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) squared

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.

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