into Uniform Undulations of Flat Wavelets. 277 



correspond to the wave-frequencies 4> x — 8/2 and <jb 2 + S/2. 

 Let us suppose that u £ w's of all the wave-frequencies 

 between these limits are present, and further assume that 

 they are so related to one another that they are all polarized 

 alike and o£ equal intensity, and that they had all reached 

 the plane YZ in the same phase when the solidification took 

 place. These would be represented on the spectrum by a 

 band extending from m to n. Let this band be divided into 

 equal parts by the line c. Then the whole of the light can 

 be divided into pairs of rays, one in the left-hand half of the 

 band, and the other in the right-hand half, and with 8 as 

 the difference of wave-frequency between the two components 

 of each pair. Every such pair, if it alone were present, 

 would produce the distribution represented in fig. 10, with 

 loops of waves from —e x to +<? l5 from e x to e 2 , and so on, 

 and with their positions of cipher intensity occurring on 

 the dotted planes. Accordingly, as each pair if acting 

 separately would produce cipher intensity on the dotted 

 planes, they all when simultaneously acting produce cipher 

 intensity on those planes. We may make use of strip a. 

 of fig. 6 to represent this distribution. And as we are 

 equally at liberty to divide the band mn on the spectrum 

 into 4 parts, or 8, or 16, &c, and thus to group the whole of 

 the light of the band into pairs of rays with a difference of 

 wave-frequency which we can make either 8/2, or S/4, or 

 8/8, &c. ; we by this process find that when the whole band 

 of light is in action there is cipher intensity on the dotted 

 lines of strip /3, and on those of strip 7, &c. of fig. 6, as well 

 as on those of strip a. This means that there is cipher 

 intensity on all the planes, dotted or undotted, that are 

 represented in fig. 10, except the plane YZ. On this plane 

 the intensity is a maximum, because by hypothesis all the 

 u f w's corresponding to the individual rays of the band 

 reached it in the same phase when the solidification took 

 place. Up to the present we have followed the same line of 

 argument as in § 13, and by continuing it to the end, we 

 learn that nearly the whole intensity of what results from 

 the coexistence of all the u f w's is to be found in the central 

 loop between the planes — e 1 and +e u while beyond those 

 limits there may be intensities that correspond to glimmers 

 of light, which become fainter the wider the band mn in the 

 spectrum is, and which, for their total extinction, would 

 require the cooperation of certain appendage rays other than 

 the rays of which we have yet taken account, which appendages, 

 however, in many of the practical cases that arise supply so 

 little light that they may be ignored. 



