o + pr’      pr’
w = ------- =  ------
r’ — r     r’ — r
Since
W = Z — 2w,
we have
rs + pr’       pr’
W = -------- = 2 ------,
r’ — r      r’ — r
or
rs — pr’
W = --------            (1)
r’ — r

[Illustration:  Fig 3.]

Fig. 3 shows how to derive W directly (as Z was derived) from the geometrical relations of pendulum and sectors.  Let r, r’, s, p, and t, be as before, but now let

width of the band (i.e., the arc BA’) = W;

that is, the band, instead of extending as before from where P begins to hide the green sector to where P ceases to hide the same, is now to extend from the point at which P ceases to hide any part of the red sector to the point where it just commences again to hide the same.

Then
                           W + p
                      t = ------- ,
                             r
and
                           W + s
                      t = ------- ,
                             r’

therefore
                 W + p W + s
                ------- = ------- ,
                   r r’

r’(W + p) = r(W + s) ,

W (r’ — r) = rs — pr’ ,
and, again,
rs — pr’
W = -------- .
r’ — r

Before asking if this pure-color band W can be identified with the bands observed in the illusion, we have to remember that the value which we have found for W is true only if disc and pendulum are moving in the same direction; whereas the illusion-bands are observed indifferently as disc and pendulum move in the same or in opposite directions.  Nor is any difference in their width easily observable in the two cases, although it is to be borne in mind that there may be a difference too small to be noticed unless some measuring device is used.

From Fig. 4 we can find the width of a pure-color band (W) when pendulum and disc move in opposite directions.  The letters are used as in the preceding case, and W will include no transition-band.

[Illustration:  Fig. 4]

We have

W + p
t = -----,
r
and
s — W
t = -----,
r’

r’(W + p) = r(s — W) ,

W(r’ + r) = rs — pr’ ,

rs — pr’
W = -------- .           (2)
r’ + r

Now when pendulum and disc move in the same direction,

rs — pr’
W = --------- ,          (1)
r’ — r

so that to include both cases we may say that

rs — pr’
W = -------- .           (3)
r’ +- r

The width (W) of the transition-bands can be found, similarly, from the geometrical relations between pendulum and disc, as shown in Figs. 5 and 6.  In Fig. 5 rod and disc are moving in the same direction, and

w = BB’.

Now
             W — p
        t = ------- ,
              r’

w
t = —–­ ,
r’

r’(w-p) = rw ,