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.]