It has been remarked, that when two systems of waves differ half an undulation in their progress, two of the semiundulations must escape from interference; that six must be exempt when the difference amounts to three semiundulations; and that, in general, the number of undulations exempt from interference is equal to the number of lengths of a semiundulation separating the corresponding points of the two systems. While this number is very small in proportion to that of the waves contained in each system, the motion must be nearly destroyed, as in the case of the exemption of a single undulation. But it may be imagined that, as we increase the difference of the progress of the two pencils, the undulations exempted from interference may become a material portion of each group, and that it may finally become so great as to separate the groups entirely from each other; and in this case the phenomena of interference would no longer be observable. If, for example, the groups of undulations consisted but of a thousand each, a difference of one-twentieth of an inch in their routes would be much more than sufficient to prevent the interference of the rays of all kinds. [p128] But there is another much more powerful reason which prevents our perceiving the effects of the mutual influence of the systems of waves when the difference of their routes is considerable; which is the impossibility of rendering the light sufficiently homogeneous: for the most simple light that we can obtain consists still of an infinity of heterogeneous rays, which have not exactly the same length of undulation; and however slight the difference may be, when it is repeated a great number of times, it produces of necessity, as we have already seen, an opposition between the modes of interference of the various rays, which then compensates for the weakening of some by the strengthening of others; [while the shades of colour are not sufficiently distinct to allow the eye to remark the difference.] This is without doubt the principal reason why the effects of the mutual interference of the rays of light become insensible when the difference of the routes is very considerable, so as to amount to 50 or 60 times the length of an undulation. It has already been laid down as one of the conditions necessary for the appearance of the phenomena of interference, that the rays which are combined should have issued at first from a common source: and it is easy to account for the necessity of this condition by the theory which has now been explained.