,

( Ȼ)

 

, , , .

 

1.

, . , .

, . , , .

, .

, , , .

-, , .

, , , .

 

1.1.

XIX-XX .

XIX - - .

- , .

- , , , .. . , - .

1881, 1886÷1887 . . " ". , , .

1889 . , V l´ :

l´ = l [1 (V2 / c2)]1/2 ( 1 )

: c - ,

l - .

1892 . . "" t´, "" t :

t´ = t [(x v) / c2] ( 2 )

: v - x.

. :

x1 = β ( x2 V t2 ) ( 3 )

y1 = y2 ( 4 )

z1 = z2 ( 5 )

t1 = β { t2 [(x V) / c2]} ( 6 )

"" β :

β = 1 / {[1 (V2 / c2)]1/2} ( 7 )

(3)÷(6) " ".

1881 . , , v, , , :

= / {[1 (v2 / c2)]1/2} ( 8 )

 

1.2.

1905 . , : - ( ), - .

: , .. () - , , .

: , .. .

, . , :

x1 = [x2 + (V t2)] / [1 (V2 / c2)]1/2 ( 9 )

x2 = [x1 (V t1)] / [1 (V2 / c2)]1/2 ( 10 )

y1 = y2 ( 11 )

z1 = z2 ( 12 )

: x1, y1, z1 t1 O1x1y1z1;

x2, y2, z2 t2 O2x2y2z2 ( . 1).

t1 = {t2 + [( V x2) / c2]} / [(1 V2/ c2)1/2] ( 13 )

t2 = {t1 [( V x1) / c2]}/ [(1 V2 / c2)1/2] ( 14 )

(9)÷(14), vx2, vy2 vz2 O2x2y2z2 vx1, vy1 vz1 O1x1y1z1 :

vx1 = (vx2 + V) / {1 + [(V vx2)/ c2)]} ( 15 )

vx2 = (vx1 V) / {1 [(V vx1)/ c2)]} ( 16 )

vy1 = {vy2 [1 (V2 / c2)]1/2} / {1 + [(V vx2)/ c2)]} ( 17 )

vy2 = {vy1 [1 (V2 / c2)]1/2} / {1 [(V vx1)/ c2)]} ( 18 )

vz1 = {vz2 [1 (V2 / c2)]1/2} / {1 + [(V vx2)/ c2)]} ( 19 )

vz2 = {vz1 [1 (V2 / c2)]1/2} / {1 [(V vx1)/ c2)]} ( 20 )

(V) , (V) (V) , V, :

(V) = / [1 (V2 / c2)]1/2 ( 21 )

(V) = ( V ) / [1 (V2 / c2)]1/2 ( 22 )

(V) = c2 {{1 / [1 + (V2 / c2)]1/2} 1} ( 23 )

: - .

, (. .), ( ).

 

2.

 

2.1. " "

, " ".

, , (.. ).

: .

(.. ).

, : O1x1y1z1 O2x2y2z2, . 1 :

-   O1x1y1z1 O2x2y2z2 ;

-   O2x2y2z2 O1x1y1z1 V2 Ox1;

-   (t1=0 t2=0) , O1 O2 .

( ), O1x1y1z1 O2x2y2z2 :

x1 = β1 ( x2 + V1 t2 ) ( 24 )

x2 = β2 ( x1 + V2 t1 ) ( 25 )

y1 = β3 y2 ( 26 )

y2 = β4 y1 ( 27 )

z1 = β5 z2 ( 28 )

z2 = β6 z1 ( 29 )

: x1, y1, z1 x2, y2, z2 O1x1y1z1 O2x2y2z2,;

t1 t2 - O1x1y1z1 O2x2y2z2, ;

β1, β2, β3, β4, β5 β6 - ;

V1 - O1x1y1z1 O2x2y2z2.

 

y1

 

 

t2

 

2

 

x1

 

x2

 

 

V

 

y2

 

 

 

 

12

 

x1

 

 

x2

 

V

 

y2

 

y1

 

 

t1

 

 

t1 = t2 = 0

 

 

 

1

 

. 1

 

 

:

V1 = V2 = V ( 30 )

β1 = β2 = β ( 31 )

β3 = β4 = 1 ( 32 )

β5 = β6 = 1 ( 33 )

(24)÷(29) :

x1 = β ( x2 + V t2 ) ( 34 )

x2 = β ( x1 V t1 ) ( 35 )

y1 = y2 ( 36 )

z1 = z2 ( 37 )

β x1, y1, z1, x2, y2, z2 t1 t2, V O1x1y1z1 O2x2y2z2 .

(34) (35) t1 t2 :

t1 = {[(β2 1) x2] / (β V)} + (β t2) ( 38 )

t2 = {[(1 β2 ) x1] / (β V)} + (β t1) ( 39 )

β (34) (35) :

- , β ;

-         β 1 V = 0 ( );

-         β 1, β V;

-   O1x1y1z1 O2x2y2z2 β 0 , β O1x1 O2x2;

-   β > 1 , , , , ;

-   0 < β < 1 , , , , ;

-         , β V β V (.. V β).

(24)÷(29) x1, y1 z1 t1 O1x1y1z1 x2, y2 z2 t2 O2x2y2z2.

(24)÷(39), vx2, vy2 vz2 O2x2y2z2 vx1, vy1 vz1 O1x1y1z1 :

vx1 = (vx2 + V) / {{[(β2 1) vx2] / (β2 V)} + 1} ( 40 )

vx2 = (vx1 V) / {{[(1 β2) vx1] / (β2 V)} + 1} ( 41 )

vy1 = vy2 / {{[(β2 1) vx2] / (β V)} + β} ( 42 )

vy2 = vy1 / {{[(1 β2) vx1] / (β V)} + β} ( 43 )

vz1 = vz2 / {{[(β2 1) vx2] / (β V)} + β} ( 44 )

vz2 = vz1 / {{[(1 β2) vx1] / (β V)} + β} ( 45 )

(40) , β > 1 V, vx1, vx2, , :

- vx2:

vx1 (vx2 + V) ( 46 )

- vx2:

vx1 (vx2 + V) ( 47 )

(46) (47) , β > 1 vx1 O1x1y1z1, vx2 O2x2y2z2.

(40) , 0 < β < 1 V, vx1, vx2, , :

- vx2:

vx1 (vx2 + V) ( 48 )

V0 : vx1 > vx2 ( 49 )

- vx2:

vx1 (vx2 + V) ( 50 )

(48)÷(50) , 0 < β < 1 vx1 O1x1y1z1, vx2 O2x2y2z2.

(38)÷(45) ax2, ay2 az2 O2x2y2z2 ax1, ay1 az1 O1x1y1z1 :

ax1 = (ax2 β-3) / {{[(β2 1) vx2] / (β2 V)} + 1}3 ( 51 )

ax2 = (ax1 β-3) / {{[(1 β2) vx1] / (β2 V)} + 1}3 ( 52 )

(ay2 {{[(β2 1) vx2] / (β V)} + β}) {[(β2 1) vy2 ax2] / (β V)}

ay1 = ( 53 )

{{[(β2 1) vx2] / (β V)} + β}3

(ay1 {{[(1 β2) vx1] / (β V)} + β}) {[(1 β2) vy1 ax1] / (β V)}

ay2 = ( 54 )

{{[(1 β2) vx1] / (β V)} + β}3

(az2 {{[(β2 1) vx2] / (β V)} + β}) {[(β2 1) vz2 ax2] / (β V)}

az1 = ( 55 )

{{[(β2 1) vx2] / (β V)} + β}3

(az1 {{[(1 β2) vx1] / (β V)} + β}) {[(1 β2) vz1 ax1] / (β V)}

az2 = ( 56 )

{{[(1 β2) vx1] / (β V)} + β}3

 

2.2.

, Vx vx1 O1x1y1z1, vx2 O2x2y2z2, Vx, .. :

vx1 = vx2 = Vx ( 57 )

(57) (40) (41), :

Vx2 = (β2 V2) / ( β2 1 ) ( 58 )

(58) Vx V β V:

Vx = (β V) / ( β2 1 )1/2 ( 59 )

, β β ≥ 1 , , Vx ( (46) (47)) :

Vx = vx1 = (β V) / ( β2 1 )1/2 ( 60 )

: vx1 - , .

, β 0 < β ≤ 1 , , Vx ( (48)÷(50), .. 0 < β ≤ 1) :

Vx = ί vx2 = β V) / (1 β2 )1/2 ( 61 )

: vx2 - , ,

ί :

ί =( 1 )1/2 ( 62 )

(58) β V V :

β2 = 1 / [1 (V2 / Vx2)] ( 63 )

(63) (60) β, β ≥ 1 β>, :

β> 2 = 1 / [1 (V2 / vx12)] ( 64 )

(63) (61) β, 0<β≤ 1 β< , :

β< 2 = 1 / [1 + (V2 / vx22)] ( 65 )

 

2.3.

: O1x1y1z1 O2x2y2z2 O3x3y3z3, . 2 :

-   O1x1y1z1 , O2x2y2z2 O3x3y3z3 ;

-   O2x2y2z2 O1x1y1z1 V2 Ox1 ;

-   O3x3y3z3 O1x1y1z1 V3 Ox1 ;

-   (t1=0 , t2=0 t3=0) , O1 , O2 O3 .

 

t3

 

V3

 

t2

 

t1

 

V2

 

 

y3

 

 

y2

 

y1

 

3

 

2

 

1

 

 

x3

 

x2

 

 

x1

 

t1 = t2 = t3 = 0

 

 

V3

 

V2

 

123

 

 

x3

 

 

x2

 

x1

 

y3

 

y2

 

 

y1

 

 

 

 

 

 

 

. 2

 

 

(41), V23 O3 O2:

V23 = (V3 V2) / {{[(1 β22) V3] / (β22 V2)} + 1} ( 66 )

V32 O2 O3:

V32 = (V2 V3) / {{[(1 β32 ) V2] / (β32 V3)} + 1} ( 67 )

: β2 β3 - , V2 V3, .

, O3 O2 , , O2 O3 , ..:

V32 = V23 ( 68 )

(68) (66) (67), :

{[(1 β22) V3] / (β22 V2)} + 1 = {[(1 β32) V2] / (β32 V3)} + 1 (69 )

β2 β3 :

β32 = (β22 V2) / [V32 (β22 V3) + (β22 V2)] ( 70 )

 

2.4. β

(69) :

(β22 1) / (β22 V22) = (β32 1 ) / (β32 V32) ( 71 )

β2 β3 , V2 V3, , V2 V3 ( ), , :

22 1) / 22 V22) = 32 1 ) / 32 V32) = = Const ( 72 )

.. , :

(β2 1) / (β2 V2) = = Const ( 73 )

: - , V (V2 V3) β (β2 β3) , .

(73), β :

- = 0 β 1,

- , β 1, .. β ≥ 1,

- , β 1, .. 0 < β ≤ 1.

V β, V , , .. V β β ≥ 1 0 < β ≤ 1.

, β ≥ 1 0 < β ≤ 1 β, .. β V β ≥ 1 0 < β ≤ 1.

, β V ( β V).

(73) β:

β 2 = 1 / [1 ( V2)] ( 74 )

(63):

β2 = 1 / [1 (V2 / Vx2)] ( 63 )

(74), , :

= 1 / Vx2 ( 75 )

.. Vx2 , V β.

(74) (75), , , β 1, Vx ( ) , .

(74), β:

- β ≥ 1:

β> 2 = 1 / [1 (V2 / vx12)] ( 64 )

- 0<β≤ 1:

β< 2 = 1 / [1 + (V2 / vx22)] ( 65 )

vx1 vx2 , V β, ..:

vx1 = Const ( 76 )

vx2 = Const ( 77 )

( V, 0, β 1) , V 0 β 1, , (64) (65), , :

vx1 0 ( 78 )

vx2 0 ( 79 )

, β V (.. β = Const = 1), :

vx1 = ( 80 )

vx2 = ( 81 )

 

2.5.

(63) (34), (35), (38)÷(39), (40)÷(45) (51)÷(56), :

x1 = [x2 + (V t2)] / [1 (V2 / Vx2)]1/2 (82 )

x2 = [x1 (V t1>)] / [1 (V2 / Vx2)]1/2 (83)

t1 = {t2 + [( V x2) / Vx2]} / [(1 V2/ Vx2)1/2] (84)

t2 = {t1 [( V x1) / Vx2 ]}/ [(1 V2/ Vx2)1/2] ( 85 )

vx1 = (vx2 + V) / {1 + [(V vx2)/ Vx2)]} ( 86 )

vx2 = (vx1 V) / {1 [(V vx1)/ Vx2)]} ( 87 )

vy1 = {vy2 [1 (V2 / Vx2)]1/2} / {1 + [(V vx2)/ Vx2)]} ( 88 )

vy2 = {vy1 [1 (V2 / Vx2)]1/2} / {1 [(V vx1)/ Vx2)]} ( 89 )

vz1 = {vz2 [1 (V2 / Vx2)]1/2} / {1 + [(V vx2)/ Vx2)]} (90)

vz2 = {vz1 [1 (V2 / Vx2)]1/2} / {1 [(V vx1)/ Vx2)]} (91)

ax1 = {ax2 [1 (V2 / Vx2)]3/2}/ {1 + [(V vx2)/ Vx2)]}3 (92)

ax2 = {ax1 [1 (V2 / Vx2)]3/2}/ {1 [(V vx1)/ Vx2)]}3 (93)

[{{1+[(Vvx2)/ Vx2)]}ay2}} [(Vvy2 ax2)/ Vx2]] [1 (V2/ Vx2)]

ay1 = (94)

{1 + [(V vx2)/ Vx2)]}3

[{{1[(Vvx1)/ Vx2)]}ay1}} +[(Vvy1 ax1)/ Vx2]] [1 (V2/ Vx2)]

ay2 = (95)

{1 [(V vx1)/ Vx2)]}3

[{{1+[(Vvx2)/ Vx2)]}az2}} [(Vvz2 ax2)/ Vx2]] [1 (V2/v Vx2)]

az1 = (96)

{1 + [(V vx2)/ Vx2)]}3

[{{1[(Vvx1)/ Vx2)]}az1}} +[(Vvz1 ax1)/ Vx2]] [1 (V2/ Vx2)]

az2 = (97)

{1 [(V vx1)/ Vx2)]}3

 

2.6. β ≥ 1

(64) (34), (35), (38)÷(39), (40)÷(45) (51)÷(56), β= β>:

x1> = [x2> + (V t2>)] / [1 (V2 / vx12)]1/2 ( 98 )

x2> = [x1> (V t1>)] / [1 (V2 / vx12)]1/2 ( 99 )

t1> = {t2> + [( V x2>) / vx12]} / [(1 V2/vx12)1/2] ( 100 )

t2> = {t1> [( V x1>) / vx12 ]}/ [(1 V2/vx12)1/2] ( 101)

vx1> = (vx2> + V) / {1 + [(V vx2>)/ vx12)]} ( 102 )

vx2> = (vx1> V) / {1 [(V vx1>)/ vx12)]} ( 103 )

vy1> = {vy2> [1 (V2 / vx12)]1/2} / {1 + [(V vx2>)/ vx12)]} ( 104 )

vy2> = {vy1> [1 (V2 / vx12)]1/2} / {1 [(V vx1>)/ vx12)]} ( 105 )

vz1> = {vz2> [1 (V2 / vx12)]1/2} / {1 + [(V vx2>)/ vx12)]} ( 106 )

vz2> = {vz1> [1 (V2 / vx12)]1/2} / {1 [(V vx1>)/ vx12)]} ( 107 )

ax1> = {ax2> [1 (V2 / vx12)]3/2}/ {1 + [(V vx2>)/ vx12)]}3 ( 108 )

ax2> = {ax1> [1 (V2 / vx12)]3/2}/ {1 [(V vx1>)/ vx12)]}3 ( 109 )

 

[{{1+[(Vvx2>)/vx12)]}ay2>}} [(Vvy2> ax2>)/vx12]] [1 (V2/vx12)]

ay1> = (110)

{1 + [(V vx2>)/ vx12)]}3

[{{1[(Vvx1>)/vx12)]}ay1>}} +[(Vvy1> ax1>)/vx12]] [1 (V2/vx12)]

ay2> = (111)

{1 [(V vx1>)/ vx12)]}3

[{{1+[(Vvx2>)/vx12)]}az2>}} [(Vvz2> ax2>)/vx12]] [1 (V2/vx12)]

az1> = (112)

{1 + [(V vx2>)/ vx12)]}3

[{{1[(Vvx1>)/vx12)]}az1>}} +[(Vvz1> ax1>)/vx12]] [1 (V2/vx12)]

az2> = (113)

{1 [(V vx1>)/ vx12)]}3

 

2.7. 0 < β < 1

(65) (34), (35), (38)÷(39), (40)÷(45) (51)÷(56), , β = β<:

x1< = [x2< + (V t2<)] / [1 + (V2 / vx22)]1/2 ( 114 )

x2< = [x1< (V t1<)] / [1 + (V2 / vx22)]1/2 ( 115 )

t1< = {t2< [( V x2<) / vx22]} / [(1 + V2/vx22)1/2] ( 116 )

t2< = {t1< + [( V x1<) / vx22 ]}/ [(1 + V2/vx22)1/2] ( 117 )

vx1< = (vx2< + V) / {1 [(V vx2<)/ vx22)]} ( 118 )

vx2< = (vx1< V) / {1 + [(V vx1<)/ vx22)]} ( 119 )

vy1< = {vy2< [1 + (V2 / vx22)]1/2} / {1 [(V vx2<)/ vx22)]} ( 120 )

vy2< = {vy1< [1 + (V2 / vx22)]1/2} / {1 + [(V vx1<)/ vx22)]} ( 121 )

vz1< = {vz2< [1 + (V2 / vx22)]1/2} / {1 [(V vx2<)/ vx22)]} ( 122 )

vz2< = {vz1< [1 + (V2 / vx22)]1/2} / {1 + [(V vx1<)/ vx22)]} ( 123 )

ax1< = {ax2< [1 + (V2 / vx22)]3/2}/ {1 [(V vx2<)/ vx22)]}3 ( 124 )

ax2< = {ax1< [1 + (V2 / vx22)]3/2}/ {1 + [(V vx1<)/ vx22)]}3 ( 125 )

[{{1[(Vvx2<)/vx22)]}ay2<}}+[(Vvy2< ax2<)/vx22]] [1 + (V2/vx22)]

ay1< = (126)

{1 [(V vx2<)/ vx22)]}3

[{{1+[(Vvx1<)/vx22)]}ay1<}}[(Vvy1< ax1<)/vx22]] [1 + (V2/vx22)]

ay2< = (127)

{1 + [(V vx1<)/ vx22)]}3

[{{1[(Vvx2<)/vx22)]}az2<}}+[(Vvz2< ax2<)/vx22]] [1 + (V2/vx22)]

az1< = (128)

{1 [(V vx2<)/ vx22)]}3

[{{1+[(Vvx1<)/vx22)]}az1<}}[(Vvz1< ax1<)/vx22]] [1 + (V2/vx22)]

az2< = (129)

{1 + [(V vx1<)/ vx22)]}3

 

, Vx (vx1 vx2) , .

 

3.

 

- - , , , .. , , .

- , .

, .. .

.

, :

- ( , );

- ( , , .. );

- ( , , .. ).

 

3.1.

:

-   : ( ) ;

-   : ( , ) , : , , , , - , , , ( , , , ): .

, .

, (V) , V, :

(V) = f (V) ( 130 )

: ;

f(V) , V.

(130), (V) , V, :

(V) = f (V) V ( 131 )

(131) (V) , V:

v

(V) = ∫ {[ f (V) V ] + [ f′ (V) V2]} dV ( 132 )

0

: f′ (V) f (V) .

( f (V)) , ( ) ( ), .

, f (V), .

 

3.1.1. 1

, , , . 1, - O1x1y1z1 O2x2y2z2 , V O1x1 O1x1y1z1.

, , 1 2 ( . 3) , 1 2 .

 

t2 > t2c

 

 

v22

 

2

 

V

 

x1

 

x2

 

y2

 

y1

 

2

 

1

 

t2 < t2c

 

v22

 

v21

 

 

2

 

1

 

V

 

 

2

 

1

 

y2

 

x2

 

x1

 

y1

 

1

 

v21

 

 

 

 

. 3

 

 

 

 

O2x2y2z2 1 2 t2 O2x2 v21x v22x .

- t2 1 2 .

t2 1 2 O2x2 v21x v22x .

, 1 2 , , 1 2 O2x2y2z2 , t2 :

[1 f (V= v21x) v21x] + [2 f (V= v22x) v22x] = [1 f (V= v21x) v21x] + [2 f (V= v22x) v22x] ( 133 )

, 1 2 , 1 2 O2x2y2z2 , t2, , 1 2 :

v21x v22x

{ 1 ∫ {[f (V) V] + [f′ (V) V2]}dV} + { 2 ∫ {[f (V) V] + [f′ (V) V2]}dV} =

0 0

v21x v22x

{ 1 ∫ {[f (V)V] +[f′ (V)V2]}dV} + { 2 ∫ {[f (V)V] +[f′ (V)V2]}dV} (134)

0 0

1 2 O2x2y2z2 1 2 O1x1y1z1, , :

- 1 2 t1, t2 O2x2y2z2,

- 1 v11x v11x , v21x v21x,

- 2 v12x v12x , v22x v22x.

(133) (134) 1 2 O1x1y1z1 , t1, , 1 2 :

[1 f (V= v11x) v11x] + [2 f (V= v12x) v12x] = [1 f (V= v11x) v11x] + [2 f (V= v12x) v12x] ( 135 )

v11x v12x

{ 1 ∫ {[f (V) V] + [f′ (V) V2]}dV} + { 2 ∫ {[f (V) V] + [f′ (V) V2]}dV} =

0 0

v11x v12x

{ 1 ∫ {[f (V)V] +[f′ (V)V2]}dV} + { 2 ∫ {[f (V)V] +[f′ (V)V2]}dV} (136)

0 0

 

3.1.2. 2

2 1 , O2x2y2z2 1 2 O2x2, O2y2, . 4.

O2x2y2z2 1 2 t2 O2y2 v21y v22y .

 

 

 

 

 

1

 
 

 

 

: 2

y1

 

. 4

 
 

 


t2 1 2 O2y2 v21y v22y .

1 2 O2x2y2z2 , t2, , 1 2 :

[1 f (V= v21y) v21y] + [2 f (V= v22y) v22y] = [1 f (V= v21y) v21y] + [2 f (V= v22y) v22y] ( 137 )

v21y v22y

{ 1 ∫ {[f (V) V] + [f′ (V) V2]}dV} + { 2 ∫ {[f (V) V] + [f′ (V) V2]}dV} =

0 0

v21y v22y

{ 1 ∫ {[f (V)V] +[f′ (V)V2]}dV} + { 2 ∫ {[f (V)V] +[f′ (V)V2]}dV} (138)

0 0

( O1x1 O1y1) 1 2 O1x1y1z1 , t1, , 1 2 :

{1 f [V = (v11y2+ V2)1/2] V } + {2 f [V = (v12y2+ V2)1/2] V } =

{1 f [V = (v11y2+ V2)1/2] V } + {2 f [V = (v12y2+ V2)1/2] V } ( 139 )

{1 f [V = (v11y2+ V2)1/2] v11y} + {2 f [V = (v12y2+ V2)1/2] v12y} =

{1 f [V = (v11y2+ V2)1/2] v11y} + {2 f [V = (v12y2+ V2)1/2] v12y} ( 140 )

(v11y2+ V2)1/2 (v12y2+ V2)1/2

{ 1 ∫ {[f (V) V] + [f′ (V) V2]}dV} + { 2 ∫ {[f (V) V] + [f′ (V) V2]}dV} =

0 0

(v11y2+ V2)1/2 (v12y2+ V2)1/2

{ 1 ∫{[f (V)V] +[f′ (V)V2]}dV} + { 2 ∫{[f (V)V] +[f′ (V)V2]}dV} ( 141)

0 0

 

3.1.3.

:

[1 f (V= v21x) v21x] + [2 f (V= v22x) v22x] = [1 f (V= v21x) v21x] + [2 f (V= v22x) v22x] ( 133 )

v21x v22x

{ 1 ∫ {[f (V) V] + [f′ (V) V2]}dV} + { 2 ∫ {[f (V) V] + [f′ (V) V2]}dV} =

0 0

v21x v22x

{ 1 ∫{[f (V)V] +[f′ (V)V2]}dV} + { 2 ∫{[f (V)V] +[f′ (V)V2]}dV} ( 134)

0 0

[1 f (V= v11x) v11x] + [2 f (V= v12x) v12x] = [1 f (V= v11x) v11x] + [2 f (V= v12x) v12x] ( 135 )

v11x v12x

{ 1 ∫ {[f (V) V] + [f′ (V) V2]}dV} + { 2 ∫ {[f (V) V] + [f′ (V) V2]}dV} =

0 0

v11x v12x

{ 1 ∫{[f (V)V] +[f′ (V)V2]}dV} + { 2 ∫{[f (V)V] +[f′ (V)V2]}dV} ( 136)

0 0

[1 f (V= v21y) v21y] + [2 f (V= v22y) v22y] = [1 f (V= v21y) v21y] + [2 f (V= v22y) v22y] ( 137 )

v21y v22y

{ 1 ∫ {[f (V) V] + [f′ (V) V2]}dV} + { 2 ∫ {[f (V) V] + [f′ (V) V2]}dV} =

0 0

v21y v22y

{ 1 ∫{[f (V)V] +[f′ (V)V2]}dV} + { 2 ∫{[f (V)V] +[f′ (V)V2]}dV} (138)

0 0

{1 f [V = (v11y2+ V2)1/2] V } + {2 f [V = (v12y2+ V2)1/2] V } =

{1 f [V = (v11y2+ V2)1/2] V } + {2 f [V = (v12y2+ V2)1/2] V } ( 139 )

{1 f [V = (v11y2+ V2)1/2] v11y} + {2 f [V = (v12y2+ V2)1/2] v12y} =

{1 f [V = (v11y2+ V2)1/2] v11y} + {2 f [V = (v12y2+ V2)1/2] v12y} ( 140 )

(v11y2+ V2)1/2 (v12y2+ V2)1/2

{ 1 ∫ {[f (V) V] + [f′ (V) V2]}dV} + { 2 ∫ {[f (V) V] + [f′ (V) V2]}dV} =

0 0

(v11y2+ V2)1/2 (v12y2+ V2)1/2

{ 1 ∫{[f (V)V] +[f′ (V)V2]}dV} + { 2 ∫{[f (V)V] +[f′ (V)V2]}dV} ( 141)

0 0

1 2 O2x2y2z2 O1x1y1z1 , (86) (88):

v11x = (v21x + V ) / {1 + [(V v21x) / Vx2)]} ( 142 )

v12x = (v22x + V ) / {1 + [(V v22x) / Vx2)]} ( 143 )

v11x = (v21x + V ) / {1 + [(V v21x) / Vx2)]} ( 144 )

v12x = (v22x + V ) / {1 + [(V v22x) / Vx2)]} ( 145 )

v11y = v21y [1 (V2 / Vx2)]1/2 ( 146 )

v12y = v22y [1 (V2 / Vx2)]1/2 ( 147 )

v11y = v21y [1 (V2 / Vx2)]1/2 ( 148 )

v12y = v22y [1 (V2 / Vx2)]1/2 ( 149 )

, 17 , 12 .

f(V), 17 , :

f (V) = 1 / [1 (V2 / Vx2)]1/2 ( 150 )

, (130)÷(132), (V), (V) (V) V:

(V) = / [1 (V2 / V2)]1/2 ( 151 )

(V) = ( V ) / [1 (V2 / Vx2)]1/2 ( 152 )

(V) = vx12 {{1 / [1 (V2 / Vx2)]1/2} 1} ( 153 )

 

3.1.4. β>1

, β β>1 , (150)÷(153) (60) f(V)>, (V)>, (V)>, (V)> V, :

f (V)> = 1 / [1 (V2 / vx12)]1/2 ( 154 )

(V)> = / [1 (V2 / vx12)]1/2 ( 155 )

(V)> = ( V ) / [1 (V2 / vx12)]1/2 ( 156 )

(V)> = vx12 {{1 / [1 (V2 / vx12)]1/2} 1} ( 157 )

 

3.1.4.1. (150) β>1

( 1 2)

(133)÷(141) (60) (154)÷(157):

{(1 v21x) / [1 (v21x2 / vx12)]1/2} + {(2 v22x) / [1 (v22x2 / vx12)]1/2} = {(1 v21x) / [1 (v21x2/vx12)]1/2} + {(2 v22x) / [1 (v22x2/vx12) ]1/2} (158)

{1 / [1 (v21x2 / vx12)]1/2} + {2 / [1 (v22x2 / vx12)]1/2} = {1 / [1 (v21x2 / vx12)]1/2} + {2 / [1 (v22x2 / vx12)]1/2} ( 159 )

{(1 v11x) / [1 (v11x2 / vx12)]1/2} + {(2 v12x) / [1 (v12x2 / vx12)]1/2} = {(1 v11x)/[1 (v21x2/vx12)]1/2} + {(2 v12x)/[1 (v12x2/vx12)]1/2} ( 160 )

{1 / [1 (v11x2 / vx12)]1/2} + {2 / [1 (v12x2 / vx12)]1/2} = {1 / [1 (v21x2 / vx12)]1/2} + {2 / [1 (v12x2 / vx12)]1/2} ( 161 )

{(1 v21y) / [1 (v21y2 / vx12)]1/2} + {(2 v22y) / [1 (v22y2 / vx12)]1/2} = {(1 v21y) / [1 (v21y2/vx12)]1/2} + {(2 v22y) / [1 (v22y2/vx12)]1/2} (162)

{1 / [1 (v21y2 / vx12)]1/2} + {2 / [1 (v22y2 / vx12)]1/2} = {1 / [1 (v21y2 / vx12)]1/2} + {2 / [1 (v22y2 / vx12)]1/2} ( 163 )

{(1 V)/{1 [(v11y2+V2)/vx12]}1/2} + {(2 V)/{1 [(v12y2+V2)/vx12]}1/2} = {(1V)/{1 [(v11y2+V2)/vx12]}1/2}+ {(2V)/{1 [(v12y2+V2)/vx12]}1/2} (164)

{(1v11y)/{1[(v11y2+V2)/vx12]}1/2}+{(2v12y)/{1 [(v12y2+V2)/vx12]}1/2} =

{(1v11y)/{1[(v11y2+V2)/vx12]}1/2}+{(2v12y)/{1[(v12y2+V2)/vx12]}1/2}(165)

{1 /{1 [(v11y2 + V2 )/ vx12]}1/2} + {2 /{1 [(v12y2+ V2)/ vx12]}1/2} =

{1 / {1 [(v11y2+V2)/ vx12]}1/2} + {2 /{1 [ (v12y2+V2)/ vx12]}1/2} ( 166 )

, (60) (142)÷(149):

v11x = (v21x + V ) / {1 + [(V v21x) / vx12)]} ( 167 )

v12x = (v22x + V ) / {1 + [(V v22x) / vx12)]} ( 168 )

v11x = (v21x + V ) / {1 + [(V v21x) / vx12)]} ( 169 )

v12x = (v22x + V ) / {1 + [(V v22x) / vx12)]} ( 170 )

v11y = v21y [1 (V2 / vx12)]1/2 ( 171 )

v12y = v22y [1 (V2 / vx12)]1/2 ( 172 )

v11y = v21y [1 (V2 / vx12)]1/2 ( 173 )

v12y = v22y [1 (V2 / vx12)]1/2 ( 174 )

, 1 = 1 , 2 = 0,5 , V / vx1 = 0,5 , v21x / vx1 = = v21y / vx1 = 0,9 , v22x / vx 1 = v22y / vx1 = 0,6 .

1:

I. O2x2y2z2 :

1)      1 :

) v21x / vx1 = 0,9 , 21 = 2,294157338706, 21 / vx1 = 2,064741604835, 21 / vx12 = 1,294157338706;

) v21x / vx1 = 0,7360143377, 21 = 1,477179174242, 21 / vx1 = 1,087225051595, 21 / vx12 = 0,477179174242;

2)      2 :

) v22x / vx1 = 0,6 , 22 = 0,625 , 22 / vx1 = 0,375, 22 / vx12 = 0,125;

) v22x / vx1 = 0,937959108239, 22 = 1,441978164463, 22 / vx1 = 1,35251655324, 22 / vx12 = 0,941978164463;

3)      1 2 :

) (21 + 22) = 2,919157338706 , (21 + 22) / vx1 = 2,439741604835, (21 + 22) /vx12 = 1,419157338706;

) (21 + 22) = 2,919157338706 , (21 + 22) / vx1 = 2,439741604835, (21 + 22) / vx12 = 1,419157338706;

II. O1x1y1z1 :

1)      1 :

) v11x / vx1 = 0,965517241379 , 11 = 3,841143835489, 11 / vx1 = 3,708690599782, 11 / vx12 = 2,841143835489;

) v11x / vx1 = 0,903514517939, 11 = 2,333409263988, 11 / vx1 = 2,108269146306, 11 / vx12 = 1,333409263988;

2)      2 :

) v12x / vx1 = 0,846153846154 , 12 = 0,938194187433 , 12 / vx1 = 0,793856620136, 12 / vx12 = 0,438194187433;

) v12x / vx1= 0,978882996844, 12 = 2,445928758933, 12 / vx1 = 2,394278073612, 12 / vx12 = 1,945928758933;

3)      1 2 :

) (11 + 12) = 4,779338022922 , (11 + 12) / vx1 = 4,502547219918, (11 + 12) / vx12= 3,279338022922 ;

) (11 + 12) = 4,779338022922 , (11 + 12) / vx1 = 4,502547219918, (11 + 12) / vx12 = 3,279338022922.

2 :

I. O2x2y2z2 :

1)      1 :

) v21y / vx1= 0,9 , 21 = 2,294157338706, 21 / vx1 = 2,064741604835, 21 / vx12 = 1,294157338706;

) v21y / vx1 = 0,7360143377, 21 = 1,477179174242, 21 / vx1 = 1,087225051595, 21 / vx12 = 0,477179174242;

2)      2 :

) v22y / vx1 = 0,6 , 22 = 0,625 , 22 / vx1 = 0,375, 22/ vx12 = 0,125;

) v22y / vx1= 0,937959108239, 22 = 1,441978164463, 22 / vx1 = 1,35251655324, 22 / vx12 = 0,941978164463;

3)      1 2 :

) (21 + 22) = 2,919157338706 , (21 + 22) / vx1 = 2,439741604835, (21 + 22) / vx12= 1,419157338706;

) (21 + 22) = 2,919157338706 , (21 + 22) / vx1 = 2,439741604835, (21 + 22) / vx12 = 1,419157338706;

II. O1x1y1z1 :

1)      1 :

) v11x / vx1 = 0,5 v11y / vx1= 0,779422863406 , 11 = 2,64906471413 , 11x / vx1 = 1,324532357065 11y / vx1 = 2,064741604835, 11 / vx12 = 1,64906471413;

) v11x / vx1 = 0,5 , v11y / vx1 = 0,637407113998 , 11 = 1,70569958778 , 11x / vx1 = 0,85284979389 , 11y / vx1 = 1,087225051595 , 11 / vx12 = 0,70569958778;

2)      2 :

) v12x / vx1 = 0,5 , v12y / vx1 = 0,519615242271 , 12 = 0,721687836487 , 12x / vx1 = 0,360843918244 , 12y /vx1 = 0,375 , 12 / vx12 = 0,221687836487;

) v12x / vx1= 0,5, v12y / vx1= 0,812296415446, 12 = 1,665052962837, 12x /vx1 = 0,832526481418 , 12y /vx1 = 1,35251655324 , 12/ vx12 = 1,165052962837;

3)      1 2 :

) (11 + 12) = 3,370752550617 , (11x + 12x) / vx1 = 1,685376275309 (11y + 12y) / vx1 = 2,439741604835 , (11 + 12) / vx12 = 1,870752550617;

) (11 + 12) = 3,370752550617 , (11x + 12x) / vx1 = 1,685376275309 (11y + 12y) / vx1 = 2,439741604835 , (11 + 12) / vx12 = 1,870752550617.

: 1 2 O2x2y2z2 O1x1y1z1 , 1 2 .

, (150)÷(153) , β > 1, (133)÷(141).

 

3.1.5. 0 < β < 1

, β 0 < β < 1 , , (150)÷(153) (61) f(V)<, (V)<, (V)<, (V)< V, :

f (V)< = 1 / [1 + (V2 / vx22)]1/2 ( 175 )

(V)< = / [1 + (V2 / vx22)]1/2 ( 176 )

(V)< = ( V ) / [1 + (V2 / vx22)]1/2 ( 177 )

(V)< = vx22 { 1 {1 / [1 + (V2 / vx22)]1/2}} ( 178 )

 

3.1.5.1. (150) 0 < β < 1

( 1 2)

(133)÷(141) (61) (175)÷(178):

{(1 v21x) / [1 + (v21x2 / vx22)]1/2} + {(2 v22x) / [1 + (v22x2 / vx22)]1/2} = {(1 v21x)/[1 + (v21x2/vx22)]1/2} + {(2 v22x)/[1 + (v22x2/vx22) ]1/2} ( 179 )

{1 / [1 + (v21x2 / vx22)]1/2} + {2 / [1 + (v22x2 / vx22)]1/2} = {1 / [1 + (v21x2 / vx22)]1/2} + {2 / [1 + (v22x2 / vx22)]1/2} ( 180 )

{(1 v11x) / [1 + (v11x2 / vx22)]1/2} + {(2 v12x) / [1 + (v12x2 / vx22)]1/2} = {(1 v11x)/[1 + (v21x2/vx22)]1/2} + {(2 v12x)/[1 + (v12x2/vx22)]1/2} ( 181 )

{1 / [1 + (v11x2 / vx22)]1/2} + {2 / [1 + (v12x2 / vx22)]1/2} = {1 / [1 + (v21x2 / vx22)]1/2} + {2 / [1 + (v12x2 / vx22)]1/2} ( 182 )

{(1 v21y) / [1 + (v21y2 / vx22)]1/2} + {(2 v22y) / [1 + (v22y2 / vx22)]1/2} = {(1 v21y) / [1 + (v21y2/vx22)]1/2} + {(2 v22y) / [1 + (v22y2/vx22)]1/2} (183)

{1 / [1 + (v21y2 / vx22)]1/2} + {2 / [1 + (v22y2 / vx22)]1/2} = {1 / [1 + (v21y2 / vx22)]1/2} + {2 / [1 + (v22y2 / vx22)]1/2} ( 184 )

{(1 V)/{1 + [(v11y2+V2)/vx22]}1/2} + {(2 V)/{1 + [(v12y2+V2)/vx22]}1/2} = {(1V)/{1 + [(v11y2+V2)/vx22]}1/2}+ {(2V)/{1 + [(v12y2+V2)/vx22]}1/2} (185)

{(1v11y)/{1+[(v11y2+V2)/vx22]}1/2}+{(2v12y)/{1+ [(v12y2+V2)/vx22]}1/2} =

{(1v11y)/{1+[(v11y2+V2)/vx22]}1/2}+{(2v12y)/{1+[(v12y2+V2)/vx22]}1/2}(186)

{1 /{1 + [(v11y2 + V2 )/ vx22]}1/2} + {2 /{1 + [(v12y2+ V2)/ vx22]}1/2} =

{1 / {1 + [(v11y2+V2)/ vx22]}1/2} + {2 /{1 + [ (v12y2+V2)/ vx22]}1/2} ( 187 )

, (61) (142)÷(149):

v11x = (v21x + V ) / {1 [(V v21x) / vx22)]} ( 188 )

v12x = (v22x + V ) / {1 [(V v22x) / vx22)]} ( 189 )

v11x = (v21x + V ) / {1 [(V v21x) / vx22)]} ( 190 )

v12x = (v22x + V ) / {1 [(V v22x) / vx22)]} ( 191 )

v11y = v21y [1 + (V2 / vx22)]1/2 ( 192 )

v12y = v22y [1 + (V2 / vx22)]1/2 ( 193 )

v11y = v21y [1 + (V2 / vx22)]1/2 ( 194 )

v12y = v22y [1 + (V2 / vx22)]1/2 ( 195 )

, 1 = 1 , 2 = 0,5 , V / vx2 = 0,5 , v21x / vx2 = = v21y / vx2 = 0,9 , v22x / vx2 = v22y / vx2 = 0,6 .

1:

I. O2x2y2z2 :

1) 1 :

) v21x / vx2= 0,9 , 21 = 0,743294146247 , 21 / vx2 = 0,668964731622 , 21 / vx22 = 0,256705853753;

) v21x / vx2 = 0,691099932748, 21 = 0,822656908881, 21 / vx2 = 0,568538134403, 21 / vx22 = 0,177343091119;

2)      2 :

) v22x / vx2 = 0,6 , 22 = 0,428746462856 , 22 / vx2 = 0,257247877714, 22 / vx22 = 0,071253537144;

) v22x / vx2= 1,023729712365, 22 = 0,349383700222 , 22 / vx2 = 0,357674474934 , 22 / vx22 = 0,150616299778;

3)      1 2 :

) (21 + 22) = 1,172040609103 , (21 + 22) / vx2 = 0,926212609336 , (21 + 22) / vx22 = 0,327959390897;

) (21 + 22) = 1,172040609103 , (21 + 22) / vx2 = 0,926212609336 , (21 + 22) / vx22 = 0,327959390897;

II. O1x1y1z1 :

1)      1 :

) v11x / vx2= 2,545454545455 , 11 = 0,365652372423 , 11 / vx2 = 0,93075149344 , 11 / vx22 = 0,634347627577;

) v11x / vx2 = 1,820001331727 , 11 =0,481548724902, 11 /vx2 = 0,876419320614 , 11 / vx22 = 0,518451275098;

2)      2 :

) v12x / vx2 = 1,571428571429 , 12 = 0,268437746097 , 12 / vx2 = 0,421830743866 , 12 / vx22 = 0,231562253903;

) v12x / vx2= 3,121532492927 , 12 = 0,152541393617, 12 / vx2 = 0,476162916693 , 12 / vx22 = 0,347458606383;

3)      1 2 :

) (11 + 12) = 0,63409011852 , (11 + 12) / vx2 = 1,352582237306 , (11 + 12) / vx22 = 0,86590988148;

) (11 + 12) = 0,63409011852 , (11 + 12) / vx2 = 1,352582237306 , (11 + 12) / vx22 = 0,86590988148.

2 :

I. O2x2y2z2 :

1)      1 :

) v21y / vx2 = 0,9 , 21 = 0,743294146247, 21 / vx2 = 0,668964731622 , 21 / vx22 = 0,256705853753;

) v21y / vx2 = 0,691099932748, 21 = 0,822656908881, 21 / vx2 = 0,568538134403, 21 / vx22 = 0,177343091119;

2)      2 :

) v22y / vx2 = 0,6 , 22 = 0,428746462856 , 22 / vx2 = 0,257247877714, 22 / vx22 = 0,071253537144;

) v22y / vx2= 1,023729712365, 22 = 0,349383700222 , 22 / vx2 = 0,357674474934 , 22 / vx22 = 0,150616299778 ;

3)      1 2 :

) (21 + 22) = 1,172040609103 , (21 + 22) / vx2 = 0,926212609336 , (21 + 22) / vx22 = 0,327959390897;

) (21 + 22) = 1,172040609103, (21+ 22) / vx2 = 0,926212609336 , (21 + 22) / vx22 = 0,327959390897;

II. O1x1y1z1 :

1)      1 :

) v11x / vx2 = 0,5 v11y / vx2 = 1,006230589875 , 11 = 0,664822495315 , 11x / vx2 = 0,332411247657 11y / vx2 = 0,668964731622, 11 / vx22 = 0,335177504685;

) v11x / vx 2 = 0,5 v11y / vx 2 = 0,772673214435 , 11 = 0,735806708167 , 11x / vx2 = 0,367903354084 , 11y / vx2 = 0,568538134403 , 11 / vx22 = 0,264193291833;

2)      2 :

) v12x / vx2 = 0,5 , v12y / vx2 = 0,67082039325 , 12 = 0,383482494424 , 12x / vx2 = 0,191741247212 , 12y / vx2 = 0,257247877714 , 12 / vx22 = 0,116517505576;

) v12x / vx2= 0,5, v12y / vx2= 1,144564613718, 12 = 0,312498281571 , 12x / vx2 = 0,156249140785 , 12y / vx2 = 0,357674474934 , 12 / vx22 = 0,187501718429;

3)      1 2 :

) (11 + 12) = 1,048304989738 , (11x + 12x) / vx2 = 0,524152494869 (11y + 12y) / vx2 = 0,926212609336 , (11 + 12) / vx22 = 0,451695010262;

) (11 + 12) = 1,048304989738 , (11x + 12x) / vx2 = 0,524152494869 (11y + 12y) / vx2 = 0,926212609336 , (11 + 12) / vx22 = 0,451695010262.

: 1 2 O2x2y2z2 O1x1y1z1 , 1 2 .

, (175)÷(178) , 0 < β < 1 , (133)÷(141).

 

3.1.6. (155)÷(157) (176)÷(178).

(155)÷(157):

(V)> = / [1 (V2 / vx12)]1/2 ( 155 )

(V)> = ( V ) / [1 (V2 / vx12)]1/2 ( 156 )

(V)> = vx12 {{1 / [1 (V2 / vx12)]1/2} 1} ( 157 )

(V)> , (V)> (V)> V , β > 1, :

- V, vx1 :

(V)> = , (V)> = V , (V)> = ( V2) /2 ;

- V = vx1 : (V)> = ∞ , (V)> = ∞ , (V)> = ∞ ;

- V < vx1 : (V)> , (V)> (V)> - ;

- V > vx1 : (V)> , (V)> (V)> - .

(176)÷(178):

(V)< = / [1 + (V2 / vx22)]1/2 ( 176 )

(V)< = ( V ) / [1 + (V2 / vx22)]1/2 ( 177 )

(V)< = vx22 { 1 {1 / [1 + (V2 / vx22)]1/2}} ( 178 )

(V)<, (V)< (V)< V , 0 < β < 1 , :

- V, vx2 :

(V)< = , (V)< = V, (V)< = ( V2) /2;

- V = vx2 : (V)< = (2)-1/2 , (V)< = vx2 (2)-1/2 (V)< = vx22 [1 (2)-1/2] ;

- V < vx2 : (V) (V)< , (V)< (V)< - ;

- V > vx2 : (V)< , (V)< (V)< - ;

- V = : (V)< , (V)< = vx2 , (V)< = vx22 .

, β > 1 0 < β < 1 ( ).

 

3.2. β

(155) (176) V , β - β > 1 0 < β < 1 , .. β V.

, ( ) , ( ), , .

.

 

3.2.1. 3

, , , . 1 - O1x1y1z1 O2x2y2z2, V O1x1 O1x1y1z1.

, , . 5 1 2, , 3.

1 2 () 3, .

ω

 

ω

 

3

 

1

 

2

 

 

R

 

R

 

 

. 5