Highly Magnified Images. By J. W. Gordon. 17 



lapping antipoints are attuned and to the same phase ; (2) that 



they are attuned, but with a phase difference A <f> = - ; and 



A 



(3) that they are independent as to phase. The result of con- 

 sidering case (2) is very remarkable, for it then appears that, how- 

 ever close the centres may be, the antipoints will be seen, if at all, 

 as separate objects. If they were to coincide exactly it is obvious 

 that the light of the one would quench that of the other, and 

 if they were separately of equal brightness the one to the other 

 they would become invisible. In any case, if the centres be 

 separated by a distance, how small soever, the middle point 

 between those centres must be a point of darkness and, there- 

 fore, a dark boundary must separate the two illuminated areas. 

 Here then we have an unlimited resolving power. It thus appears 

 that — 



X. The limit of resolving power is not simply a question of 

 the propinquity of luminous objects, but depends in a material 

 degree upon the phase relations of the light by which they are 

 severally rendered visible, and from this it follows as a practical 

 inference that the expedient of controlling the phase relations 

 of adjacent antipoints — if we can find the means of applying it — 

 will give us command of a resolving power beyond the Helmholtz 

 limit, and possibly beyond any limit that can be assigned. To 

 this point we shall have occasion to recur upon a later page in 

 this paper. 



There is still another point in respect of which Helmholtz' 

 result invites criticism. As already stated, the limit which he 

 named was not put forward as an exact or calculated limit, but as 

 a result of which actual practice must always and necessarily fall 

 short, and fall short by a considerable measure. He took the anti- 

 point as the extreme case of a very small surface, and argued that 

 if two antipoints could not be separated from one another, the two 

 finite surfaces on the confines of which these antipoints lay must 

 in like manner be inextricably fused together. There is here a 

 very singular oversight, the nature of which may be exhibited by a 

 diagram. In fig. 9 {ante, p. 15) any one of the six curves shown 

 represents the light amplitude curve of an antipoint, but that of a 

 luminous area in which antipoints stand side by side and close to one 

 another, overlapping as completely as may be, will be represented 

 by the curve of the following figure (fig. 10). It is clear that 

 the full brightness of the luminous area is not developed at its 



very edge, but at a distance = _ . - measured inward from the 

 J ° ' 2 sin u 



edge. Moreover, the light intensity here, even if the antipoints 



have no determinate phase relation inter se, will be double the 



intensity at the true focus of a single antipoint, and if, therefore, we 



Feb. 15th, 1905 c 



