Abstract

A ray of light is propagated in the XY plane and its angle with the X-axis is determined by two observers, S and S′, who are in relative motion to each other along the X-axis at a velocity q (expressed as a fraction of the velocity of light). If the angles measured by them are δ and δ′, then it is proved that tan12δ/tan12δ=cos(45°+12α)/cos(45°-12α), where Sin α=q. This relationship is interpreted geometrically by means of a cone in oblique coordinates. The axes of coordinates are X, Y, and time T. The T and X axes are different for the two observers, and the four axes T, X, T′, X′, form a Lorentzian Plane to which Y is perpendicular. The properties of the cone are considered in detail and a mechanical model of the cone, built of wood and iron, is described. At a desired velocity q, one can compare directly the angles δ and δ′. It is also shown that the relationship between the angles δ, δ′, and α can be represented on a pyramid. References are made to other articles by the same writer, in which oblique axes are also used for the solution of problems in the theory of relativity.

© 1924 Optical Society of America

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References

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  1. For a popular description of this model see Science and Invention,  11, p. 442; 1923.
  2. The Physical Review,  23, p. 239; 1924.
    [Crossref]
  3. See, for example, O. W. Richardson, The Electron Theory of Matter, p. 300; 1916.
  4. See, for example, Snyder and Sisam, Analytic Geometry of Space, index under “Cone.”
  5. The foregoing proof is, of course, but a particular case of the invariancy of the expression x2+y2+z2−c2t2 when z=0.
  6. See, for example, Seaver’s Mathematical Handbook, p. 143, formulae 1342.
  7. See, for example, Richardson, loc. cit., p. 306, eq. (14).

1924 (1)

The Physical Review,  23, p. 239; 1924.
[Crossref]

1923 (1)

For a popular description of this model see Science and Invention,  11, p. 442; 1923.

1916 (1)

See, for example, O. W. Richardson, The Electron Theory of Matter, p. 300; 1916.

Richardson,

See, for example, Richardson, loc. cit., p. 306, eq. (14).

Richardson, O. W.

See, for example, O. W. Richardson, The Electron Theory of Matter, p. 300; 1916.

Sisam,

See, for example, Snyder and Sisam, Analytic Geometry of Space, index under “Cone.”

Snyder,

See, for example, Snyder and Sisam, Analytic Geometry of Space, index under “Cone.”

Science and Invention (1)

For a popular description of this model see Science and Invention,  11, p. 442; 1923.

The Electron Theory of Matter (1)

See, for example, O. W. Richardson, The Electron Theory of Matter, p. 300; 1916.

The Physical Review (1)

The Physical Review,  23, p. 239; 1924.
[Crossref]

Other (4)

See, for example, Snyder and Sisam, Analytic Geometry of Space, index under “Cone.”

The foregoing proof is, of course, but a particular case of the invariancy of the expression x2+y2+z2−c2t2 when z=0.

See, for example, Seaver’s Mathematical Handbook, p. 143, formulae 1342.

See, for example, Richardson, loc. cit., p. 306, eq. (14).

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Figures (6)

Fig. 1
Fig. 1

Te problem of aberration of light.

Fig. 2
Fig. 2

The Lorentzian Plane.

Fig. 3
Fig. 3

The bisector cone.

Fig. 4
Fig. 4

The angles which determine the direction of a ray of light.

Fig. 5
Fig. 5

The aberration pyramid.

Fig. 6
Fig. 6

A mechanical model of aberration of light.

Equations (48)

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t = β [ t + ( q / c 2 ) x ]
x = β ( x + q t )
y = y
β = c / ( c 2 - q 2 ) 0.5
sin α = q
t cos α = t + x sin a
x cos α = x + t sin α
E A = C A + E C
O L = O C + C L
x 2 + y 2 = t .
r = c t .
x 2 + y 2 = t 2
x 2 + y 2 = t 2
x sin 2 γ = b sin γ - n cos γ
t sin 2 γ = b sin γ + n cos γ
γ = 45 ° + 1 2 α
γ = 45 ° - 1 2 α
y 2 sin 2 γ = 2 b n
y 2 cos α = 2 b n
b = ( n + b ) / 2
n = ( n - b ) / 2
y 2 cos α + b 2 = n 2
cos ( 180 ° - 2 γ δ ) = cos ( 180 ° - 2 γ ) cos δ
cos 2 γ δ = cos 2 γ cos δ
γ δ = 45 ° + 1 2 α δ
sin α δ = sin α cos δ
1 / sin ( γ + θ ) = cos δ / sin ( γ - θ ) = O H / sin 2 γ
cos δ sin ( γ - θ ) / sin ( γ + θ )
O H = sin δ / tan θ y
sin δ = cos α tan θ y / sin ( γ + θ )
cos δ = ( sin γ cos θ - cos γ sin θ ) / ( sin γ cos θ + cos γ sin θ )
tan θ = tan γ ( 1 - cos δ ) / ( 1 + cos δ ) = tan γ tan 2 1 2 δ
tan θ = tan γ ( 1 - cos δ ) / ( 1 + cos δ ) = tan γ tan 2 1 2 δ
tan 1 2 δ / tan 1 2 δ = cos γ / cos γ
tan γ ( 1 - cos δ ) / ( 1 + cos δ ) = tan γ ( 1 - cos δ ) / ( 1 + cos δ )
tan γ = tan ( 45 ° + 1 2 α ) = ( 1 + sin α ) / ( 1 - sin α )
( 1 + sin α ) ( 1 - cos δ ) / ( 1 + cos δ ) = ( 1 - sin α ) ( 1 - cos δ ) / ( 1 + cos δ )
cos δ = ( cos δ - sin α ) / ( 1 - sin α cos δ )
cos δ = ( cos δ - q ) / ( 1 - q cos δ )
tan 1 2 δ = D C / O D ;
tan 1 2 δ = D C / O D ;
O D = O A / cos γ ;
O D = O A / cos γ ;
D C = D C
tan 1 2 ( δ + Δ δ ) tan 1 2 δ = tan ( 45 ° + 1 2 α ) tan 45 °
tan 1 2 ( δ + Δ δ ) - tan 1 2 δ 1 2 Δ δ · 1 1 2 α tan 1 2 δ = tan ( 45 ° + 1 2 α ) - tan 45 ° 1 2 α · 1 1 2 Δ δ tan 45 °
1 2 α tan 1 2 δ cos 2 1 2 δ = 1 2 Δ δ tan 45 ° cos 2 45 ° ,
Δ δ = q sin δ