o c 48o4a<8 2:;NG8BL c2eBo48o4K o?Bpo=o2, a 7aBe<
?po2ecB8 =a1;N4e=8e 87<e=e=8O ?po2o48<ocB8 DoBope78cBopo2 o?Bpo-
=o2 2o 2pe<e=8, <K <o3;8 1K y284eBL =a M:pa=e ocF8;;o3paDa c83=a;K,
c2O7a==Ke c Bo:o< Gepe7 48o4K, B.e. c2O7a==Ke c 87<e=e=8e< Op:ocB8
c2eGe=8O I c2eBo48o4o2 2 7a28c8<ocB8 oB 2pe<e=8.

P8c. 8. 8o4=Ke Bo:8 2 7a<:=yBo9 8 pa7o<:=yBo9 Fe?Ox
B :8px3oDo2c:o9 Fe?8 o=8 4o;6=K 7a3opaBLcO Gepe7 ?ep8o4 co c4283o<
=a ?o;o28=y ?ep8o4a, :a: 87o1pa6e=o =a p8c. 8. p8 =a;8G88 pe78cBopa
<e64y BoG:a<8 A 8 B (c<. p8c. 6) 8 e<:ocB8, ?p8coe48=e==o9 ?apa;;e;L-
=o9 e<y, Gepe7 48o4 D1 1y4eB ?poBe:aBL 7apO4=K9 Bo:, a Gepe7 48o4 D2 
Bo: pa7pO4a e<:ocB8.
B =e:8px3oDo2c:o9 Fe?8 (pa7pK2 <e64y BoG:a<8 A 8 B (p8c. 6)) c2eBo-
48o4K 4o;6=K 1y4yB 7a683aBLcO c8=xpo==o c y42oe==o9 GacBoBo9 (Gepe7
?o;o28=y ?ep8o4a) :a: ?pe4c:a7K2aeB Beop8O.
Ha;8G8e 2 28;:e A2pa<e=:o ?y;Lc8pyNIe3o Bo:a, o1yc;o2;e==o3o M;e:-
Bpo=a<8 ?po2o48<ocB8, 8 ?epe<e==o3o Bo:a, o1pa7o2a==o3o 1e7K=epF8-
a;L=K<8 7apO4a<8, ?p82o48B : :a6yIe<ycO =apyHe=8N 7a:o=a O<a 8
M=ep3eB8Gec:8x cooB=oHe=89 2 28;:e A2pa<e=:o.
14
6. 2e <o4e;8
cc;e4yO ?po1;e<K M;e:Bpo48=a<8:8, <K 8cxo48;8 87 c;e4yNI8x <aBe-
<aB8Gec:8 cBpo3o ycBa=o2;e==Kx ?o;o6e=89. Bo-?ep2Kx, 7a4aGa oH8
4;O ypa2=e=89 2 GacB=Kx ?po872o4=Kx =e 8<eeB e4u=cm2e==o3o peHe=uO.
Bo 2BopKx, :a;81po2:a ope=Fa =e M:282a;e=B=a :y;o=o2c:o9 :a;81po2-
:e, B.e. 3pa48e=B=aO 8=2ap8a=B=ocBL 2 o1Ie< c;yGae =e 8<eeB <ecBa.
A=a;87 ?po1;e< ?p82e; : c;e4yNI8< pe7y;LBaBa<.
1. �pa48e=B=aO 8=2ap8a=B=ocBL 8<eeB <ecBo Bo34a 8 Bo;L:o Bo34a, :o34a
Bo:8 8 7apO4K, 2xo4OI8e 2 ypa2=e=8O Ma:c2e;;a, y4o2;em2opONm 2o;-
=o2o<y ypa2=e=uN. Ta:8< o1pa7o<, ypa2=e=8O Ma:c2e;;a =e <o3ym
o?8cK2aBL :2a78cBaB8Gec:8e O2;e=8O M;e:Bpo<a3=eB87<a 8, cooB2eBcB-
2e==o, o=8 =e <o3yB o?8cK2aBL 87;yGe=8e 2o;=K u=epFua;L=K<u 7apO-
4a<u. =epF8a;L=Ke 7apO4K =e 87;yGaNB M;e:Bpo<a3=8B=Kx 2o;=.
2. apO4K, 2xo4OI8e 2 ypa2=e=8O Ma:c2e;;a, O2;ONBcO 1e7K=epFua;L=K-
<u. O4=a:o 8x =e;L7O pacc<ampu2amL :a: 2o;=y, ?oc:o;L:y, xoBO ux
<acca no:oO pa2=a =y;N, 8x M;e:Bpo<a3=8B=aO <acca oB;8G=a oB =y;O.
=K<8 c;o2a<8, o=8 8<eNB cmpy:mypy GacmuFK.
3. -;e:Bpo<a3=8B=aO 2o;=a Dop<8pyeBcO 8 87;yGaeBcO 1;a3o4apO 87<e=e-
=8N 2o 2pe<e=8 M;e:mpuGec:o3o no;O, co74a==o3o 1e7K=epF8a;L=K<8
7apO4a<8.
-B8 2K2o4K <o3yB c;y68BL oc=o2o9 4;O pa74e;e=8O O2;e=89 M;e:Bpo<a3-
=eB87<a =a 42e 3py??K.
ep2aO 3py??a, :oBopyN <o6=o =a72aBL :2a7ucmamuGec:o9 M;e:mpo4u-
=a<u:o9, 8<eeB 4e;o c :2a78cBaB8Gec:8<8 O2;e=8O<8.  =8< <o6=o oB=e-
cB8 M;e:BpocBaB8:y, <a3=8BocBaB8:y, :8px3oDo2c:yN Beop8N M;e:Bp8Ge-
c:8x Fe?e9 8 B.?. Teop8O :2a78cBaB8Gec:8x O2;e=89 pa7282a;acL =a<8 2
pa1oBe [1].
BBopaO 3py??a, :oBopyN <o6=o =a72aBL 2o;=o2o9 M;e:mpo4u=a<u:o9,
8<eeB 4e;o c =e:8px3oDo2c:8<8 Fe?O<8 (4;8==Ke ;8=88, 2o;=o2o4K 8
B.4.) 8 Beop8e9 87;yGe=8O M;e:Bpo<a3=8B=Kx 2o;=.
7;o6e==Ke pe7y;LBaBK ycBa=o2;e=K <aBe<aB8Gec:8 :oppe:B=o 8 =e co-
4ep6am 3unome7 u;u no;o6e=u9, :omopKe =e no4m2ep64e=K M:cnepu-
15
<e=ma;L=K<u pe7y;Lmama<u. epe94e< Be?epL : 38?oBe7e, o1JOc=ONIe9
?p8po4y 1e7K=epF8a;L=Kx 7apO4o2.
a: 872ecB=o, M;e:Bp8Gec:89 7apO4 ecBL c2o9cm2o, ?p8=a4;e6aIee =e:o-
Bopo<y <amepua;L=o<y =ocume;N. B cyIecB2yNI8x Beop8Ox =oc8Be;O<8
7apO4a (:a: c2o9cB2a <aBep8a;L=o3o o1Je:Ba) O2;ONBcO 8=epF8a;L=Ke
GacB8FK: M;e:Bpo=K, ?poBo=K, ?o78Bpo=K 8 B.4. <e==o 74ecL ?o;e7=o
1K;o 1K o1cy48BL c;e4yNI89 2o?poc, :a:o9 <aBep8a;L=K9 =oc8Be;L oB-
2eGaeB 7a 1e7K=epF8a;L=Ke 7apO4K? Ma:poc:o?8Gec:aO Beop8O =e ?o72o-
;OeB BoG=o cDop<y;8po2aBL oB2eB. 4ecL <K <o6e< Bo;L:o o1cy48BL 2o7-
<o6=Ke 2ap8a=BK 38?oBe7, :omopKe =y64aNmcO 2 4a;L=e9He9 npo2ep:e.
O4=a:o ?pe64e, Ge< o?8cK2aBL 2o7<o6=Ke 2ap8a=BK, <K 4o;6=K 2Kc:a-
7aBL c;e4yNIee ?pe4?o;o6e=8e. MK ?pe4?o;a3ae<, GBo ;N1aO 8=epF8-
a;L=aO GacB8Fa (o1;a4aeB o=a 7apO4o<, 8;8 =eB,  =e cBo;L 2a6=o) o:py-
6e=a =e:oBopo9 cy1cBa=F8e9 (�Hy1o9), ?o4o1=o e<;e, o:py6e==o9 aB-
<ocDepo9. <e==o MBa �Hy1a oB2eBcB2e==a 7a ?poO2;e=8e (po64e=8e 8
y=8GBo6e=8e) 1e7K=epF8a;L=Kx 7apO4o2 8 8x 4286e=8e co c:opocBLN
c2eBa.
B B2ep4o< Be;e :a64K9 aBo< o:py6e= MBo9 cy1cBa=F8e9. B ?po2o4=8:ax
cy1cBa=F88 oB4e;L=Kx 8o=o2 :p8cBa;;8Gec:o9 peHeB:8 c<K:aNBcO, o1-
pa7yO <e64y co1o9 <ocB8:8 8, 2 :o=eG=o< cGeBe, o1pa7yO ?pocBpa=cB2e=-
=yN peHeB:y. O=a O2;OeBcO =e Bo;L:o 8cBoG=8:o< 1e7K=epF8a;L=Kx 7a-
pO4o2, =o 8 co74aeB ?yB8 4;O 8x pac?pocBpa=e=8O. 4ecL 2o7<o6=K 42e
<o4e;8 MBo9 cy1cBa=F88.
o;e2aO <o4e;L. =epF8a;L=Ke GacB8FK o:py6aeB =e:aO cy1cBa=F8O,
B.e. ecBL o1pa7o2a=8e, o1;a4aNIee c2o9cB2a<8 ?o;O. N1oe M;e:Bpo<a3-
=8B=oe 2o74e9cB28e =a MBy cy1cBa=F8N 2K7K2aeB 2 =e9 2o7<yIe=8O. Bo7-
<yIe=8O ?poO2;ONBcO :a: Bo:8 1e7K=epF8a;L=Kx 7apO4o2, pac?pocBpa-
=ONI8ecO co c:opocBLN c2eBa ?o �<ocB8:a<.
8c:peB=aO <o4e;L. O:py6aNIaO 8=epF8a;L=Ke GacB8FK cy1cBa=F8O
?pe4cBa2;OeB co1o9 co2o:y?=ocBL =e9Bpa;L=Kx, ?o;o68Be;L=Kx 8 oBp8-
FaBe;L=Kx 1e7K=epF8a;L=Kx 7apO6e==Kx GacB8F, y4ep682ae<Kx =e:8<8
c8;a<8 21;878 8=epF8a;L=Kx GacB8F (=a?p8<ep, 8o=o2). 283aOcL ?o
<ocB8:a< o1Je<=o9 peHeB:8, 1e7K=epF8a;L=Ke 7apO4K co74aNB Bo:8.
C;e4yeB oB<eB8BL, GBo 2 MBo9 <o4e;8, GBo1K coxpa=8BL :o;8GecB2o 2eIe-
16
cB2a, <K 4o;6=K cG8BaBL, GBo oB Dpo=Ba 2o;=K o1paB=o : 8cBoG=8:y
4o;6=K 4283aBLcO =e9Bpa;L=Ke 1e7K=epF8a;L=Ke GacB8FK.
Ta:o2a ?pe42ap8Be;L=aO :apB8=a 42yx <o4e;e9 ?p8po4K 1e7K=epF8a;L-
=Kx 7apO4o2 8 Bo:o2.
a:;NGe=8e
4eo;o38Gec:89 By?8:, 2 :oBopK9 7a3=a;8 ca<8 ce1O 8cc;e4o2aBe;8, ?p8-
2e; 8x : :a=Bo2c:o<y a3=ocB8F87<y, :oBopK9 7a;o6e= 2 872ecB=o< Be78-
ce: O2;e=uO <u:po<upa =e <o3ym 1KmL o1JOc=e=K O7K:o< :;accuGec:ux
npe4cma2;e=u9. o3<aB87<, 3oc?o4cB2yNI89 2 co2pe<e==o9 D878:e, <e-
HaeB 2Kxo4y 87 MBo3o By?8:a [4].
Ho2aO 8=Bep?peBaF8O O2;e=89 :;acc8Gec:o9 M;e:Bpo48=a<8:8, o?8paN-
IaOcO =a cBpo389 <aBe<aB8Gec:89 a=a;87, =a 8c?pa2;e=8e cyIecB2yNI8x
<aBe<aB8Gec:8x, D878Gec:8x 8 3=oceo;o38Gec:8x oH81o:, ?o72o;OeB =a-
4eOBLcO, GBo :2a=Bo2aO <exa=8:a 8 :2a=Bo2aO M;e:Bpo48=a<8:a Bo6e 8c-
?KBaNB o?pe4e;e==Ke :o=Fe?Bya;L=Ke 87<e=e=8O, 8 O7K: :;accuGec:ux
meopu9 cBa=eB 8x O7K:o<.
�py??a �A=a;87 cBa28;a c2oe9 Fe;LN a=a;87 oH81o: 2 cyIecB2yNI8x
Beop8Ox 8 8c?pa2;e=8e 8x, GBo1K ?o43oBo28BL oc=o2K 4;O ?ocBpoe=8O
=o2o9 D878:8. Ta:aO oc=o2a Be?epL cyIecB2yeB. c?o;L7yO =aH8 c:po<-
=Ke 2o7<o6=ocB8, <K 1y4e< ?po4o;6aBL pacH8pOBL 8 y3;y1;OBL MBoB
?;aF4ap<.

C?8co: ;8BepaBypK
1. y;838= B.A., y;838=a �.A.. Mexa=8:a :2a78=e9Bpa;L=Kx c8cBe< 7apO6e==Kx Gac-
B8F 8 7a:o=K coxpa=e=8O =epe;OB828cBc:o9 M;e:Bpo48=a<8:8. Bopo=e6. y=-B, Bopo-
=e6,1986. e?. 2 BHT 09.04.86, !6451-B8.
2. ae2 H.E. C2epx?po2o4=8:8 8=6e=epa A2pa<e=:o. Tex=8:a  Mo;o4e68, !1, M.,
1991.
3. ae2 H.E., A2pa<e=:o C.B., 8c8= B.H. 7<epe=8O Bo:a ?po2o48<ocB8, 2o71y64ae<o-
3o ?o;Op87aF8o==K< Bo:o<. Pycc:aO D878Gec:aO <Kc;L, !2, PeyBo2o, Moc:o2c:o9
o1;., 1990.
17
4. y;838= B.A., y;838=a �.A.. op=e2a M.B., $878:a 8 D8;ocoD8O D878:8 / Bopo=e6.
y=-B, Bopo=e6, 2001. e?. 2 BHT 26.03.01, !729-B2001.
C<. Ba:6e:
1. y;838= B.A., y;838=a �.A.. op=e2a M.B. p878c pe;OB828cBc:8x Beop89, 'acBL 5
(-;e:Bpo<a3=8B=aO <acca) 8 'acBL 6 (Ma3=8B=Ke 27a8<o4e9cB28O 4286yI8xcO 7apO-
4o2). H8T, 2001.
2. y;838= B.A., y;838=a �.A., op=e2a M.B. $878:a 8 D8;ocoD8O D878:8. H8T, 2001.
3. ae2 H.E. O4=o?po2o4=aO - . oGe<y c?OB 7a:o=K? P !10/94.
aBa ?y1;8:aF88: -;e:Bpo==aO 2epc8O:
13 8N=O 2002 3o4a � �Hay:a 8 Tex=8:a, www.n-t.org
18