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)

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

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 pa.s.sed 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 ident.i.ty.

As to the characteristic effects incident to the interception-bands; in Fig. 7 (Plate V.), let _A'C'_ represent at a given moment _M_, the total circ.u.mference of a color-disc, _A'B'_ represent a green sector of 90, and _B'C'_ a red complementary sector of 270. If the disc is supposed to rotate from left to right, it is clear that a moment previous to _M_ the two sectors and their intersection _B_ will have occupied a position slightly to the left. If distance perpendicularly above _A'C'_ is conceived to represent time previous to _M_, the corresponding previous positions of the sectors will be represented by the oblique bands of the figure. The narrow bands (_GG_, _GG_) are the loci of the successive positions of the green sector; the broader bands (_RR_, _RR_), of the red sector.

In the figure, 0.25 mm. vertically = the unit of time = 1[sigma]. The successive stimulations given to the retina by the disc _A'C'_, say at a point _A'_, during the interval preceding the moment _M_ will be

green 10[sigma], red 30[sigma], green 10[sigma], red 30[sigma], etc.

Now a certain number of these stimulations which immediately precede _M_ will determine the characteristic effect, the fusion color, for the point _A'_ at the moment _M_. We do not know the number of unit-stimulations which contribute to this characteristic effect, nor do we need to, but it will be a constant, and can be represented by a distance _x_ = _A'A_ above the line _A'C'_. Then _A'A_ will represent the total stimulus which determines the characteristic effect at _A'_.

Stimuli earlier than _A_ are no longer represented in the after-image.

_AC_ is parallel to _A'C'_, and the characteristic effect for any point is found by drawing the perpendicular at that point between the two lines _A'C_ and _AC_.

Just as the movement of the disc, so can that of the concealing pendulum be represented. The only difference is that the pendulum is narrower, and moves more slowly. The slower rate is represented by a steeper locus-band, _PP'_, than those of the swifter sectors.

We are now able to consider geometrically deduced bands as 'characteristic effects,' and we have a graphic representation of the color-deduction determined by the interception of the pendulum. The deduction-value of the pendulum is the distance (_xy_) which it intercepts on a line drawn perpendicular to _A'C'_.

Lines drawn perpendicular to _A'C'_ through the points of intersection of the locus-band of the pendulum with those of the sectors will give a 'plot' on _A'C'_ of the deduction-bands. Thus from 1 to 2 the deduction is red and the band green; from 2 to 3 the deduction is decreasingly red and increasingly green, a transition-band; from 3 to 4 the deduction is green and the band red; and so forth.

We are now prepared to continue our identification of these geometrical interception-bands with the bands observed in the illusion. It is to be noted in pa.s.sing that this graphic representation of the interception-bands as characteristic effects (Fig. 7) is in every way consistent with the previous equational treatment of the same bands. A little consideration of the figure will show that variations of the widths and rates of sectors and pendulum will modify the widths of the bands exactly as has been shown in the equations.

The observation next at hand (p. 174, No. 5) is that "The total number of bands seen at any one time is approximately constant, howsoever the widths of the sectors and the width and rate of the rod may vary. But the number of bands is inversely proportional (Jastrow and Moorehouse) to the time of rotation of the disc; that is, the faster the disc, the more bands."

[Ill.u.s.tration: PSYCHOLOGICAL REVIEW. MONOGRAPH SUPPLEMENT, 17. PLATE V.

Fig. 7. Fig. 8. Fig. 9.]

This is true, point for point, of the interception-bands of Fig. 7. It is clear that the number of bands depends on the number of intersections of _PP'_ with the several locus-bands _RR_, _GG_, _RR_, etc. Since the two sectors are complementary, having a constant sum of 360, their relative widths will not affect the number of such intersections. Nor yet will the width of the rod _P_ affect it. As to the speed of _P_, if the locus-bands are parallel to the line _A'C'_, that is, of the disc moved _infinitely_ rapidly, there would be the same number of intersections, no matter what the rate of _P_, that is, whatever the obliqueness of _PP'_. But although the disc does not rotate with infinite speed, it is still true that for a considerable range of values for the speed of the pendulum the number of intersections is constant. The observations of Jastrow and Moorehouse were probably made within such a range of values of _r_. For while their disc varied in speed from 12 to 33 revolutions per second, that is, 4,320 to 11,880 degrees per second, the rod was merely pa.s.sed to and fro by hand through an excursion of six inches (J. and M., _op.

cit._, pp. 203-5), a method which could have given no speed of the rod comparable to that of the disc. Indeed, their fastest speed for the rod, to calculate from certain of their data, was less than 19 inches per second.

The present writer used about the same rates, except that for the disc no rate below 24 revolutions per second was employed. This is about the rate which v. Helmholtz[4] gives as the slowest which will yield fusion from a bi-sectored disc in good illumination. It is hard to imagine how, amid the confusing flicker of a disc revolving but 12 times in the second, Jastrow succeeded in taking any reliable observations at all of the bands. Now if, in Fig. 8 (Plate V.), 0.25 mm. on the base-line equals one degree, and in the vertical direction equals 1[sigma], the locus-bands of the sectors (here equal to each other in width), make such an angle with _A'C'_ as represents the disc to be rotating exactly 36 times in a second. It will be seen that the speed of the rod may vary from that shown by the locus _P'P_ to that shown by _P'A_; and the speeds represented are respectively 68.96 and 1,482.64 degrees per second; and throughout this range of speeds the locus-band of _P_ intercepts the loci of the sectors always the same number of times. Thus, if the disc revolves 36 times a second, the pendulum may move anywhere from 69 to 1,483 degrees per second without changing the number of bands seen at a time.

[4] v. Helmholtz, H.: 'Handbuch d. physiolog. Optik,' Hamburg u. Leipzig, 1896, S. 489.

And from the figure it will be seen that this is true whether the pendulum moves in the same direction as the disc, or in the opposite direction. This range of speed is far greater than the concentrically swinging metronome of the present writer would give. The rate of Jastrow's rod, of 19 inches per second, cannot of course be exactly translated into degrees, but it probably did not exceed the limit of 1,483. Therefore, although beyond certain wide limits the rate of the pendulum will change the total number of deduction-bands seen, yet the observations were, in all probability (and those of the present writer, surely), taken within the aforesaid limits. So that as the observations have it, "The total number of bands seen at any one time is approximately constant, howsoever ... the rate of the rod may vary." On this score, also, the illusion-bands and the deduction-bands present no differences.

But outside of this range it can indeed be _observed_ that the number of bands does vary with the rate of the rod. If this rate (_r_) is increased beyond the limits of the previous observations, it will approach the rate of the disc (_r'_). Let us increase _r_ until _r_ = _r'_. To observe the resulting bands, we have but to attach the rod or pendulum to the front of the disc and let both rotate together. No bands are seen, _i.e._, the number of bands has become zero. And this, of course, is just what should have been expected from a consideration of the deduction-bands in Fig. 8.

One other point in regard to the total number of bands seen: it was observed (page 174, No. 5) that, "The faster the disc, the more bands." This too would hold of the deduction-bands, for the faster the disc and sectors move, the narrower and more nearly parallel to _A'C'_ (Fig. 7) will be their locus-bands, and the more of these bands will be contained within the vertical distance _A'A_ (or _C'C_), which, it is remembered, represents the age of the oldest after-image which still contributes to the characteristic effect. _PP'_ will therefore intercept more loci of sectors, and more deduction-bands will be generated.

6. "The colors of the bands (page 175, No. 6) approximate those of the two sectors; the transition-bands present the adjacent 'pure colors'

merging into each other. But _all_ the bands are modified in favor of the moving rod. If, now, the rod is itself the same in color as one of the sectors, the bands which should have been of the other color are not to be distinguished from the fused color of the disc when no rod moves before it."