The bands persist as after-images while new ones are being generated.  The very oldest, however, disappear pari-passu with the generation of the new.  We have already seen (p. 169) how well these authors have shown this, in proving that the number of bands seen, multiplied by the rate of rotation of the disc, is a constant bearing some relation to the duration of a retinal image of similar brightness to the bands.  It is to be noted now, however, that as soon as the rod has produced a band and passed on, the after-image of that band on the retina is exposed to the same stimulation from the rotating disc as before, that is, is exposed to the fused color; and this would tend to obliterate the after-images.  Thus the oldest bands would have to disappear more quickly than an unmolested after-image of the same original brightness.  We ought, then, to see somewhat fewer bands than the formula of Jastrow and Moorehouse would indicate.  In other words, we should find on applying the formula that the ’duration of the after-image’ must be decreased by a small amount before the numerical relations would hold.  Since Jastrow and Moorehouse did not determine the relation of the after-image by an independent measurement, their work neither confirms nor refutes this conjecture.

What they failed to emphasize is that the real origin of the bands is not the intermittent appearances of the rod opposite the lighter sector, as they seem to believe, but the successive eclipse by the rod of each sector in turn.

If, in Fig. 2, we have a disc (composed of a green and a red sector) and a pendulum, moving to the right, and if P represents the pendulum at the instant when the green sector AOB is beginning to pass behind it, it follows that some other position farther to the right, as P’, will represent the pendulum just as the last part of the sector is passing out from behind it.  Some part at least of the sector has been hidden during the entire interval in which the pendulum was passing from P to P’.  Clearly the arc BA’ measures the band BOA’, in which the green stimulation from the sector AOB is thus at least partially suppressed, that is, on which a relatively red band is being produced.  If the illusion really depends on the successive eclipse of the sectors by the pendulum, as has been described, it will be possible to express BA’, that is, the width of a band, in terms of the widths and rates of movement of the two sectors and of the pendulum.  This expression will be an equation, and from this it will be possible to derive the phenomena which the bands of the illusion actually present as the speeds of disc and rod, and the widths of sectors and rod, are varied.

[Illustration:  Fig 2.]

Now in Fig. 2 let the
  width of the band (i.e., the arc BA’) = Z
  speed of pendulum = r degrees per second;
  speed of disc = r’ degrees per second;
  width of sector AOB (i.e., the arc AB) = s degrees of arc;
  width of pendulum (i.e., the arc BC) = p degrees of arc;
  time in which the pendulum moves from P to P’ = t seconds.