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 ,

w(r'-r) = pr' ,

pr'

w = ------- . (4) r'-r

[Ill.u.s.tration: Fig. 5]

[Ill.u.s.tration: 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)

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