1G Transactions of the Society. 



but Helmholtz, not putting this forward as an exact solution, did 

 not in fact state it with so much precision. He was content to say, 

 using a merely approximative figure, the limit of resolving power 



must necessarily be > c = . . This is the well-known limit 



2 sm u 



which on his authority and that of Prof. Abbe has been adopted 

 by almost all subsequent writers upon die Microscope as the ulti- 

 mate and necessary limit beyond which its performance can never 

 go. It is a curious circumstance that both Helmholtz and Abbe 

 should have fixed upon this expression. It is not, as has just been 

 shown, the true result of Helmholtz' theory, but a figure arbitrarily 

 selected as lying within the true limit, and Abbe was led to it by 

 considering the rather fanciful question as to how a picture could 

 be formed of an object illuminated by a beam of light having a 

 divergence angle = 0. It is a mere coincidence, but a very strange 

 one, that two such widely different attempts to solve the problem 

 should both lead to the same result, and that an erroneous result. 

 It is perhaps less surprising that the error so authenticated should 

 have passed undetected and even unchallenged until 1896. 



In the last-named year Lord Eayleigh published a paper in 

 which the whole subject was reviewed, the inadequacy of Prof. 

 Abbe's treatment of it was pointed out, and a very pertinent 

 inquiry started as to whether Helmholtz' method of obtaining the 

 values of his total light curves (see fig. 9 above) took due account 

 of the phase relations of contiguous antipoints. So long as we 

 concern ourselves only with light intensities (ignoring the light 

 amplitudes) no question of phase relation and resulting inter- 

 ference can arise ; and it is commonly assumed by physicists that 

 unless two beams of light originate in the same incandescent 

 particle they must be independent as to phase, and cannot, there- 

 fore, exhibit the phenomena of regular interference. This is only 

 very imperfectly true, and Lord Piayleigh in this paper showed 

 that — 



IX. If two beams of light, although originating in independent 

 sources of light, follow very closely adjacent and nearly parallel 

 paths, so that they interpenetrate one another, they will modify one 

 another where they interpenetrate, and may thereby become attuned 

 to one another almost as if they had had a common origin, and so as 

 to be capable of exhibiting all the phenomena of interference.* It 

 now appears that the results of overlapping must be more complex 

 than Helmholtz had assumed, and Lord Eayleigh illustrates this 

 fact by taking three typical cases. He assumes (1) that the over- 



* This interference of light beams from independent sources would seem to have 

 been illustrated by a very elegant experiment devised by Dr. Johnstone Stoney. and 

 demonstrated by him at a Meeting of the British Association. See Rep. B.A., 

 1901, p. 574. 



