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  1. L. Silberstein, J. Opt. Soc. Am. 32, 474–485 (1942).
    [Crossref]
  2. L. Silberstein and A. P. H. Trivelli, J. Opt. Soc. Am. 28, 441 (1938).
    [Crossref]
  3. For the four shorter, and αG(y)+βF(y/m) for the two longest developments.
  4. For all but the shortest (1 min.) development which calls for αF(y)+βG(y/m), α:β=9:8.

1942 (1)

1938 (1)

J. Opt. Soc. Am. (2)

Other (2)

For the four shorter, and αG(y)+βF(y/m) for the two longest developments.

For all but the shortest (1 min.) development which calls for αF(y)+βG(y/m), α:β=9:8.

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Figures (3)

Fig. 1
Fig. 1

From left to right, one-, two-, and three-quantic curves.

Fig. 2
Fig. 2

D = 1 2 D m { F ( y ) + F ( y / m ) }, m=16. Inflection point marked by eyelet.

Fig. 3
Fig. 3

D = 1 2 D m { F ( y ) + F ( y / m ) }, m=256. Inflection points marked by eyelet.

Equations (79)

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k a = f ( a ) d a { 1 - e - ϵ a n ( 1 + ϵ a n + ( ϵ a n ) 2 2 ! + + ( ϵ a n ) r - 1 ( r - 1 ) ! ) } ,
d k a ϵ d n = ϵ r - 1 a r n r - 1 ( r - 1 ) ! e - ϵ a n f ( a ) d a .
D = M a k a ,             M = Log e ,
d D d n = M a d k a d n ,
d D ϵ d n = M ( ϵ n ) r - 1 ( r - 1 ) ! 0 a r + 1 e - ϵ a n f ( a ) d a .
f ( a ) = ( N / ā ) e - a / ā .
1 D m d D d y = y r - 1 ( r - 1 ) ! 0 x r + 1 e - ( 1 + y ) x d x
D ( y ) D m = y r - 1 ( r - 1 ) ! ( 1 + y ) r + 2 0 u r + 1 e - u d u .
D ( y ) = D m r ( r + 1 ) y r - 1 ( 1 + y ) r + 2 .
D ( y ) D m = y r ( r + 1 + y ) ( 1 + y ) r + 1 .
y ˜ = r 2 ,             D ˜ = ( 2 + 3 r ) r r ( r + 2 ) r + 1 D m ,
γ = D m M · ( r + 1 ) r r + 1 2 r ( 1 + r / 2 ) r + 2 ,
D ( 1 ) D m = 3 4 , 1 2 , 5 16 , 3 16 ,
D ( y ) + D ( 1 y ) = D m ,
D = D m X ( y ) ,
D = α X ( y ) + β Y ( y / m ) ,             α + β = D m ,
ϵ = h c / λ ā E 1 ,
ϵ = 1.965 + 10 - 4 λ ā E 1 .
E ( y ) = y ( 2 + y ) ( 1 + y ) 2 ,             F ( y ) = y 2 ( 3 + y ) ( 1 + y ) 3 , G ( y ) = y 3 ( 4 + y ) ( 1 + y ) 4 .
r = 1 2 3 γ / D m = 0.682 0.863 0.955 D m / γ = 1.47 1.16 1.05.
D = 0.40 F ( y ) ,             Log E 1 = 0.69.
ϵ = 1.3 · 10 - 4 .
Log E 1 = 0.70 0.68 0.65 0.61 0.63
0.09 9 0.00 7 0.01 1 0.01 2 - 0.03 3 .
ϵ = 1.3 1.3 1.4 1.6 1.5 · 10 - 4 .
0.02 8 - 0.01 0.01 2 0.01 2 0.02 0.04 7 .
ϵ = 1.3 1.4 1.6 1.8 2.1 2.2 · 10 - 4 ,
D = 1.01 F ( y ) ,             Log E 1 = 0.98
D = 147 F ( y ) ,             Log E 1 = 0.89
D = 0.88 G ( y ) ,             Log E 1 = 0.84 ,
D = 0.43 E ( y ) ,             Log E 1 = 1.17 ,
D = D m E ( y ) ;
ϵ = 8.2 · 10 - 4 / E 1 .
D = 0.94 F ( y ) ,             D = 1.27 F ( y ) ,             D = 1.41 G ( y )
D = 1.44 y 5 ( 6 + y ) ( 1 + y ) 6 ,             D = 1.48 y 8 ( 9 + y ) ( 1 + y ) 9 , D = 1.53 y 16 ( 17 + y ) ( 1 + y ) 17 .
Δ = 0.00 2 .00 6 - .00 2 .00 5 .00 5 .00 1 - .00 3 .00 1 .00 .00 ,
Δ = 0.00 .00 - .01 5 - .02 9 .00 6 - .03 - .03 - .02 - .01 .00.
r =     5 for t =     6 minutes , r = 16 for t = 18 minutes ,
D ( y ) = α X ( y ) + β Y ( y / m ) ,             α + β = D m ,
D ( y ) = α F ( y ) + β F ( y / m ) .
D ( y ) = 1 2 D m { F ( y ) + F ( y / m ) }
D ( y * ) = 1 2 D m { F ( m / y ) + F ( 1 / y ) }
D ( y ) + D ( y * ) = 1 2 D m { F ( y ) + F ( 1 / y ) + F ( y / m ) + F ( m / y ) } = 1 2 D m { 1 + 1 }
D ( y ) + D ( y * ) = D m ,
m 2 ( 1 - y ) ( 1 + y ) 5 + 1 - y / m ( 1 + y / m ) 5 = 0
γ = 3 D m M y ˜ 2 [ 1 ( 1 + y ˜ ) 4 + 1 m 2 ( 1 + y ˜ / m ) 4 ] .
γ / D m = 3 / M m · 1 + m 2 / ( 1 + m ) 4 .
M ( γ / D m ) = 0.375 , 0.357 , 0.157.
D 1 ( y ) = 0.98 F ( y ) .
F ( 4 / m ) = 1 2 , very nearly , or 4 / m = 1 ,
F ( 8 / m ) = 0.764 , 8 / m = 2.17 , or 4 / m = 1.08 5 ,
D = 0.98 F ( y ) + 0.22 F ( y 3.85 ) ,             Log E 1 = 1 ¯ .87.
ϵ 1 = 8.65 · 10 - 4 , ϵ 2 = ( 1 / m ) ϵ 1 = 2.24 · 10 - 4 .
ϵ = 1.3 · 10 - 4 .
D = α F ( y ) + β F ( y / m ) ,
F ( 2 y ) / F ( y ) = 0.32 / 0.17 = 1.88.
D 1 ( y ) = 0.64 F ( y ) , with E 1 = 1 2 .
F ( 2 / m ) = 0.105 ,             2 / m = 0.250 ,
F ( 4 / m ) = 0.265 ,             4 / m = 0.510 ,             2 / m = 0.255 ,
D = 0.64 F ( 2 E ) + 0.96 F ( 1 4 E ) ,
ϵ 1 = 11.3 · 10 - 4 ,             ϵ 2 = 1 8 ϵ 1 .
D = 0.47 F ( y ) + 1.17 G ( 0.21 y ) ,             Log E 1 = 1 ¯ .91.
D = 0.58 F ( y ) + 1.44 G ( y / 7 ) ,             Log E 1 = 1 ¯ .69 ,
D = α E ( y ) + β F ( y / m ) ,
D = 1.02 E ( y ) + 0.40 F ( y / 4 ) ,             Log E 1 = 0.43 ,
D = α E ( y ) + β E ( y / m ) ,
D = D m F ( y ) ,
D = 1.28 F ( y ) + 0.67 F ( y / 8.5 ) ,             Log E 1 = 1 ¯ .74             for             T = 1 6
D = 1.39 F ( y ) + 0.60 F ( y / 8.7 ) ,             Log E 1 = 0.05             for             T = 1 7 4 .
α F ( y ) + β F ( y / m ) ,             i d e m ,             α F ( y ) + β G ( y / m ) ,             i d e m
α F ( y ) + β F ( y / m ) ,             i d e m ,             α F ( y ) + β G ( y / m ) ,
α G ( y ) + β G ( y / m ) .
D = D m G ( y ) ,
ϵ = 0.17 5 0.19 0.18 0.18 0.18 0.18 5 κ .
D = D m G ( y ) with D m = 1.10 to 1.52
ϵ = 0.20 0.20 0.20 0.20 0.21 0.24 κ .
D = α F ( y ) + β F ( y / m ) .
ϵ 1 = 0.045 0.040 0.040 0.044 0.042 0.050 κ ,
α : β = 20 / 7 5 / 2 9 / 4 9 / 4 9 / 4 24 / 11 m = 22 20 17 19 16 16 ϵ 1 = 0.015 0.014 5 0.014 5 0.015 0.014 5 0.019 κ .