14 Transact ions of the Society. 



precisely like the difference between drawing an object with a fine- 

 pointed pencil and with a stick of charcoal. The finer antipoint 

 will obviously yield the better resolved picture. 



This way of viewing the matter led Helmholtz to enter upon 

 some very interesting speculations concerning the ultimate limit of 



1 *2 \ 

 resolving power. For it is evident that the expression r- , which 

 ° sin u 



expresses the diameter of the false disc of an antipoint produced 



by a circular aperture, cannot be infmitesimally small. The 



value of the wave-length \ will be somewhere in the region 



of 5oooo mcn > an( i sm u cannot be greater than 1. If then we 



1*2 



write 2 p = 1 • 2 X = — inch, we shall have the smallest anti- 



50,000 

 point that can by any possibility be obtained with green light of 

 the wave-length mentioned. "What, then, must be the minimum 

 separation of two bright objects which are by means of such an 

 antipoint to be separately delineated ? This is the much discussed 

 and profoundly interesting problem of the limit of resolving power. 

 Professor Helmholtz, although he approached this problem, as 

 we have seen, by a series of most masterly attacks upon what may 

 be called its outworks, did not drive his attack home or succeed in 

 capturing the citadel itself. It is not difficult to realise what re- 

 mained to be done. The form, dimensions and illumination of the 

 antipoint being taken to be known, it becomes in the next place 

 necessary to consider how the overlapping of adjacent antipoints will 

 affect the appearance of the field in which they lie. This, clearly, 

 is a problem of great complexity, for antipoints may overlap in all 

 imaginable degrees, from complete coincidence, as one extreme case, 

 to complete separation as the opposite extreme. Moreover, any 

 number of antipoints may overlap, and with varying degrees of 

 encroachment upon the common area, thus giving rise to still 

 further complexity. Helmholtz did not essay the regular solution 

 of this problem ; it appears, indeed, from a postscript appended to 

 his paper, that the necessary time was not at his command. But 

 he thought that the extreme case could be very simply stated, and 

 in effect he stated it as follows. Let A x A 2 in the following 

 diagram (fig. 8) be two adjacent antipoints winch encroach upon 

 one another. What is the smallest distance between their centres 

 at which they can be discerned as separate objects ? The figure 

 shows pairs of antipoints. The members of the first pair may be 

 assumed to be indistinguishably merged in one another. The mem- 

 bers of the third pair may be taken to be unmistakably distinct. 

 If we assume the second pair to be at the limit of resolving-power, 

 what will be the calculated distance of their centre points from 

 one another? This would be the exact statement of the problem 

 of the resolving limit as Helmholtz conceived it. But putting aside 



