# Page:TolmanEmission.djvu/5

140
RICHARD C. TOLMAN.

Consider now a source of light approaching the slit with the velocity v. If τ' is the period of the source which now produces a bright line at D and Δt' the time interval between departure from the source of two light impulses

which now arrive simultaneously at D, we evidently have the relation

 ${\displaystyle i\tau '=\Delta t'={\frac {L'-L_{2}+v\Delta t'}{c+v}}+{\frac {L_{2}}{c+v_{3}}}-{\frac {L_{4}}{c+v_{4}}},}$ (4)

where c+v[1] in accordance with the Stewart theory is the velocity of the light before reflection and ${\displaystyle L_{3}}$ and ${\displaystyle L_{4}}$ are the components which must be added to c to give the velocity of light along the paths ${\displaystyle BD=L_{3}}$ and ${\displaystyle CD=L_{4}}$ after its reflection.

According to the Stewart theory ${\displaystyle v_{3}}$ and ${\displaystyle v_{4}}$ will be equal to the components in the direction BD and CD of the velocities of the mirror images of the original source. An idea of the size of these components is most easily obtained graphically. Considering, for example, the point of reflection C as a portion of a plane mirror EF which is tangent to the concave mirror at C, the position of the image ${\displaystyle I_{2}}$ can be found by the usual construction, the line ${\displaystyle AI_{2}}$ connecting source and image being perpendicular to EF and the distances AE and ${\displaystyle EI_{2}}$ equal. Both the original source and the image will evidently be moving towards the point F with the same velocity v. By a similar construction, which has been omitted to avoid confusion, the image ${\displaystyle I_{1}}$ produced by reflection from B is found to be located as shown, and moves also with the velocity v in the direction of the corresponding arrow.

It can be seen from the construction that in the arrangement shown the motion of the image ${\displaystyle I_{1}}$ and the corresponding reflected ray BD are

1. See note I. p. 138.