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  1. Analytic Geometry of Three Dimensions, Fifth Edition, Revised by R. A. P. Rogers. J, Chapter X. London, 1912.
  2. F. Becke, Denkschriften d. k. Akad. d. Wissen zu Wien, Math.-Wissensch. Kl. 75, p. 68; 1904. A. Beer in 1852 considered briefly in his Einleitung in die höhere Optik analogous cones and called them Geschwindigkeitskegel erster und zweiter Art. He bases his conclusions, however, on equations analogous to 15a and 15b and not to 3 and 11.
  3. For a summary of these methods of computation, which are standard, see F. E. Wright, The transmission of light, etc.Am. J. Sci., (4) 31, pp. 157–211; 1911.
    [CrossRef]
  4. See F. E. Wright, The methods of petrographic microscope research. Carnegie Institution of Washington, Pub. 158, pp. 75–76; 1911.
  5. Denkschriften der K. K. d. Wissenschaften. Wien. Math.-Naturwissen. Klasse,  70; 1904.
  6. Am. J. Sci. (4) 24, p. 336; 1907.
  7. Tschermak’s Min. Petrogr. Mitteil. N. F. 28, pp. 290–293; 1909.
  8. Am. J. Sci.,  4, pp. 157–211; 1911.
  9. Fortschritte der Mineralogie,  3, pp. 141–158; 1913.
  10. Uber die Becke-Wright’sche Streitfrage. ZS. f. Krystallogr. 54, pp. 113–119; 1914.
  11. Am. J. Sci., (4),  31, pp. 186–197; 1911.
  12. Mineralogical Magazine,  16, pp. 347–351; 1913.

1914 (1)

Uber die Becke-Wright’sche Streitfrage. ZS. f. Krystallogr. 54, pp. 113–119; 1914.

1913 (2)

Fortschritte der Mineralogie,  3, pp. 141–158; 1913.

Mineralogical Magazine,  16, pp. 347–351; 1913.

1911 (4)

Am. J. Sci., (4),  31, pp. 186–197; 1911.

Am. J. Sci.,  4, pp. 157–211; 1911.

For a summary of these methods of computation, which are standard, see F. E. Wright, The transmission of light, etc.Am. J. Sci., (4) 31, pp. 157–211; 1911.
[CrossRef]

See F. E. Wright, The methods of petrographic microscope research. Carnegie Institution of Washington, Pub. 158, pp. 75–76; 1911.

1909 (1)

Tschermak’s Min. Petrogr. Mitteil. N. F. 28, pp. 290–293; 1909.

1907 (1)

Am. J. Sci. (4) 24, p. 336; 1907.

1904 (2)

F. Becke, Denkschriften d. k. Akad. d. Wissen zu Wien, Math.-Wissensch. Kl. 75, p. 68; 1904. A. Beer in 1852 considered briefly in his Einleitung in die höhere Optik analogous cones and called them Geschwindigkeitskegel erster und zweiter Art. He bases his conclusions, however, on equations analogous to 15a and 15b and not to 3 and 11.

Denkschriften der K. K. d. Wissenschaften. Wien. Math.-Naturwissen. Klasse,  70; 1904.

Becke, F.

F. Becke, Denkschriften d. k. Akad. d. Wissen zu Wien, Math.-Wissensch. Kl. 75, p. 68; 1904. A. Beer in 1852 considered briefly in his Einleitung in die höhere Optik analogous cones and called them Geschwindigkeitskegel erster und zweiter Art. He bases his conclusions, however, on equations analogous to 15a and 15b and not to 3 and 11.

Wright, F. E.

For a summary of these methods of computation, which are standard, see F. E. Wright, The transmission of light, etc.Am. J. Sci., (4) 31, pp. 157–211; 1911.
[CrossRef]

See F. E. Wright, The methods of petrographic microscope research. Carnegie Institution of Washington, Pub. 158, pp. 75–76; 1911.

Am. J. Sci. (1)

Am. J. Sci.,  4, pp. 157–211; 1911.

Am. J. Sci. (4) (1)

Am. J. Sci. (4) 24, p. 336; 1907.

Am. J. Sci., (4) (2)

For a summary of these methods of computation, which are standard, see F. E. Wright, The transmission of light, etc.Am. J. Sci., (4) 31, pp. 157–211; 1911.
[CrossRef]

Am. J. Sci., (4),  31, pp. 186–197; 1911.

Carnegie Institution of Washington, Pub. (1)

See F. E. Wright, The methods of petrographic microscope research. Carnegie Institution of Washington, Pub. 158, pp. 75–76; 1911.

Denkschriften d. k. Akad. d. Wissen zu Wien, Math.-Wissensch. Kl. (1)

F. Becke, Denkschriften d. k. Akad. d. Wissen zu Wien, Math.-Wissensch. Kl. 75, p. 68; 1904. A. Beer in 1852 considered briefly in his Einleitung in die höhere Optik analogous cones and called them Geschwindigkeitskegel erster und zweiter Art. He bases his conclusions, however, on equations analogous to 15a and 15b and not to 3 and 11.

Denkschriften der K. K. d. Wissenschaften. Wien. Math.-Naturwissen. Klasse (1)

Denkschriften der K. K. d. Wissenschaften. Wien. Math.-Naturwissen. Klasse,  70; 1904.

Fortschritte der Mineralogie (1)

Fortschritte der Mineralogie,  3, pp. 141–158; 1913.

Mineralogical Magazine (1)

Mineralogical Magazine,  16, pp. 347–351; 1913.

Tschermak’s Min. Petrogr. Mitteil. N. F. (1)

Tschermak’s Min. Petrogr. Mitteil. N. F. 28, pp. 290–293; 1909.

ZS. f. Krystallogr. (1)

Uber die Becke-Wright’sche Streitfrage. ZS. f. Krystallogr. 54, pp. 113–119; 1914.

Other (1)

Analytic Geometry of Three Dimensions, Fifth Edition, Revised by R. A. P. Rogers. J, Chapter X. London, 1912.

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Figures (23)

Fig. 1
Fig. 1

Orthographic projection of the equivibration curves (spheroconics) of a crystal having the principal refractive indices: α=1.580, β=1.5882, γ=1.600. Only half of the projection is shown. The curve labeled 1.592 shows, for example, the positions in space of all radii vectors (vibration directions) of the index ellipsoid of length 1.592. Similarly for the other curves. The contour interval between two successive curves is 5 per cent of the difference, γα.

Fig. 2
Fig. 2

In this figure BEB, BFB are the two cyclic planes; A1, A2, two optic axes; Z the acute bisectrix; ρ, ρ2, the two vibration directions of the diametral plane ρ2a1 ρ1a2, of which P is the pole; ρ1A1=D1 and ρ1A2=D2, perpendiculars let fall from ρ1 on BEB and BFB respectively; PA1=θ1, PA2=θ2, the angular distances of P from A1 and A2 respectively.

Fig. 3
Fig. 3

Upper half: Equirefringence curves in orthographic projection on the xy plane of a crystal having the principal refractive indices α=1.580, β=γ=1.600, 2VZ=180°. The curve, 1.586, gives, for example, the position in space of all directions of propagation having the refractive index 1.586; similarly for the other curves. The contour interval between any two successive curves is 5 per cent of γα. Lower half: Equibirefringence curves for same crystal in the same projection. The curve, .012, for example, passes through all directions within the crystal for which the birefringence is .012; likewise the remaining curves. The interval between any two successive curves is 5 per cent of the total birefringence, γα. The letters α, β, γ indicate the directions of the principal axes of the index ellipsoid.

Fig. 4
Fig. 4

Left half: Equirefringence curves, projected as in Fig. 3, of a crystal having α=1.580, β=1.5993, γ=1.600, 2VZ=160°. Fig. 4A. Equibirefringence curves for same crystal and in the same projection.

Fig. 5
Fig. 5

Equirefringence curves in orthographic projection as in Fig. 3 for a crystal having α=1.580, β=1.5976, γ=1.600, 2VZ=140°. Fig. 5A. Equibirefringence curves for same crystal and in same projection.

Fig. 6
Fig. 6

Equirefringence curves in orthographic projection as in Fig. 3 for a crystal for which α=1.580, β=1.5949, γ=1.600, 2VZ=120°. Fig. 6A. Equibirefringence curves for same crystal and in same projection.

Fig. 7
Fig. 7

Equirefringence curves in orthographic projection as in Fig. 3 for a crystal for which α=1.580, β=1.5916, γ=1.600, 2VZ=100°. Fig. 7A. Equibirefringence curves for same crystal and in same projection.

Fig. 8
Fig. 8

Equirefringence curves in orthographic projection as in Fig. 3 for a crystal for which α=1.580, β=1.882, γ=1.600, 2VZ=80°. Fig. 8A. Equibirefringence curves for same crystal and in same projection.

Fig. 9
Fig. 9

Equirefringence curves in orthographic projection, as in Fig. 3, for a crystal for which α=1.580, β=1.5849, γ=1.600, 2VZ=60°. Fig. 9A. Equibirefringence curves for same crystal and in same projection.

Fig. 10
Fig. 10

Equirefringence curves in orthographic projection, as in Fig. 3, for a crystal having refractive indices α=1.580, β=1.5823, γ=1.600, 2VZ=40°. Fig. 10A. Equibirefringence curves for same crystal and in same projection.

Fig. 11
Fig. 11

Equirefringence curves in orthographic projection, as in Fig. 3, for a crysal for which α=1.580, β=1.5806, γ=1.600, 2VZ=20°. Fig. 11A. Equibirefringence curves for tsame crystal and in same projection.

Fig. 12
Fig. 12

Upper half. Equirefringence curves in orthographic projection, as in Fig. 3, for a crystal for which α=β=1.5800, γ=1.600, 2VZ=0°. Lower half. Corresponding equibirefringence curves in the same projection.

Fig. 13
Fig. 13

Plot for converting graphically directions observed in an interference figure from a crystal plate of average refractive index n=1.59 and weak to medium birefringence, to corresponding directions within the crystal plate or vice versa. The radii vectors in the S.E. quadrant are n times the lengths of the corresponding radii vectors in the N.W. quadrant.

Fig. 14
Fig. 14

Graphical plot for correcting for change in length of path of waves transmitted through a crystal plate at an angle with the normal to the plate. In the N.E. quadrant the percentage increase in length of path for a given inclination whose angular value is shown in the S.W. quadrant.

Fig. 15
Fig. 15

Diagram to illustrate the change in azimuth angle of the plane of vibration of a plane polarized beam of light on entering an isotropic substance LMML at the plane surface MOM′. Fig. 15A. The geometrical relations of Fig. 15 represented in stereographic projection.

Fig. 16
Fig. 16

A. Top view and B, side view of a liquid prism of variable angle. M; segment of a brass cylinder, 6 cm diameter and 4 cm high. The ends are sealed by brass disks and out of the brass box thus formed two segments are cut off. One of these openings is again sealed by cementing over it a plane parallel piece of optical glass, as indicated in Fig. 16.I. H is a brass cylinder soldered to M; it slips over the central bearing of the spectrometer and serves as support for M. The liquid with horizontal surface is shown at L.

Fig. 17
Fig. 17

Orthographic projection of the computed (equations 21, 21a) azimuth angles which the planes of vibration make with the plane of incidence after refraction at a single surface (lower half of Fig. 17) and after refraction at two plane parallel surfaces (upper half of Fig. 17) as encountered on transmission through an isotropic glass plate. In the figure the straight E-W and N-S lines are small circle coordinates spaced 10° apart. The initial plane of vibration is represented by the N-S line. It should be noted that the distances from the pole (center of projection) represent the angles (i-r).

Fig. 18
Fig. 18

Curves in orthographic Projection representing data of observation of Table 4 on the rotation of the plane of vibration of a beam of plane polarized light in mercury light (546μμ) on transmission through the condenser (aplanatic) and objective (4 mm apochromatic) of the microscope (N.W.) quadrant; through the above lens system plus an intervening cover glass mounted with Canada balsam on an object glass (S.W. quadrant); through lens system plus 6 unmounted cover glasses between objective and condenser (S.E. quadrant); through lens system plus 12 unmounted cover glasses (N.E. quadrant). The slope angles in the projection are the angles of inclination of the points observed in the conoscopic field.

Fig. 19
Fig. 19

Diagram showing in stereographic projection the planes of vibration of a dark point, A, in the interference figure and the corresponding beam, B, within the crystal plate. OC is the plane of vibration of the polarizer, AHC the plane of vibration of the dark point A, and BHD the plane of vibration of the corresponding beam within the crystal plate.

Fig. 20
Fig. 20

Diagram illustrating Professor Becke’s method for ascertaining the direction of vibration B of a dark point P in the interference figure. The great circle PB in stereographic projection is tangent to the dark point P. B is on the great circle KG polar to P. B is also at the intersection of the polar circle KG and the straight line PC; OC is the plane of vibration of the polarizer.

Fig. 21
Fig. 21

Diagram illustrating the writer’s first method for ascertaining the direction of vibration of the dark point P. This direction is represented by G, at the intersection of the plane of vibration, OC, of the polarizer and the great circle KB, polar to P. This point, however, is the point of vibration for a wave traveling in reverse direction, namely, from the observer down through the crystal plate. The point H at the intersection of the polar circle PC and the polar circle KB is the direction of vibration for waves which travel toward the observer from the crystal plate. H is also at the intersection of the polar plane KB and the straight line from P to Q, where Q is the end of the radius OBQ.

Fig. 22
Fig. 22

Diagram illustrating the step-down effect of the microscope lens system on the rotation of the plane of vibration on a transmitted beam of plane polarized light. OC is the plane of vibration of the polarizer. A and A′ observed dark points in an interference figure. On the left side of the figure the inclination A is supposed to have been reached by a succession of 5 refractions of 10° each (ir=10°); on the right side of the figure the slope angle A′ is supposed to have been reached by 10 successive refractions of 5° each (ir=5°). By the process the direction of vibration for A is found to be at L, for A′ at L′, whereas for a single refraction of 50° (ir=50°) H and H′ are found to be the two vibration directions; D and D′ are the vibration directions as determined by the Becke method which presupposes an infinite number of very small refractions. The arc LD is greater than the arc LD′ corresponding to the smaller number of refraction. By this method the vibration directions for B and B′ are indicated by the points M and M′ respectively. For the case of a single refraction they are at N and N′.

Fig. 23
Fig. 23

New method proposed for ascertaining the vibration direction of a dark point P in the interference figure. OC in stereographic projection is the plane of vibration of polarizer, GH polar circle to P; B is the intersection of the straight line PC with GH; H is the intersection of the polar circle GH and the straight line from P to Q at the end of the radius OB. The desired point N is then somewhere between B and H and, as a first approximation midway between them. In the case of a number of refractions (apochromatic objective) it is nearer B than H.

Tables (5)

Tables Icon

Table 1 Rotation of vibration plane of beam of light on refraction at a single surface of glass. n=1.5087; i-r=30°; ϵ=azimuth of vibration plane of incident wave; δ=azimuth of vibration plane of refracted wave.

Tables Icon

Table 2 In this table i-r is the angle between the incident and the refracted beam; ϵ is the azimuth angle of the plane of vibration of the incident beam; δ, that of the beam after refraction at a single surface; δ′, that after transmission through a plane parallel plate.

Tables Icon

Table 3 In this table ϵ is the azimuth angle of the plane of vibration of a beam incident on a glass plate (n=1.51) under an angle, i=51°24′; δ′, the corresponding azimuth angle after transmission through the glass plate; δR, the azimuth angle for the twice reflected, emergent beam.

Tables Icon

Table 4 1 Observed angular rotations of the plane of vibration for different points in the conoscopic field of the microscope for : I. Lens system (aplanatic condenser and 4 mm objective) alone; II. Lens system+mounted cover glass; III. Lens system +6 unmounted cover glasses; IV. Lens system +12 unmounted cover glasses.

Tables Icon

Table 5 In this table the angles of maximum rotation of the plane of vibration for different slope-difference angles, i-r and for both incident and refracted beams on refraction at a single surface (ϵ and δ respectively) and after transmission through a plane parallel plate (angles ϵ′ and δ′).

Equations (51)

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x 2 α 2 + y 2 β 2 + z 2 γ 2 = 1
x 2 ρ 2 + y 2 ρ 2 + z 2 ρ 2 = 1
x 2 ( 1 α 2 - 1 ρ 2 ) + y 2 ( 1 β 2 - 1 ρ 2 ) + z 2 ( 1 γ 2 - 1 ρ 2 ) = 0
x 2 ( 1 α 2 - 1 β 2 ) - z 2 ( 1 β 2 - 1 γ 2 ) = 0
( x 1 α 2 - 1 β 2 + z 1 β 2 - 1 γ 2 ) ( x 1 α 2 - 1 β 2 - z 1 β 2 - 1 γ 2 ) = 0
( l x + n z ) ( l x - n z ) = 0
l = 1 α 2 - 1 β 2 1 α 2 - 1 γ 2 ; n = 1 β 2 - 1 γ 2 1 α 2 - 1 γ 2
sin 2 V z = 1 α 2 - 1 β 2 1 α 2 - 1 γ 2 ;             cos 2 V z = 1 β 2 - 1 γ 2 1 α 2 - 1 γ 2 ;             tan 2 V z = 1 α 2 - 1 β 2 1 β 2 - 1 γ 2
x x ( 1 α 2 - 1 ρ 2 ) + y y ( 1 β 2 - 1 ρ 2 ) + z z ( 1 γ 2 - 1 ρ 2 ) = 0
l x + m y + n z = 0
l = k · x ( 1 α 2 - 1 ρ 2 ) ;             m = k y ( 1 β 2 - 1 ρ 2 ) ;             n = k · z ( 1 γ 2 - 1 ρ 2 )
l 1 α 2 - 1 ρ 2 = k · x 1 α 2 - 1 ρ 2 ;             m 1 β 2 - 1 ρ 2 = k · y 1 β 2 - 1 ρ 2 ; n 1 γ 2 - 1 ρ 2 = k · z 1 γ 2 - 1 ρ 2
l 2 1 α 2 - 1 ρ 2 + m 2 1 β 2 - 1 ρ 2 + n 2 1 γ 2 - 1 ρ 2 = 0
1 ρ 4 - 1 ρ 2 ( l 2 + m 2 γ 2 + m 2 + n 2 α 2 + n 2 + l 2 β 2 ) + α 2 l 2 + β 2 m 2 + γ 2 n 2 α 2 β 2 γ 2 = 0
1 ρ 1 2 + 1 ρ 2 2 = l 2 + m 2 γ 2 + m 2 + n 2 α 2 + n 2 + l 2 β 2
1 ρ 1 2 × 1 ρ 2 2 = α 2 l 2 + β 2 m 2 + γ 2 n 2 α 2 β 2 γ 2
x 2 1 α 2 - 1 ρ 2 + y 2 1 β 2 - 1 ρ 2 + z 2 1 γ 2 - 1 ρ 2 = 0
x 2 + y 2 + z 2 = x 2 1 α - 1 β 2 1 α 2 - 1 ρ 2 + z 2 1 γ 2 - 1 β 2 1 γ 2 - 1 ρ 2 = 1
= y 2 1 β 2 - 1 γ 2 1 β 2 - 1 ρ 2 + x 2 1 α 2 - 1 γ 2 1 α 2 - 1 ρ 2 = 1
= z 2 1 γ 2 - 1 α 2 1 γ 2 - 1 ρ 2 + y 2 1 β 2 - 1 α 2 1 β 2 - 1 ρ 2 = 1
x 2 + y 2 + z 2 = x 2 1 β 2 - 1 α 2 1 β 2 - 1 ρ 2 + z 2 1 β 2 - 1 γ 2 1 β 2 - 1 ρ 2 = 1
= y 2 1 γ 2 - 1 β 2 1 γ 2 - 1 ρ 2 + x 2 1 γ 2 - 1 α 2 1 γ 2 - 1 ρ 2 = 1
= z 2 1 α 2 - 1 γ 2 1 α 2 - 1 ρ 2 + y 2 1 α 2 - 1 β 2 1 α 2 - 1 ρ 2 = 1
( x 1 α 2 - 1 β 2 1 α 2 - 1 γ 2 + z 1 β 2 - 1 γ 2 1 α 2 - 1 γ 2 ) ( x 1 α 2 - 1 β 2 1 α 2 - 1 γ 2 - z 1 β 2 - 1 γ 2 1 α 2 - 1 γ 2 ) = ρ 2 1 ρ 2 - 1 β 2 1 α 2 - 1 γ 2
1 ρ 2 - 1 β 2 = ( 1 α 2 - 1 γ 2 ) sin D 1 sin D 2
sin D 1 = sin ρ 1 a 1 sin θ 1 sin D 2 = - sin ρ 1 a 2 sin θ 2 Therefore sin D 1 · sin D 2 = - sin 2 ρ 1 a 1 sin θ 1 sin θ 2 = 1 ρ 2 - 1 β 2 1 α 2 - 1 γ 2
cos 2 1 2 A 1 P A 2 · sin θ 1 · sin θ 2 = 1 β 2 - 1 ρ 2 1 α 2 - 1 γ 2
cos 2 1 2 A 1 P A 2 sin θ 1 sin θ 2 = sin S sin ( S - V z ) = sin 2 θ 2 + θ 1 2 - sin 2 V z = sin 2 θ 2 + θ 1 2 - 1 α 2 - 1 β 2 1 α 2 - 1 γ 2
1 α 2 - 1 γ 2 = ( 1 α 2 - 1 γ 2 ) sin 2 θ 2 + θ 1 2 , or 1 γ 2 - 1 γ 2 = ( 1 α 2 - 1 γ 2 ) cos 2 θ 2 + θ 1 2
1 α 2 - 1 α 2 = ( 1 α 2 - 1 γ 2 ) sin 2 θ 2 - θ 1 2 , or 1 α 2 - 1 γ 2 = ( 1 α 2 - 1 γ 2 ) cos 2 θ 2 - θ 1 2
1 α 2 - 1 γ 2 = ( 1 α 2 - 1 γ 2 ) sin θ 1 · sin θ 2
γ - α = ( γ - α ) sin θ 1 sin θ 2
x 2 sin 2 1 2 ( θ 2 + θ 1 ) + y 2 cos 2 V z - cos 2 1 2 ( θ 2 + θ 1 ) - z 2 cos 2 1 2 ( θ 2 + θ 1 ) = 0
x 2 sin 2 1 2 ( θ 2 - θ 1 ) + y 2 cos 2 V z - cos 2 1 2 ( θ 2 - θ 1 ) - z 2 cos 2 1 2 ( θ 2 - θ 1 ) = 0
x 2 sin 2 V z cos 2 1 2 ( θ 2 + θ 1 ) + z 2 · cos 2 V z sin 2 1 2 ( θ 2 + θ 1 ) = 1
y 2 cos 2 V z cos 2 V z - sin 2 1 2 ( θ 2 + θ 1 ) + x 2 1 cos 2 1 2 ( θ 2 + θ 1 ) = 1
z 2 1 sin 1 2 ( θ 2 + θ 1 ) + y 2 sin 2 V z sin 2 V z - cos 2 1 2 ( θ 2 + θ 1 ) = 1
x 2 · sin 2 V z cos 2 1 2 ( θ 2 - θ 1 ) + z 2 cos 2 V z sin 2 1 2 ( θ 2 - θ 1 ) = 1
y 2 cos 2 V z cos 2 V z - sin 2 1 2 ( θ 2 - θ 1 ) + x 2 1 cos 2 1 2 ( θ 2 - θ 1 ) = 1
z 2 1 sin 2 1 2 ( θ 2 - θ 1 ) + y 2 sin 2 V z sin 2 V z - cos 2 1 2 ( θ 2 - θ 1 ) = 1
K - d 2 C 2 - 2 C = 0 or             [ α 2 x 2 ( β 2 + γ 2 - d 2 ) + β 2 γ 2 ( γ 2 + α 2 - d 2 ) + γ 2 z 2 ( α 2 + β 2 - d 2 ) ] 2 - 4 α 2 β 2 γ 2 ( α 2 x 2 + β 2 y 2 + γ 2 z 2 ) = 0
d 2 = K - 2 C C 2 or d 2 = α 2 β 2 ( l 2 + m 2 ) + β 2 γ 2 ( m 2 + n 2 ) + γ 2 α 2 ( n 2 + l 2 ) - 2 α β γ α 2 l 2 + β 2 m 2 + γ 2 n 2 α 2 l 2 + β 2 m 2 + γ 2 n 2
1 = sec α
tan δ = cos ( i - r ) · tan ϵ
tan ( ϵ - δ ) = 1 - cos ( i - r ) 2 cos ( i - r )
tan δ R = cos 2 ( i - r ) · cos 2 2 r · tan ϵ
tan δ n = cos 2 n ( i - r ) tan ϵ
tan δ = cos 2 ( i - r ) · tan ϵ
tan δ n = cos 2 n i - r n · tan ϵ
tan δ N = cos A . cos B , cos C , . . . . cos N . tan ϵ
cot δ h = cos ( i - r ) cot δ h ± sin 2 r h tan s h sin ( i + r h ) sin ϵ h