Abstract

The general theory of the integrating sphere as used in the measurement of total spectral reflectance is formulated as an integral equation and solved for five special cases. Two methods of measuring reflectance are considered. In the substitution method, sample and standard are placed in turn at the sample aperture and the ratio of the respective photocell readings determined. In the comparison method, both sample and standard are in place at all times at their respective apertures. The beam is switched from sample to standard and the ratio of the respective photocell readings determined. The efficiency and error, and their interdependence are discussed for both methods and for different geometries of the sphere. The design of a comparison type sphere with maximum efficiency and low error is considered.

© 1955 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. T. Walsh, Photometry (Constable & Co. Ltd., London, 1953), second edition.
  2. E. Karrer, Sci. Papers Bur. Standards. No. 415, pp. 203–225 (1921).
    [Crossref]
  3. H. J. McNicholas, Bur, Standards J. Research 1, 29 (1928).
    [Crossref]
  4. E. B. Rosa and A. H. Taylor, Sci. Papers Bur. Standards No. 447, pp. 281–325 (1922).
  5. A. H. Taylor, Sci. Papers Bur. Standards, No. 391, pp. 421–436 (1920).
    [Crossref]
  6. A. C. Hardy and O. W. Pineo, J. Opt. Soc. Am. 21, 502 (1931).
    [Crossref]
  7. J. S. Preston, Trans. Opt. Soc. (London) 31, 15 (1929–30).
    [Crossref]
  8. P. Moon, J. Opt. Soc. Am. 30, 195 (1940).
    [Crossref]
  9. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, New York, 1950), fourth edition.
  10. W. V. Lovitt, Linear Integral Equations (Dover Publications Inc., New York, 1950), first edition.
  11. K. S. Gibson, Natl. Bur. Standards, Circular No. 484, (1949).

1949 (1)

K. S. Gibson, Natl. Bur. Standards, Circular No. 484, (1949).

1940 (1)

1931 (1)

1928 (1)

H. J. McNicholas, Bur, Standards J. Research 1, 29 (1928).
[Crossref]

1922 (1)

E. B. Rosa and A. H. Taylor, Sci. Papers Bur. Standards No. 447, pp. 281–325 (1922).

1921 (1)

E. Karrer, Sci. Papers Bur. Standards. No. 415, pp. 203–225 (1921).
[Crossref]

1920 (1)

A. H. Taylor, Sci. Papers Bur. Standards, No. 391, pp. 421–436 (1920).
[Crossref]

Gibson, K. S.

K. S. Gibson, Natl. Bur. Standards, Circular No. 484, (1949).

Hardy, A. C.

Karrer, E.

E. Karrer, Sci. Papers Bur. Standards. No. 415, pp. 203–225 (1921).
[Crossref]

Lovitt, W. V.

W. V. Lovitt, Linear Integral Equations (Dover Publications Inc., New York, 1950), first edition.

McNicholas, H. J.

H. J. McNicholas, Bur, Standards J. Research 1, 29 (1928).
[Crossref]

Moon, P.

Pineo, O. W.

Preston, J. S.

J. S. Preston, Trans. Opt. Soc. (London) 31, 15 (1929–30).
[Crossref]

Rosa, E. B.

E. B. Rosa and A. H. Taylor, Sci. Papers Bur. Standards No. 447, pp. 281–325 (1922).

Taylor, A. H.

E. B. Rosa and A. H. Taylor, Sci. Papers Bur. Standards No. 447, pp. 281–325 (1922).

A. H. Taylor, Sci. Papers Bur. Standards, No. 391, pp. 421–436 (1920).
[Crossref]

Walsh, J. W. T.

J. W. T. Walsh, Photometry (Constable & Co. Ltd., London, 1953), second edition.

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, New York, 1950), fourth edition.

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, New York, 1950), fourth edition.

Bur, Standards J. Research (1)

H. J. McNicholas, Bur, Standards J. Research 1, 29 (1928).
[Crossref]

J. Opt. Soc. Am. (2)

Natl. Bur. Standards, Circular No. 484 (1)

K. S. Gibson, Natl. Bur. Standards, Circular No. 484, (1949).

Sci. Papers Bur. Standards No. 447 (1)

E. B. Rosa and A. H. Taylor, Sci. Papers Bur. Standards No. 447, pp. 281–325 (1922).

Sci. Papers Bur. Standards, No. 391 (1)

A. H. Taylor, Sci. Papers Bur. Standards, No. 391, pp. 421–436 (1920).
[Crossref]

Sci. Papers Bur. Standards. No. 415 (1)

E. Karrer, Sci. Papers Bur. Standards. No. 415, pp. 203–225 (1921).
[Crossref]

Trans. Opt. Soc. (London) (1)

J. S. Preston, Trans. Opt. Soc. (London) 31, 15 (1929–30).
[Crossref]

Other (3)

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, New York, 1950), fourth edition.

W. V. Lovitt, Linear Integral Equations (Dover Publications Inc., New York, 1950), first edition.

J. W. T. Walsh, Photometry (Constable & Co. Ltd., London, 1953), second edition.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Relation of coordinate system on cavity wall to Cartesian coordinate system.

Fig. 2
Fig. 2

Vector relations for the calculation of the nucleus for a generalized cavity.

Fig. 3
Fig. 3

Vector relations for the calculation of the nucleus for a spherical cavity.

Fig. 4
Fig. 4

Geometry of substitution type sphere with flat sample and standard.

Fig. 5
Fig. 5

Geometry of comparison type sphere with flat sample and standard.

Fig. 6
Fig. 6

Geometry of comparison type sphere with small specular sample.

Tables (1)

Tables Icon

Table I Error in measurement of total reflectance.a

Equations (67)

Equations on this page are rendered with MathJax. Learn more.

π N = r ( u , v ) { H 0 ( u , v ) + H ( u , v ) } .
r ( u , v ) π { H 0 ( u , v ) + H ( u , v ) } d a ( - ϱ · n ) ρ d a ( ϱ · n ) ρ 3 .
H ( u , v ) = 1 π s H 0 ( u , v ) r ( u , v ) ( - ϱ · n ) ( ϱ · n ) ρ 4 d a + 1 π s H ( u , v ) r ( u , v ) ( - ϱ · n ) ( ϱ · n ) ρ 4 d a .
H ( u , v ) = f ( u , v ) + λ a b c d K ( u , v , u , v ) H ( u , v ) d u d v ,
H ( u , v ) = f ( u , v ) + λ s K ( u , v , u , v ) H ( u , v ) d a .
D 0 = 1 ,             D 0 ( u , v , u , v ) = K ( u , v , u , v ) ,
D m = s D m - 1 ( u , v , u , v ) d a
D m ( u , v , u , v ) = K ( u , v , u , v ) D m - m s K ( u , v , u , v ) D ( u , v , u , v ) d a
D ( λ ) = n = 0 ( - 1 ) n n ! D n λ n ,
D ( u , v , u , v ; λ ) = n = 0 ( - 1 ) n n ! D n ( u , v , u , v ) λ n .
K ( u , v , u , v ; λ ) = D ( u , v , u , v ; λ ) D ( λ ) .
H ( u , v ) = f ( u , v ) + λ s K ( u , v , u , v ; λ ) f ( u , v ) d a .
B = b H ( u , v ) d a ,
( - ϱ · n ) ( ϱ · n ) ρ 4 = 1 / ( 4 R 2 ) .
H ( θ , ϕ ) = 1 4 π R 2 s r ( θ , ϕ ) H 0 ( θ , ϕ ) d a + 1 4 π R 2 s r ( θ , ϕ ) H ( θ , ϕ ) d a .
H ( θ , ϕ ) = r s S c H 0 ( θ , ϕ ) d a + 1 S s r ( θ , ϕ ) H ( θ , ϕ ) d a .
c H 0 ( θ , ϕ ) d a = P ,
K ( θ , ϕ , θ , ϕ ) = { r S , ( θ , ϕ ) S ( θ , ϕ ) d r s S , ( θ , ϕ ) S ( θ , ϕ ) c 0 , ( θ , ϕ ) S ( θ , ϕ ) a b .
D ( λ ) = 1 - ( r d / S ) - ( r s c / S ) ,
D ( θ , ϕ , θ , ϕ ; λ ) = K ( θ , ϕ , θ , ϕ ) .
H ( θ , ϕ ) = P r s S 1 1 - ( r d / S ) - ( r s c / S ) ,
B s = P r s b / S 1 - ( r d / S ) - ( r s c / S ) .
B s B s t = r s r s t { 1 - ( r s t - r s ) c / S 1 - ( r d / S ) - ( r s c / S ) } .
K ( θ , ϕ , θ , ϕ ) = { r S , ( θ , ϕ ) S ( θ , ϕ ) d r s S , ( θ , ϕ ) S ( θ , ϕ ) c s r s t S , ( θ , ϕ ) S ( θ , ϕ ) c s t 0 , ( θ , ϕ ) S ( θ , ϕ ) a b .
D ( λ ) = 1 - ( r d / S ) - ( r s c / S ) - ( r s t c / S ) ,
D ( θ , ϕ , θ , ϕ ; λ ) = K ( θ , ϕ , θ , ϕ ) .
B s = P r s b / S 1 - ( r d / S ) - ( r s c / S ) - ( r s t c / S ) ,
B s / B s t = r s / r s t .
( - ϱ · n ) ( ϱ · n ) ρ 4 ( R - L cos θ ) ( L - R cos θ ) ( R 2 + L 2 - 2 R L cos θ ) 2 = g ( θ ) .
H ( θ , ϕ ) = P r s π g ( θ ) + s r ( θ , ϕ ) π g ( θ ) H ( θ , ϕ ) d a .
f ( θ , ϕ ) = { P r s π g ( θ ) , ( θ , ϕ ) a b d 0 , ( θ , ϕ ) c f .
K ( θ , ϕ , θ , ϕ ) = { r S , ( θ , ϕ ) a b d ( θ , ϕ ) d 0 , ( θ , ϕ ) S ( θ , ϕ ) a b r s π g ( θ ) , ( θ , ϕ ) a b d ( θ , ϕ ) c f 0 , ( θ , ϕ ) c f ( θ , ϕ ) c f r S c c f , ( θ , ϕ ) c f ( θ , ϕ ) d .
a = a g ( θ ) d a b = b g ( θ ) d a d = d g ( θ ) d a
D ( λ ) = 1 - ( r d / S ) - ( r s r d c / S π ) ,
D ( θ , ϕ , θ , ϕ ; λ ) = { r S + r S r s c π g ( θ ) , ( θ , ϕ ) a b d ( θ , ϕ ) d 0 ( θ , ϕ ) S ( θ , ϕ ) a b r s π g ( θ ) - r S r s π [ d g ( θ ) - d ] , ( θ , ϕ ) a b d ( θ , ϕ ) c f r S c c f r s d π , ( θ , ϕ ) c f ( θ , ϕ ) c f r S c c f , ( θ , ϕ ) c f ( θ , ϕ ) d .
H ( θ , ϕ ) = P r s π g ( θ ) + P r s d D ( λ ) [ ( r / S ) + ( r r s c g ( θ ) / S π ) ] .
B s = P r s π { b + ( b d - b d ) ( r / S ) 1 - ( r d / S ) - ( r s r d c / S π ) }
B s B s t = r s r s t { 1 - ( r s t - r s ) ( r d c / S π ) 1 - ( r d / S ) - ( r s r d c / S π ) } .
( - ϱ · n ) ( ϱ · n ) ρ 4 { g ( θ ) , ( θ , ϕ ) a b d ( θ , ϕ ) c s g ( χ ) , ( θ , ϕ ) a b d ( θ , ϕ ) c s t .
a = a g ( θ ) d a = a g ( χ ) d a , b = b g ( θ ) d a = b g ( χ ) d a , d = d g ( θ ) d a = d g ( χ ) d a , c = C s t s p g ( θ ) d a = C s s p g ( χ ) d a .
f ( θ , ϕ ) = { P r s π g ( θ ) , ( θ , ϕ ) a b d 0 , ( θ , ϕ ) c s P r s π c c f , ( θ , ϕ ) c s t .
f ( θ , ϕ ) = P r s π 1 c f C s t s p g ( θ ) d a = P r s π c c f ,             ( θ , ϕ ) c s t .
K ( θ , ϕ , θ , ϕ ) = { r S , ( θ , ϕ ) a b d ( θ , ϕ ) d 0 , ( θ , ϕ ) S ( θ , ϕ ) a b r s π g ( θ ) , ( θ , ϕ ) a b d ( θ , ϕ ) c s r s t π g ( χ ) , ( θ , ϕ ) a b d ( θ , ϕ ) c s t 0 , ( θ , ϕ ) c s ( θ , ϕ ) c s 0 , ( θ , ϕ ) c s t ( θ , ϕ ) c b t r S c c f , ( θ , ϕ ) c s ( θ , ϕ ) d r S c c f , ( θ , ϕ ) c s t ( θ , ϕ ) d r s t π c c f , ( θ , ϕ ) c s ( θ , ϕ ) c s t r s π c c f , ( θ , ϕ ) c s t ( θ , ϕ ) c s .
D ( λ ) = 1 - ( r d / S ) - ( r d c ( r s + r s t ) / S π ) - ( r s r s t ( c ) 2 / π 2 ) - ( 2 r r s r s t c c d / S π 2 ) + ( r r s r s t ( c ) 2 d / S π 2 ) ,
D ( θ , ϕ , θ , ϕ ; λ ) = { r S + r c π S { r s g ( θ ) + r s t g ( χ ) } + r r s r s t S π 2 { c c [ g ( θ ) + g ( χ ) ] - ( c ) 2 } , ( θ , ϕ ) a b d ( θ , ϕ ) d 0 , ( θ , ϕ ) S ( θ , ϕ ) a b r s π g ( θ ) - r d S r s π g ( θ ) + r r s d S π + r s r s t π 2 c g ( χ ) + r S r s r s t π 2 { c d + c d g ( χ ) - c d g ( θ ) - c d g ( χ ) } , ( θ , ϕ ) a b d ( θ , ϕ ) c s r s t π g ( χ ) - r d r s t S π g ( χ ) + r r s t S π d + r s r s t π 2 c g ( θ ) + r r s r s t S π 2 { c d + c d g ( θ ) - c d g ( χ ) - c d g ( θ ) } , ( θ , ϕ ) a b d ( θ , ϕ ) c s t r r s S π c c f d + r s r s t π 2 ( c ) 2 c f + r S r s r s t π 2 { 2 c c f c d - ( c ) 2 c f d } , ( θ , ϕ ) c s ( θ , ϕ ) c s r S r s t π c c f d + r s r s t ( c ) 2 π 2 c f + r S r s r s t π 2 { 2 c c f c d - ( c ) 2 c f d } , ( θ , ϕ ) c s t ( θ , ϕ ) c s t r S c c f + r S r s t π c c c f , ( ϕ , θ ) c s ( θ , ϕ ) d r S c c f + r S r s π c c c f , ( θ , ϕ ) c s t ( θ , ϕ ) d r s t π c c f - r s t π c c f r d S + r S r s t π c d c f , ( θ , ϕ ) c s ( θ , ϕ ) c s t r s π c c f - r s π c c f r d S + r S r s π c c f d , ( θ , ϕ ) c s t ( θ , ϕ ) c s .
B s = P r s π D ( λ ) { 1 + r s t π c } { b + r S ( b d - b d ) } .
f ( θ , ϕ ) = { P r s t π g ( χ ) , ( θ , ϕ )             a b d P r s t π c c f , ( θ , ϕ )             c s 0 , ( θ , ϕ )             c s t .
B s t = P r s t π D ( λ ) { 1 + r s c π } { b + r S ( b d - b d ) } ,
B s B s t = r s r s t { 1 + ( r s t - r s ) ( c / π ) 1 + ( r s c / π ) } .
b = photocell aperture , b = area reflected into b by sample , } of equal area b a = entrance aperture , a = area reflected into a by sample , } of equal area a c s t = standard , c s t = area reflected into c s t by sample , c s = sample , } each of area c d = S - b - b - a - a - c s - c s t - c s t , of area S - 2 b - 2 a - 3 c D = d + a + b + c s t , of area S - a - b - 2 c .
K ( θ , ϕ , θ , ϕ ) = { r S , ( θ , ϕ ) a b d ( θ , ϕ ) d a b c s t r S , ( θ , ϕ ) a b c s ( θ , ϕ ) d a b c s t r S , ( θ , ϕ ) c s t ( θ , ϕ ) d a b c s t r S , ( θ , ϕ ) c s t ( θ , ϕ ) d a b c s t 0 , ( θ , ϕ ) S ( θ , ϕ ) a b r s t S , ( θ , ϕ ) d a b ( θ , ϕ ) c s t r s t S , ( θ , ϕ ) a b c s ( θ , ϕ ) c s t r s t S , ( θ , ϕ ) c s t ( θ , ϕ ) c s t 0 , ( θ , ϕ ) c s t ( θ , ϕ ) c s t r r s S H ( θ , ϕ + π ) H ( 0 ) , ( θ , ϕ ) d a b ( θ , ϕ ) c s r r s S H ( θ , ϕ + π ) H ( 0 ) , ( θ , ϕ ) c s t ( θ , ϕ ) c s r s t r s S H ( θ , ϕ + π ) H ( 0 ) , ( θ , ϕ ) c s t ( θ , ϕ ) c s 0 , ( θ , ϕ ) a b c s ( θ , ϕ ) c s .
H ( 0 ) = 1 c c s H ( θ , ϕ ) d a .
D ( λ ) = 1 - r D S - r r s t c D S 2 r 2 r s c [ 1 + ( r s t c / S ) ] S 2 H ( 0 ) d H ( θ , ϕ + π ) d a - r r s r s t c [ 1 + ( r s t c / S ) ] S 2 H ( 0 ) × c s t H ( θ , ϕ + π ) d a - r r s r s t c S 2 H ( 0 ) c s t H ( θ , ϕ + π ) d a .
D ( θ , ϕ , θ , ϕ ; λ ) = { r S + r r s t c S 2 + r 2 r s S 2 H ( θ , ϕ + π ) c H ( 0 ) + r 2 r s r s t c 2 S 3 H ( 0 ) H ( θ , ϕ + π ) , ( θ , ϕ ) d a b ( θ , ϕ ) d a b c s t r S + r r s t c S 2 , ( θ , ϕ ) a b c s ( θ , ϕ ) d a b c s t r S + r r s t c S 2 + r r s r s t c S 2 H ( θ , ϕ + π ) H ( 0 ) + r r s r s t 2 c 2 S 3 H ( θ , ϕ + π ) H ( 0 ) , ( θ , ϕ ) c s t ( θ , ϕ ) d a b c s t r S + r 2 r s c S 2 H ( θ , ϕ + π ) H ( 0 ) + r 2 r s r s t c 2 S 3 H ( θ , ϕ + π ) H ( 0 ) - r 2 r s r s t S 3 H ( 0 ) c c s t H ( θ 1 , ϕ 1 + π ) d a 1 , ( θ , ϕ ) c s t ( θ , ϕ ) d a b c s t 0 , ( θ , ϕ ) S ( θ , ϕ ) a b r s t S + r r s r s t c S 2 H ( 0 ) H ( θ , ϕ + π ) , ( θ , ϕ ) d a b ( θ , ϕ ) c s t r s t S , ( θ , ϕ ) a b c s ( θ , ϕ ) c s t r s t S + r s r s t 2 S 2 H ( 0 ) c H ( θ , ϕ + π ) ( θ , ϕ ) c s t ( θ , ϕ ) c s t r r s t S 2 D + r r s r s t S 2 H ( 0 ) c H ( θ , ϕ + π ) + r 2 r s r s t S 3 H ( 0 ) c d H ( θ 1 , ϕ 1 + π ) d a 1 + 2 r r s r s t 2 S 3 H ( 0 ) c c s t H ( θ 1 , θ 1 + π ) d a 1 , ( θ , ϕ ) c s t ( θ , ϕ ) c s t r r s S H ( 0 ) H ( θ , ϕ + π ) - r 2 r s S 2 H ( 0 ) D H ( θ , ϕ + π ) + r 2 r s S 2 H ( 0 ) d H ( θ 1 , ϕ 1 + π ) d a 1 + r r s r s t S 3 H ( 0 ) c s t H ( θ 1 , ϕ 1 + π ) d a 1 + r r s r s t S 2 H ( 0 ) c s t H ( θ 1 , ϕ 1 + π ) d a 1 + r 2 r s r s t S 3 H ( 0 ) c d H ( θ 1 , ϕ 1 + π ) d a 1 + r r s r s t 2 S 3 H ( 0 ) c c s t H ( θ 1 , ϕ 1 + π ) d a 1 - r 2 r s r s t S 3 H ( 0 ) c D H ( θ , ϕ + π ) , ( θ , ϕ ) d a b ( θ , ϕ ) c s r r s S H ( 0 ) H ( θ , ϕ + π ) - r 2 r s S 2 H ( 0 ) D H ( θ , ϕ + π ) + r 2 r s S 2 H ( 0 ) d H ( θ 1 , ϕ 1 + π ) d a 1 + r r s r s t S 2 H ( 0 ) c s t H ( θ 1 , ϕ 1 + π ) d a 1 + r 2 r s r s t S 3 H ( 0 ) D c s t H ( θ 1 , ϕ 1 + π ) d a 1 - r 2 r s r s t S 3 H ( 0 ) c D H ( θ , ϕ + π ) , ( θ , ϕ ) c s t ( θ , ϕ ) c s r s r s t S H ( 0 ) H ( θ , ϕ + π ) - r s r s t r S 2 H ( 0 ) D H ( θ , ϕ + π ) + r 2 r s S 2 H ( 0 ) d H ( θ 1 , ϕ 1 + π ) d a 1 + r r s r s t S 2 H ( 0 ) c s t H ( θ 1 , ϕ 1 + π ) d a 1 + r r s r s t S 2 H ( 0 ) c s t H ( θ 1 , ϕ 1 + π ) d a 1 + r 2 r s r s t S 3 H ( 0 ) c d H ( θ 1 , ϕ 1 + π ) d a 1 + r r s r s t 2 S 3 H ( 0 ) c × c s t H ( θ 1 , ϕ 1 + π ) d a 1 - r r s r s t 2 S 3 H ( 0 ) c D H ( θ , ϕ + π ) , ( θ , ϕ ) , c s t ( θ , ϕ ) c s r 2 r s S 2 H ( 0 ) d H ( θ 1 , ϕ 1 + π ) d a 1 + r r s r s t S 2 H ( 0 ) c s t H ( θ 1 , ϕ 1 + π ) d a 1 + r r s r s t S 2 H ( 0 ) c s t H ( θ 1 , ϕ 1 + π ) d a 1 + r 2 r s r s t S 3 H ( 0 ) c d H ( θ 1 , ϕ 1 + π ) d a 1 + r r s r s t 2 S 3 H ( 0 ) c c s t H ( θ 1 , ϕ 1 + π ) d a 1 , ( θ , ϕ ) a b c s ( θ , ϕ ) c s .
f ( θ , ϕ ) = { P r s a , ( θ , ϕ ) a 0 , ( θ , ϕ ) a b d c s c s t c s t b . .
H s ( θ , ϕ ) = P r s r S D s ( λ ) { 1 + r s t c S } { 1 + r r s c S H s ( θ , ϕ + π ) H s ( 0 ) } .
f ( θ , ϕ ) = { P r s t S , ( θ , ϕ ) a b d c s c s t a b 0 , ( θ , ϕ ) c s t
H s t ( θ , ϕ ) = P r s t S D s t ( λ ) { 1 + r s t c S } { 1 + r r s c S H s t ( θ , ϕ + π ) H s t ( 0 ) } .
B s = P r r s S D s ( λ ) { 1 + r s t c S } b [ 1 + r r s c S H s ( θ , ϕ + π ) H s ( 0 ) ] d a ,
B s t = P r s t S D s t ( λ ) { 1 + r s t c S } b [ 1 + r r s c S H s t ( θ , ϕ + π ) H s t ( 0 ) ] d a .
d H ( θ , ϕ ) H ( 0 ) d a d .
η = b / S 1 - r d / S - ( r s + r s t ) c / S ,
η = b / S 1 - r + ( r a / S ) + ( r b / S ) + ( 2 r - r s - r s t ) c / S
η r > 0 ,             η ( b / s ) > 0 ,             η ( a / S ) < 0 ,
η ( c / S ) < 0 , [ provided ( 2 r - r s - r s t ) > 0 ] .
α = ( r s t - r s ) c / π 1 + r s c / π .
α ( r s t - r s ) c / S 1 + r s c / S