From this last equation it is clear that unless s(green) = s(red), W(red) cannot equal W(green). That is, if the two sectors are unequal in width, the bands are also unequal. This was the first feature of the illusion above noted.
Again, if one sector is larger, the oppositely colored bands will be larger, that is, the light-colored bands will be narrower; or, in other words, 'the narrower bands correspond in color to the larger sector.'
Finally, if the sectors are equal, the bands must also be equal.
So far, then, the bands geometrically deduced present the same variations as the bands observed in the illusion.
2. Secondly ([p. 174], No. 2), "The faster the rod moves the broader become the bands, but not in like proportions; broad bands widen relatively more than narrow ones." The speed of the rod or pendulum, in degrees per second, equals r. Now if W increases when r increases, DτW must be positive or greater than zero for all values of r which lie in question.
Now
| W = | rs - pr' , |
| r' ± r |
and
| DτW = | (r' ± r)s + (rs - pr') , |
| (r ± r')² |
or reduced,