pr’
W = --------- ,
r’ — r

But the contrary will be true when they move in opposite directions, for then

pr’
W = --------- ,
r’ + r

that is, the larger r is, the narrower is w.

The present writer could not be sure whether or not the width of transition-bands varied with r.  He did observe, however (page 174) that ’the transition-bands are broader when rod and disc move in the same, than when in opposite directions.’  This will be true likewise for the geometrical bands, for, whatever r (up to and including r = r’),

pr’ pr’
——­ > ——­
r’-r r’+r

In the observation, of course, r, the rate of the rod, was never so large as r’, the rate of the disc.

5.  We next come to an observation (p. 174, No. 5) concerning the number of bands seen at any one time.  The ’geometrical deduction of the bands,’ it is remembered, was concerned solely with the amount of color which was to be deducted from the fused color of the disc. W and w represented the widths of the areas whereon such deduction was to be made.  In observation 5 we come on new considerations, i.e., as to the color from which the deduction is to be made, and the fate of the momentarily hidden area which suffers deduction, after the pendulum has passed on.

We shall best consider these matters in terms of a concept of which Marbe[3] has made admirable use:  the ‘characteristic effect.’  The Talbot-Plateau law states that when two or more periodically alternating stimulations are given to the retina, there is a certain minimal rate of alternation required to produce a just constant sensation.  This minimal speed of succession is called the critical period.  Now, Marbe calls the effect on the retina of a light-stimulation which lasts for the unit of time, the ‘photo-chemical unit-effect.’  And he says (op. cit., S. 387):  “If we call the unit of time 1[sigma], the sensation for each point on the retina in each unit of time is a function of the simultaneous and the few immediately preceding unit-effects; this is the characteristic effect.”

   [3] ’Marbe, K.:  ‘Die stroboskopischen Erscheinungen,’ Phil. 
   Studien., 1898, XIV., S. 376.

We may now think of the illusion-bands as being so and so many different ‘characteristic effects’ given simultaneously in so and so many contiguous positions on the retina.  But so also may we think of the geometrical interception-bands, and for these we can deduce a number of further properties.  So far the observed illusion-bands and the interception-bands have been found identical, that is, in so far as their widths under various conditions are concerned.  We have now to see if they present further points of identity.