w(r’-r) = pr’ ,

pr’
w = ------- .          (4)
r’-r

[Illustration:  Fig. 5]

[Illustration:  Fig. 6]

In Fig. 6 rod and disc are moving in opposite directions, and

w = BB’,

p — w
t = ------- ,
r

w
t = —–­ ,
r’

r’(p — w) = rw ,

w(r’ + r) = pr’ ,

pr’
w = -------- .
r’ + r            (5)

So that to include both cases (of movement in the same or in opposite directions), we have that

pr’
w = -------- .
r’ +- r            (6)

VI.  APPLICATION OF THE FORMULAS TO THE BANDS OF THE ILLUSION.

Will these formulas, now, explain the phenomena which the bands of the illusion actually present in respect to their width?

1.  The first phenomenon noticed (p. 173, No. 1) is that “If the two sectors of the disc are unequal in arc, the bands are unequal in width; and the narrower bands correspond in color to the larger sector.  Equal sectors give equally broad bands.”

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) squared