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Equations (28)

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F 1 = ( n 1 ) / r 1 and F 2 = ( n 1 ) / r 2 ,
F = F 1 + F 2 c F 1 F 2 .
F V = 1 c F 1 = F 1 V F 2 .
S = Z F , S = Z F , and φ = F 1 F
S = ( 1 c φ ) { S ( 1 c φ ) 1 } 1 + c { S ( 1 c φ ) 1 } .
{ ( φ 3 X ) c 2 + ( n 1 ) ( n 3 n φ φ 2 + X n ) c n 2 ( n 1 ) 2 } S 2 + 2 { ( φ 3 X ) c 2 φ ( φ 3 X + Y φ ) c + Y } S ( 1 c φ ) { ( 1 c φ ) X + c φ 4 } = 0 ,
X = n { ( n + 2 ) φ 2 n ( 2 n + 1 ) φ + n 3 } , Y = n ( n 1 ) { ( n + 1 ) φ n 2 } .
F 1 φ = Z S = V ( 1 c F 1 ) ,
H F 1 7 + J F 1 6 + K F 1 5 + L F 1 4 + M F 1 3 + N F 1 2 + P F 1 + Q = 0 ,
H = α 1 c 7 + α 2 c 6 + α 3 c 5 + α 4 c 4 , J = β 1 c 6 + β 2 c 5 + β 3 c 4 + β 4 c 3 , K = γ 1 c 5 + γ 2 c 4 + γ 3 c 3 + γ 4 c 2 , L = δ 1 c 4 + δ 2 c 3 + δ 3 c 2 + δ 4 c , M = 1 c 3 + 2 c 2 + 3 c , N = η 1 c 2 + η 2 c + η 3 , P = θ 1 c + θ 2 , Q = λ ;
α 1 = n 2 ( 1 + V ) V 6 ; α 2 = { n 2 ( 2 n + 1 ) V n ( n + 2 ) V 2 } V 4 ; α 3 = { n ( n + 2 ) ( 1 + V ) + 2 } V 4 ; 3 α 4 = β 4 = γ 4 = 3 δ 4 = 3 V 3 ; β 1 = n 2 { 2 ( n 1 ) ( 1 + V ) 2 Z + ( 6 + 7 V ) V 2 } V 4 ; β 2 = 2 { n 2 ( 1 + V ) + n ( n 2 1 ) ( 1 + V ) V V } ( 1 + V ) V 2 Z + { 6 n ( n + 2 ) V 2 + 5 ( 2 n + 1 ) V 4 n 2 } V 4 ; β 3 = { n ( n + 2 ) ( 4 + 5 V ) + 8 } V 4 2 ( 1 + V ) V 2 Z ; γ 1 = n 2 [ ( 1 + V ) Z { ( n 1 ) 2 ( 1 + V ) 2 Z + 4 ( n 1 ) ( 2 + 3 V ) V 2 } + 3 ( 5 + 7 V ) V 4 ] V 2 ; γ 2 = { n 2 ( 1 + V 3 ) + ( 3 n 2 1 V ) ( 1 + V ) V } Z 2 2 [ n 2 ( 2 + 3 V ) Z + ( n 2 1 ) { 3 ( n + 1 ) + 4 ( 2 n + 1 ) V + 5 n V 2 } V Z + ( 5 V 3 n 2 ) V 2 + 10 n V 3 ] V 2 15 n ( n + 2 ) V 6 ; γ 3 = 2 { ( 2 + 3 V ) Z 3 ( n 2 + 2 n + 2 ) V 2 5 n ( n + 2 ) V 3 } V 2 ; δ 1 = n 2 [ ( n 1 ) 2 ( 1 + V ) 2 ( 2 + 5 V ) Z 2 + 2 ( n 1 ) ( 6 + 20 V + 15 V 2 ) V 2 Z + 5 ( 4 + 7 V ) V 4 ] V 2 ; δ 2 = 2 [ { n 2 + 3 ( n 3 + 2 n 2 n 1 ) V + 6 ( n 2 1 ) ( 2 n + 1 ) V 2 + 10 n ( n 2 1 ) V 3 } Z + { 10 n ( n + 2 ) V 2 + 5 ( 2 n + 1 ) V 2 n 2 } V 2 ] V 2 ( 1 + V ) { 3 n 2 ( 1 + V ) 1 + 3 V ) } V Z 2 ; δ 3 = 2 [ { 5 n ( n + 2 ) V + 2 ( n 2 + 2 n + 2 ) } V 2 ( 1 + 3 V ) Z ] V 2 ; 1 = n 2 { ( n 1 ) 2 ( 1 + V ) ( 1 + 8 V + 10 V 2 ) Z 2 + 8 ( n 1 ) ( 1 + 5 V + 5 V 2 ) V 2 Z + 5 ( 3 + 7 V ) V 4 } V 2 ; 2 = { 3 n 2 ( 1 + V ) ( 2 + 3 V ) } V 2 Z 2 2 { n 3 + 2 n 2 n 1 + 2 ( n 2 1 ) ( 2 + 4 n + 5 n V ) V } V 3 Z { 15 n ( n + 2 ) V 2 + 5 ( 2 n + 1 ) V n 2 } V 4 ; 3 = 2 V 3 Z { 5 n ( n + 2 ) V + n 2 + 2 n + 2 } V 4 ; η 1 = n 2 { ( n 1 ) 2 ( 3 + 12 V + 10 V 2 ) Z 2 + 2 ( n 1 ) ( 1 + 10 V + 15 V 2 ) V Z + 3 ( 2 + 7 V ) V 3 } V 3 ; η 2 = { ( 2 n + 1 ) V 2 + 6 n ( n + 2 ) V 3 + 2 ( n 2 1 ) ( 1 + 2 n + 5 n V ) V Z ( n 2 1 ) Z 2 } V 3 ; η 3 = n ( n + 2 ) V 5 ; θ 1 = n 2 { ( n 1 ) 2 ( 3 + 5 V ) Z 2 + 4 ( n 1 ) ( 1 + 3 V ) V Z + ( 1 + 7 V ) V 2 } V 4 ; θ 2 = n { ( n + 2 ) V + 2 ( n 2 1 ) Z } V 5 ; λ = n 2 { ( n 1 ) Z + V } 2 V 5 .
N 0 = n ( n + 2 ) F 5 ; P 0 = 2 n ( n 2 1 ) F 5 Z n ( n + 2 ) F 6 ; Q 0 = n 2 { ( n 1 ) Z + F } ² F 5 .
α F 1 2 + β F 1 + γ = 0 , ( c = 0 ) ,
α = n + 2 , β = 2 ( n 2 1 ) Z ( n + 2 ) F , γ = n { F + ( n 1 ) Z } 2 .
2 ( n + 2 ) φ 2 ( n 2 1 ) S ( n + 2 ) = ± 2 ( n 1 ) ( S x 1 ) ( S x 2 ) , ( F = 1 ) ,
( n + 2 ) ( n 1 ) ± n n ( n + 2 ) 2 ( n 1 ) .
2 ( n + 2 ) φ 2 ( n 2 1 ) S ( n + 2 ) = 0 ,
S = x 1 , φ = n 2 { n + ( n 1 ) n ( n + 2 ) n + 2 }
S = x 2 , φ = n 2 { n ( n 1 ) n ( n + 2 ) n + 2 } .
6.76923 φ 2.5200 S 3.3842 = ± ( S 5.1798 ) ( S + 1.6598 ) .
H = 1193114 × 10 15 ; M = 55.33859 ; J = + 1922575 × 10 12 ; N = + 7 388.700 ; K = 1139709 × 10 9 ; P = 202 292.3 L = + 3210912 × 10 7 ; Q = + 1 455 092 .
H = 1410243 × 10 13 ; M = 3 725.887 ; J = + 1702760 × 10 10 ; N = + 301 160.6 ; K = 8880752 × 10 8 ; P = 7 834 748 . ; L = + 23.81681 ; Q = + 62 794 569 .
F 2 = V F 1 1 c F 1 ,
x 2 = z γ 2 , sin α 2 = x 2 sin θ 3 r 2 , sin α 2 = sin α 2 n , θ 2 = θ 3 α 2 + α 2 , x 2 = r 2 sin α 2 sin θ 2 , a = r 2 r 1 + d , x 1 = x 2 + a , sin α 1 = x 1 sin θ 2 r 1 , sin α 1 = n sin α 1 .
D 1 = sin ( α 1 α 1 ) r 1 sin α 1 , D 2 = n · sin ( α 2 α 2 ) r 2 sin α 2 , h 1 = r 1 sin ( α 1 θ 1 ) , h 2 = r 2 sin ( α 2 θ 2 ) , t = h 2 h 1 n · sin θ 2 .
B 1 = D 1 , A 1 = D 1 cos 2 α 1 , B 2 = B 1 1 t B 1 , A 2 = A 1 1 t A 1 , B 2 = B 2 + D 2 , A 2 = P 2 cos 2 α 2 + D 2 cos 2 α 2 .
a a b b = , u = b , and u = b ,
a = a u = a u .