392 Transactions of the Society. 



form and dimensions ; and as a complete series of intermediate forms 

 of antipoint can be obtained by properly proportioning the emergent 

 wave-front and the focal length, it is not too much to say that any 

 such solution as that said to have been worked out by Dr. Eichhorn 

 can at best be only a particular case having no special bearing upon 

 the general problem.* 



There is, however, a problem of much greater practical import- 

 ance than any of those presented by optical illusions, upon which 

 the theory of the antipoint has an all-important bearing. I refer 

 to the problem of the construction of an objective with reference 

 to resolving power. Practical experience has established the very 

 high resolving power of wide-angled objectives, but of this practical 

 excellence no satisfactory explanation has yet been given. The 

 explanation generally accepted that a wide-angled objective collects 

 a larger number of diffraction spectra than a narrow-angled lens, has 

 been proved by the foregoing investigation to be neither true in fact 

 nor explanatory if true. But the theory of the antipoint suggests 

 the true explanation, which indeed becomes perfectly obvious when 

 the general laws governing the form of the antipoint are taken into 

 consideration. The problem, of course, is to make the antipoint as 

 nearly punctiform as possible. An image in which the points of 

 nature were replaced by truly punctiform antipoints, if such an image 

 could be produced, would be perfectly resolved. Now, the visible 

 geometrical form of the antipoint depends upon three influences : — 



(1) The focussing property of the wave-front segment by which it 

 is formed. 



(2) The brilliancy of its illumination. 



(3) The overlapping of the outlying parts of adjacent antipoints. 

 The first of these influences has formed the subject of this inquiry, 



and has been shown to depend upon the contour of the wave-fronr, 

 being affected both by its outline and by its curvature. Strong cur- 

 vature and wide expanse tend towards the compression of the anti- 

 point. On the other hand, a flattened form and small diameter of 

 the wave-front tend to expand the antipoint, and so to destroy the 

 resolution of the image. 



The second of these influences has received cursory notice in con- 

 nection with iigs. 81 and 90 ; and it is obvious that the subordinate 

 features of the antipoint — the margin of its central disc and its flanking 

 or encompressing images — will be more conspicuous with strong than 

 with weak illumination. This point, therefore, may pass without 

 more detailed consideration. 



* The reader will no doubt appreciate that the drawings of antipoints through- 

 out this paper are mere diagrams, any close representation by means of woodcuts 

 being impossible. Moreover, the salient points are of necessity exaggerated iu order 

 to exhibit them clearly. Thus, for example, the stellate appearance of fig. !)1 is 

 accentuated much beyond the reality. 



