We have now to take cognizance of an item thus far neglected. When the green sector has reached the position _A'B'_, that is, is just emerging wholly from behind the pendulum, the front of the red sector must already be in eclipse. The generation of a green band (red sector in eclipse) will have commenced somewhat before the generation of the red band (green sector in eclipse) has ended. For a moment the pendulum will lie over parts of both sectors, and while the red band ends at point _A'_, the green band will have already commenced at a point somewhat to the left (and, indeed, to the left by a trifle more than the width of the pendulum). In other words, the two bands _overlap_. This area of overlapping may itself be accounted a band, since here the pendulum hides partly red and partly green, and obviously the result for sensation will not be the same as for those areas where red or green alone is hidden. We may call the overlapped area a 'transition-band,' and we must then ask if it corresponds to the 'transition-bands' spoken of in the observations. Now the formula obtained for Z includes two such transition-bands, one generated in the vicinity of OB and one near OA'. To find the formula for a band produced while the pendulum conceals solely one, the oppositely colored sector (we may call this a 'pure-color' band and let its width = W), we must find the formula for the width (w) of a transition-band, multiply it by two, and subtract the product from the value for Z already found. The formula for an overlapping or transition-band can be readily found by considering it to be a band formed by the passage behind P of a sector whose width is zero. Thus if, in the expression for Z already found, we substitute zero for s, we shall get w; that is, 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) ,