# Page:TolmanEmission.djvu/3

138
RICHARD C. TOLMAN.
the only difference in length of path occurs, before reflection, i. e., ${\displaystyle AB=L_{1}>AC=L_{2}}$

Consider first a stationary source, and let τ be the period of the source which produces a bright line at D. For the production of such a line, it is evident that light impulses coming over the two paths ABD and ACD must arrive at D in the same phase. If Δt is the time interval between the departures from the source of two light impulses which arrive simultaneously at D, the condition necessary for their arrival in phase is evidently given by the equation

 ${\displaystyle i\tau =\Delta t={\frac {L_{1}-L_{2}}{c}}}$ (I)

where i is a whole number. (Note that ${\displaystyle L_{2}-L_{1}}$ with the apparatus as arranged.)

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 departures from the source of two light impulses which now arrive simultaneously at D we evidently have the relation

${\displaystyle i\tau '=\Delta t'}$.

In order to obtain an expression for Δt' in terms of ${\displaystyle L_{1}}$ and ${\displaystyle L_{2}}$, we must note that the source moves toward the slit the distance vΔt' during the interval of time between the departures of the two light impulses, and hence the difference in path which was ${\displaystyle L_{1}-L_{2}}$ for a stationary source has now become ${\displaystyle L_{1}-L_{2}+v\Delta t'}$. Furthermore we must remember that according to the theory which we are investigating the light before reflection will have the velocity c+v,[1] and hence

 ${\displaystyle i\tau '=\Delta t'={\frac {L_{1}-L_{2}+v\Delta t'}{c+v}},}$ ${\displaystyle i\tau '={\frac {L_{1}-L_{2}}{c}},}$ (2)

which by comparison with equation (i) gives us ${\displaystyle \tau '=\tau }$.

In other words if the first of the above emission theories of light is true, both before and after the source of light is set in motion, light produced by the same period of the source gives a bright line at the point D, that is, the expected Doppler effect or shifting of the lines does not occur.

In interpreting actual experimental results, it must be borne in mind that the adjustment of the grating was assumed to be such that the

1. The slight difference in direction between the rays AB and AC and the motion of the source may be neglected.