only is there discordance between the central and marginal waves, but that every line of waves such as x R’, on the one side of O R’, finds a line x’ R’ upon the other side of O R’, from which its path differs by half an undulation—­with which, therefore, it is in complete discordance.  The consequence is, that the light on the one side of the central line will completely abolish the light on the other side of that line, absolute darkness being the result of their coalescence.  The first dark interval of our series of bands is thus accounted for.  It is produced by an obliquity of direction which causes the paths of the marginal waves to be a whole wave-length different from each other.

When the difference between the paths of the marginal waves is half a wave-length, a partial destruction of the light is effected.  The luminous intensity corresponding to this obliquity is a little less than one-half—­accurately 0.4—­that of the undiffracted light.  If the paths of the marginal waves be three semi-undulations different from each other, and if the whole beam be divided into three equal parts, two of these parts will, for the reasons just given, completely neutralize each other, the third only being effective.  Corresponding, therefore, to an obliquity which produces a difference of three semi-undulations in the marginal waves, we have a luminous band, but one of considerably less intensity than the undiffracted central band.

With a marginal difference of path of four semi-undulations we have a second extinction of the entire beam, because here the beam can be divided into four equal parts, every two of which quench each other.  A second space of absolute darkness will therefore correspond to the obliquity producing this difference.  In this way we might proceed further, the general result being that, whenever the direction of wave-motion is such as to produce a marginal difference of path of an even number of semi-undulations, we have complete extinction; while, when the marginal difference is an odd number of semi-undulations, we have only partial extinction, a portion of the beam remaining as a luminous band.

A moment’s reflection will make it plain that the wider the slit the less will be the obliquity of direction needed to produce the necessary difference of path.  With a wide slit, therefore, the bands, as observed, will be closer together than with a narrow one.  It is also plain that the shorter the wave, the less will be the obliquity required to produce the necessary retardation.  The maxima and minima of violet light must therefore fall nearer to the centre than the maxima and minima of red light.  The maxima and minima of the other colours fall between these extremes.  In this simple way the undulatory theory completely accounts for the extraordinary appearance above referred to.