Highly Magnified Images. By J. IV. Gordon. 31 



to a comparatively short length of the edge of the beam. It is readily 

 •deducibie from this that the fringe will exhibit bright and dark bands 

 ■consecutively, and that the falling off in brightness of the outer bands 

 will be very rapid. It does not appear, however, that the dark bands will 

 sink to zero illumination, as in Airy's curve. 



(c) The form of (11) shows that the fringe must be a wave-front. 

 For, recurring to Fig. 23, suppose the triangle 8 S ^ P, to be shifted 

 up a distance = A along the edge of the beam. Then, since the middle 

 point D n of every successive wave-length segment emits light in the 

 same phase the series of (11) will be unchanged, except by the loss of a 

 few wholly insignificant terms at the far end of the series, representing 

 light from segments immediately below the aperture, where sin a is 

 sensibly = 1 and where consequently n is sensibly = N, and therefore 

 A ^ is sensibly = 0. It follows that in this new position of -q x the 

 values of \p x and ty g are severally identical with what they were in the 

 original position. The same result would have appeared had we moved 

 it 2 A, or 10 A, or n A, n being any integer not immoderately large, that 

 is to say, so large as to bring rj x within a few wave-lengths of the 

 aperture A . . A. Furthermore, if we take intermediate positions on 

 the ray joining all these positions of ^ we shall have corresponding 

 intermediate values for the light phase, and in any given position the 

 phase will, of course, change in time with the contemporary change in 

 the generating edge ray of the beam. Therefore, along this supposed 

 ray parallel to the edge of the beam, we have a regular succession of 

 undulations moving forward with the velocity of light. Similarly with 

 •every other ray drawn parallel to the edge of the beam. We thus see 

 that through a conical surface drawn normal to the edge of the cone we 

 have a system of rays normal to tnat conical surface along which light 

 undulations pass with the velocity of light. This seems to import that 

 the disturbance set up in the region immediately surrounding a focussed 

 beam takes the form of a conical wave-front, and from that it follows 

 by a reversal of the reasoning on p. 20 (see fig. 12), that in the focal 

 plane itself the antipoint will exhibit a zonal arrangement of phases, 

 the light-phase being most retarded at the focal point. The phenomena 

 resulting from that arrangement when the fringe has slipped off the 

 beam and forms an antipoint, are worked out above in connection with 

 figs. 13 to 20, and their experimental verification is there attempted. 



Here it may be pointed out that if we now substitute this system 

 of conical wave-fronts surrounding the true cone of the focussed beam 

 for the complicated system of interpenetrating beams given off in many 

 different directions by the radiant outer surface of the cone, we obtain 

 a clear and perfectly coherent geometrical conception of the simple 

 diffraction fringe from which the antipoint is eventually formed. 



(d) This last case leads naturally to the next. So far we have con 

 sidered only antipoints and the simple cones which give rise to them. 

 More complicated cases arise in practice when light is radiated from 

 surfaces of finite magnitude. Upon this subject I have very little to 

 offer, for as yet I have hardly broken ground in that direction. But 

 one conclusion of great importance seems obvious. It is that the 

 diffraction fringe upon the edge of a luminous area will have a regular 



