;>2 Transactions of the Society. 



gradual distribution of phases parallel to the edge of the boundary, 

 ruing from a phase equal to the external phase of the individual anti- 

 point at the outer edge of the diffraction fringe to a value which is the 

 average of the phases in one half of the antipoint over the boundary 

 itself. This is obvious, for the light on the outermost edge is unmixed, 

 and the light at every point on the boundary is the summed light of 

 one half of the antipoint. Proceeding inward from the boundary we 

 find the phase still increasing, for the region near the boundary is 

 lighted up by something more than half the antipoint, and the addi- 

 tional light consists in more than the total average proportion of the 

 light of the innermost zones of the antipoint. But when we get to a 

 point equal to the radius of the antipoint within the true boundary, we 

 reach a region where the light on every point is the integral of all the 

 light from a single antipoint. Here, then, the light phase must haw 

 returned to the phase on the boundary, since the average phase of the com- 

 plete antipoint must be the same as the average of the semi-antipoint, 

 seeing that one half of every zone enters into the semi-antipoint, 

 and the proportional value of every zone in relation to the whole illu- 

 mination is therefore the same in both cases. This consideration points, 

 to the existence of a doubly conical wave-front with unequal surfaces,. 

 and yields at once a forecast of certain very striking phenomena which 

 are, as the foregoing paper shows, very strikingly verified by experiments 

 (See above, p. 23.) 



