If, now, the eye which watches this process is kept from moving, these relations will be reproduced on the retina.  For the retinal area corresponding to the triangle p^{A}Op^{B}, there will be less stimulation from the sector A than there would have been if the pendulum had not partly hidden it.  That is, the triangle in question will not be seen of the fused color of A and B, but will lose a part of its A-component.  In the same way the triangle p^{B}OpC will lose a part of its B-component; and so on alternately.  And by as much as either component is lost, by so much will the color of the intercepting pendulum (in this case, black) be present to make up the deficiency.

We see, then, that the purely geometrical relations of disc and pendulum necessarily involve for vision a certain banded appearance of the area which is swept by the pendulum, if the eye is held at rest.  We have now to ask, Are these the bands which we set out to study?  Clearly enough these geometrically inevitable bands can be exactly calculated, and their necessary changes formulated for any given change in the speed or width of A, B, or P.  If it can be shown that they must always vary just as the bands we set out to study are observed to vary, it will be certain that the bands of the illusion have no other cause than the interception of retinal stimulation by the sectors of the disc, due to the purely geometrical relations between the sectors and the pendulum which hides them.

And exactly this will be found to be the case.  The widths of the bands of the illusion depend on the speed and widths of the sectors and of the pendulum used; the colors and intensities of the bands depend on the colors and intensities of the sectors (and of the pendulum); while the total number of bands seen at one time depends on all these factors.

V. GEOMETRICAL DEDUCTION OF THE BANDS.

In the first place, it is to be noted that if the pendulum proceeds from left to right, for instance, before the disc, that portion of the latter which lies in front of the advancing rod will as yet not have been hidden by it, and will therefore be seen of the unmodified, fused color.  Only behind the pendulum, where rotating sectors have been hidden, can the bands appear.  And this accords with the first observation (p. 167), that “The rod appears to leave behind it on the disc a number of parallel bands.”  It is as if the rod, as it passes, painted them on the disc.

Clearly the bands are not formed simultaneously, but one after another as the pendulum passes through successive positions.  And of course the newest bands are those which lie immediately behind the pendulum.  It must now be asked, Why, if these bands are produced successively, are they seen simultaneously?  To this, Jastrow and Moorehouse have given the answer, “We are dealing with the phenomena of after-images.”