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Tables (3)

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Table 1 Concentric Meniscus

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Table A Exhibiting results of numerical calculations in various special cases, most of which are referred to in the text. (Distances all given in centimeters.)

Equations (84)

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r 1 = A 1 C 1 , r 2 = A 2 C 2 , n c = d = A 1 A 2 ,
R 1 = 1 / r 1 and R 2 = 1 / r 2 .
F 1 = ( n 1 ) R 1 , F 2 = 2 n R 2 , F 3 = 2 n R 1 , F 4 = ( n 1 ) R 2 ,
A 1 F = Y 1 F , A 1 H = 1 Y 1 F , A 2 F = X 4 F , A 1 H = X 4 1 F .
1 12
1 9
11 12
19 36
10 9
4 9
13 9
7 36
2 3
5 4
2 3
F = 10 , X 4 = 5 4 , Y 1 = 2 3 ,
A 1 F = 1 15 , A 1 H = 1 30 , A 2 F = 1 8 , A 2 H = 1 40 ,
FH = FA 1 + A 1 H = H A 2 + A 2 F = 1 F = f ,
A 1 A 2 = 0.83 , A 2 C 1 = A 2 C 2 = A 2 H = A 2 H = 2.50 , A 1 F = 6.67 , A 2 F = 12.50 ;
X 4 = 1 c ( 3 F 1 + 2 F 2 + F 3 ) + c 2 ( 2 F 1 F 2 + F 2 F 3 + 2 F 3 F 1 ) c 3 F 1 F 2 F 3 ; Y 1 = 1 c ( 3 F 4 + 2 F 3 + F 2 ) + c 2 ( 2 F 4 F 3 + F 3 F 2 + 2 F 2 F 4 ) c 3 F 4 F 3 F 2 ; F = F 1 + F 2 + F 3 + F 4 c ( F 1 F 2 + 2 F 1 F 3 + F 2 F 3 + 3 F 1 F 4 + F 2 F 4 + F 3 F 4 ) + c 2 ( F 1 F 2 F 3 + 2 F 1 F 2 F 4 + 2 F 1 F 3 F 4 + F 2 F 3 F 4 ) c 3 F 1 F 2 F 3 F 4 .
n = 1.5 ,
d = 1.5 c , F 1 = 0.5 R 1 , F 2 = 3 R 2 , F 3 = 3 R 1 , F 4 = 0.5 R 2 , ( n = 1.5 ) .
= R 2 R 1 , ϕ = F R 1 , x = c R 1 ,
ϕ x = c F , ϕ R 2 = F , x = c R 2 .
X 4 = 1 + 3 ( 4 3 ) 2 x + 3 ( 1 4 ) x 2 + 9 2 x 3 , Y 1 = 1 3 ( 4 3 ) 2 x + 3 ( 4 ) x 2 9 2 2 x 3 = X 4 3 ( 1 + ) x { 1 + 2 ( 1 ) x + 3 x 2 } 2 ,
A 2 + B + C = 0 ,
A = 3 x ( 3 x 2 ) ( x 2 ) , B = 24 x 4 51 x + 14 , C = 2 ( 6 x 7 + 2 ϕ ) .
Δ = B 2 4 A C ,
= B ± ( Δ ) 1 / 2 2 A .
F 1 = 0.25000 , F 2 = 0.74820 , F 3 = 1.50000 , F 4 = 0.12470.
A 1 A 2 = 0.30000 , A 1 C 1 = 2.00000 , A 2 C 2 = 4.00962 , A 1 F = Y 1 / F = 0.57095 , A 1 H = ( 1 Y 1 ) / F = 0.42905 , A 2 F = X 4 / F = 0.82167 , A 2 H = ( X 4 1 ) / F = 0.17833.
A 1 A 2 = 3.00 , A 1 C 1 = 20.00 , A 2 C 2 = 40.10 , A 1 H = 4.29 , A 1 F = 5.71 , A 2 H = 1.78 , A 2 F = 8.22 ;
ϕ x = 1 30 , ϕ = 7 6 x 2 .
x = 0.00944722 , ϕ = r 1 = + 3.52837.
F = 10 dptr : A 1 A 2 = 0.50 , A 1 C 1 = 35.28 , A 1 C 2 = , A 1 H = 0.57 , A 1 F = 10.57 , A 2 H = 0.43 , A 2 F = 10 .43 ;
9 x 3 48 x 2 + 75 x 28 + 4 ϕ = 0 .
135 x 4 720 x 3 + 1125 x 2 420 x + 2 = 0 .
ϕ = r 1 = r 2 = 6.9099065.
X 4 = Y 1 = 0.94970 , F = 1 .
9 x 3 + 24 x 2 42 x 28 + 16 ϕ = 0.
135 x 4 + 360 x 3 360 x 2 420 x + 8 = 0 ,
X 4 = 0.971177 , Y 1 = 0.928672.
9 2 x 3 + 24 ( 1 ) x 2 + { 12 ( 1 + 2 ) 51 } x 14 ( 1 ) + 4 ϕ = 0 .
12.15 x 4 + 75.6 x 3 33.3 x 2 147 x + 2 = 0.
2 ϕ = 7 6 x ,
X 4 = Y 1 = 1 , ( c = 0 ) ;
F = F 1 + F 2 + F 3 + F 4 = 7 ( R 1 R 2 ) / 2 .
Δ ( p x 2 + q x + m ) 2 + μ x 4 + α x 3 + β x 2 + γ x + δ 48 x ( x 2 ) ( 3 x 2 ) ϕ ,
μ = 144 p 2 , α = 2 ( p q + 396 ) β = 1353 q 2 2 p m , γ = 2 ( q m + 378 ) , δ = 196 m 2 .
ϕ = μ x 4 + a x 3 + β x 2 + γ x + δ 48 x ( x 2 ) ( 3 x 2 ) ,
Δ = ( p x 2 + q x + m ) 2
p = α = 0 , q = α = 0 , q = γ = 0 and m = δ = 0 ,
ϕ = α x 2 + β x + γ 48 ( x 2 ) ( 3 x 2 ) , p = 12 α = 24 ( q + 33 ) ; and either { m = + 14 , β = 1017 q 2 , γ = 28 ( 27 + q ) , or { m = 14 , β = 1689 q 2 , γ = 28 ( 27 q ) ; Δ = ( 12 x 2 + q x + 14 ) 2 .
3 x 3 x 2
2 ( 3 x 7 ) 3 x 2
3 x 7 2 ( x 2 ) ( 3 x 2 )
10 x x 2
3 x 7 3 ( 3 x 2 )
5 ( 5 x 14 ) 2 ( x 2 ) ( 3 x 2 )
5 x 14 3 ( x 2 )
5 ( 6 x 2 4 x 49 ) 6 ( x 2 ) ( 3 x 2 )
ϕ = B x 2 + γ x + δ 48 x ( x 2 ) ( 3 x 2 ) , p = ± 12 , q = 33 , β = 24 ( 11 m ) γ = 6 ( ± 11 m 126 ) , δ = 196 m 2 ,
ϕ = x ( μ x 2 + α x + β ) 48 ( x 2 ) ( 3 x 2 ) , q = ± 27 , m = 14 , α = 18 ( 44 ± 3 p ) , β = 4 ( 15 ± 7 p ) , μ = 144 p 2 ,
144 ϕ = u x + v ,
7 6 x 2
5 ( 7 3 x ) 12
3 ( 5 x 14 ) 16
45 x 154 48
ϕ = 3 x 3 x 2 ,
= 2 3 x 2 .
ϕ = 10 x x 2 , = 2 x 2 ,
c = 2 R 1 ( 1 + 10 R 1 ) , R 2 = 1 + 10 R 1 10 , Y 1 = 50 R 1 2 15 R 1 12 50 R 1 2 X 4 = 10 ( 10 R 1 2 + 3 R 1 2 ) ( 10 R 2 1 + 1 ) .
A 1 C 1 = 5.95 , A 2 C 2 = 5.61 , A 1 H = 2.63 , A 1 F = 7.37 , A 2 H = 0.93 , A 2 F = 10.93
ϕ = 2 ( 3 x 7 ) 3 x 2 ,
c = 2 ( 7 R 1 + 1 ) 3 R 1 ( 2 R 1 + 1 ) , R 2 = 3 ( 2 R 1 + 1 ) 5 , Y 1 = 6 R 1 + 1 R 1 , X 4 = 2 R 1 2 143 R 1 22 3 ( 2 R 1 + 1 ) 2 ;
A 1 C 1 = 69.29 , A 2 C 2 = 23.43 , A 1 H = 0.71 , A 1 F = 9.29 , A 2 H = 270.60 , A 2 F = 280.60 ;
c = H 1 A 2 n
X 1 = 1 , F 11 = F 1 = F 3 , X 2 = X 1 c F 11 , F 12 = F 11 + X 2 F 2 , X 3 = X 2 c F 12 , F 13 = F 12 + X 3 F 3 ; F = F 13 , H 1 F = H 1 F = X 3 / F , H 1 H = H 1 H = ( 1 X 3 ) / F , H 1 K = ( 1 X 3 ) / F .
F 1 = F 3 = ( n 1 ) R 1 , F 2 = 2 n R 2 , A 1 F = A 1 F = X 3 / F , A 1 H = ( 1 X 3 ) / F , A 1 K = ( 1 + X 3 ) / F .
F 1 = F 3 = R 1 / 2 , F 2 = 3 R 2 , ( n = 1.5 ) .
= R 2 R 1 , ϕ = F R 1 , x = c R 1 ,
ϕ x = c F , = 2 ( 2 x 2 ϕ ) 3 ( x 2 ) 2 .
3 x 3 2 ( 6 1 ) x 2 + 4 ( 3 1 ) x + 4 / k = 0 .
A 1 A 2 = 1 cm , A 1 C 1 = 50.725 cm , A 2 C 2 = 25.363 cm , A 1 H = 0.662 cm , A 1 K = 19.338 cm .