*Project Gutenberg*. Public domain.

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