In formula 3, _W_ represents the width of a band, and _s_ the width of the _oppositely colored_ sector. Therefore, if a disc is composed, for example, of a red and a green sector, then rs(green) - pr' W(red) = ------------------ , r' ± r and rs(red) - pr' W(green) = ------------------ , r' ± r therefore, by dividing, W(red) rs(green) - pr' --------- = ------------------- . W(green) rs(red) - pr' 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, _D_{[tau]}W_ must be positive or greater than zero for all values of _r_ which lie in question. Now rs - pr' W = --------- , r' ± r and (r' ± r)s [±] (rs - pr') D_{[tau]}W = -------------------------- , (r ± r') or reduced, r'(s ± p) = ----------- (r' ± r)² Since _r'_ (the speed of the disc) is always positive, and _s_ is always greater than _p_ (cf. p. 173), and since the denominator is a square and therefore positive, it follows that D_{[tau]}W > 0 or that _W_ increases if _r_ increases. Furthermore, if _W_ is a wide band, _s_ is the wider sector. The rate of increase of _W_ as _r_ increases is r'(s ± p) D_{[tau]}W = ----------- (r' ± r)² which is larger if _s_ is larger (_s_ and _r_ being always positive). That is, as _r_ increases, 'broad bands widen relatively more than narrow ones.' 3. Thirdly (p. 174, No. 3), "The width of The bands increases if the speed of the revolving disc decreases." This speed is _r'_. That the observed fact is equally true of the geometrical bands is clear from inspection, since in rs - pr' W = --------- , r' ± r as _r'_ decreases, the denominator of the right-hand member decreases while the numerator increases.