4. We now come to the transition-bands, where one color shades over into the other. It was observed (p. 174, No. 4) that, "These partake of the colors of both the sectors on the disc. The wider the rod the wider the transition-bands." We have already seen (p. 180) that at intervals the pendulum conceals a portion of both the sectors, so that at those points the color of the band will be found not by deducting either color alone from the fused color, but by deducting a small amount of both colors in definite proportions. The locus of the positions where both colors are to be thus deducted we have provisionally called (in the geometrical section) 'transition-bands.' Just as for pure-color bands, this locus is a radial sector, and we have found its width to be (formula 6, p. 184) pr' W = --------- , r' ± r Now, are these bands of bi-color deduction identical with the transition-bands observed in the illusion? Since the total concealing capacity of the pendulum for any given speed is fixed, less of _either_ color can be deducted for a transition-band than is deducted of one color for a pure-color band. Therefore, a transition-band will never be so different from the original fusion-color as will either 'pure-color' band; that is, compared with the pure color-bands, the transition-bands will 'partake of the colors of both the sectors on the disc.' Since pr' W = --------- , r' ± r it is clear that an increase of _p_ will give an increase of _w_; _i.e._, 'the wider the rod, the wider the transition-bands.' Since _r_ is the rate of the rod and is always less than _r'_, the more rapidly the rod moves, the wider will be the transition-bands when rod and disc move in the same direction, that is, when pr' W = --------- , r' - r But the contrary will be true when they move in opposite directions, for then pr' W = --------- , r' + r that is, the larger _r_ is, the narrower is _w_. The present writer could not be sure whether or not the width of transition-bands varied with _r_. He did observe, however (page 174) that 'the transition-bands are broader when rod and disc move in the same, than when in opposite directions.' This will be true likewise for the geometrical bands, for, whatever _r_ (up to and including _r_ = _r'_), pr' pr' ---- > ---- r'-r r'+r In the observation, of course, _r_, the rate of the rod, was never so large as _r'_, the rate of the disc.