7.  The bands are more strikingly visible when the two sectors differ considerably in luminosity.  But Jastrow’s observation, that a difference in luminosity is necessary, could not be confirmed.  Rather, on the contrary, sectors of the closest obtainable luminosity still yielded the illusion, although faintly.

8.  A broad but slowly moving rod shows the bands overlying itself.  Other bands can be seen left behind it on the disc.

9.  But a case of a rod which is broad, or slowly-moving, or both, is a special complication which involves several other and seemingly quite contradictory phenomena to those already noted.  Since these suffice to show the principles by which the illusion is to be explained, enumeration of the special variations is deferred.

IV.  THE GEOMETRICAL RELATIONS BETWEEN THE ROD AND THE SECTORS OF THE DISC.

It should seem that any attempt to explain the illusion-bands ought to begin with a consideration of the purely geometrical relations holding between the slowly-moving rod and the swiftly-revolving disc.  First of all, then, it is evident that the rod lies in front of each sector successively.

Let Fig. 1 represent the upper portion of a color-wheel, with center at O, and with equal sectors A and B, in front of which a rod P oscillates to right and left on the same axis as that of the wheel.  Let the disc rotate clockwise, and let P be observed in its rightward oscillation.  Since the disc moves faster than the rod, the front of the sector A will at some point come up to and pass behind the rod P, say at p^{A}.  P now hides a part of A and both are moving in the same direction.  Since the disc still moves the faster, the front of A will presently emerge from behind P, then more and more of A will emerge, until finally no part of it is hidden by P.  If, now, P were merely a line (having no width) and were not moving, the last of A would emerge just where its front edge had gone behind P, namely at p^{A}.  But P has a certain width and a certain rate of motion, so that A will wholly emerge from behind P at some point to the right, say p^{B}.  How far to the right this will be depends on the speed and width of A, and on the speed and width of P.

Now, similarly, at p^{B} the sector B has come around and begins to pass behind P.  It in turn will emerge at some point to the right, say p^{C}.  And so the process will continue.  From p^{A} to p^{B} the pendulum covers some part of the sector A; from p^{B} to p^{C} some part of sector B; from p^{C} to P^{D} some part of A again, and so on.

[Illustration:  Fig. 1.]