Propagation of Cusped Waves. 591 



(fig. 1). The upper branch of the cusp is concave upward, 

 and is the portion of the wave which left the reflecting 

 surface first and has passed through a focus. The lower 

 branch is concave downward, or in the direction of pro- 

 pagation, and represents the portion of the wave which has 

 just left the surface and is on the way to its focus. The 

 radius of curvature increases from zero as we go away from 

 the cusp-point along either branch, as I have said before. 

 This cusped wave moves down the mirror, the lower branch 

 being continually replenished by consecutive portions of the 

 incident wave as it encounters the mirror, the upper branch 

 being continually added to by elements of the lower branch 

 as they pass through their foci at the cusp. 



As I have said in a previous paper, the cusp traces the 

 caustic surface ; and since the wave is always coming to a 

 focus on the cusp, the increased illumination along the caustic 

 is accounted for. 



Let us now examine the relation of these reflected wave- 

 fronts to the primary and secondary focal lines. If we inspect 

 the diagram usually given to illustrate the formation of focal 

 lines (Winkelmann, p. 33, for example), it is at once apparent 

 that the wave-front between the two focal lines is expanding 

 along one meridian, and contracting along a meridian at right 

 angles to it ; in other words, the wave is convex along one 

 meridian and concave along the other. The form of the 

 surface is not unlike a small bit on the inside of an anchor- 



Consider now the diagram shown in fig. 2, remembering 

 that the complete wave-front at this stage is formed by the 

 rotation of this figure around the axis of the mirror. The 

 bowl of the crater is concave along every meridian, but it is 

 at once apparent that any portion of the outer slope has the 

 required saddle-shape, being concave in horizontal planes and 

 convex in vertical planes. From this it is evident that the 

 outer wall of the volcanic cone, before it crosses the axis of 

 the mirror, always represents the portions of the wave-front 

 between the primary and secondary focal lines. 



That this is true is evident, when we recollect that the first 

 focal line is formed by the intersection of rays on the caustic 

 surface, or, regarded from the wave point of view, by the 

 passage through their foci on the cusp of the wave of adjacent 

 elements of the wave-front. The second focal line lies on the 

 axis of the mirror ; consequently the wave-front between the 

 lines is that portion of the surface which has passed through 

 a focus on the cusp, but which has not crossed the axis. 



I have found that a small glass model of the wave-front, 



