474 Lord Rayleigh's Notes on some 



Unebenheiten, welche, in Hinsicht ihren Entfernung von 

 einander, kleiner als co sind [© denotes the wave-length], so 

 sind sie sowohl fiir durchfahrendes als zuriickgeworfenes 

 Licht ohne Nachtheil, und es konnen dadurch keine Farben 

 irgend einer Art entstehen ; auch ware es durch kein Mittel 

 moglich diese Unebenheiten sichtbar zu machen." And here 

 he appends the very pregnant note : — " Man kann daraus 

 schliessen, was moglicher Weise durch Mikroskope noch zu 

 sehen ist. Ein mikroskopischer Gegenstand z. B., dessen 

 Durchmesser = co ist, und der aus zwei Theilen besteht, kann 

 nicht mehr als aus zwei Theilen bestehend erkannt werden. 

 Dieses zeigt uns eine Granze des Sehevermogens durch 

 Mikroskope." 



Frauenhofer's views did not commend themselves to Her- 

 schel *, who regarded the " alleged limit to the powers of 

 microscopes " as not " following from the premises/'' It so 

 happens that I can give an independent opinion upon the 

 persuasiveness of Frauenhofer's reasoning, for I had occasion 

 in 1870, in connection with my own work upon the repro- 

 duction of gratings, and before the publication of the investi- 

 gations of Abbe and v. Helmholtz, to consult his writings, when 

 the note above quoted attracted my attention and fully con- 

 vinced me of the general truth of the doctrine of the micro- 

 scopic limit. It seemed evident, at any rate, that two radiant 

 points, separated by only a small fraction of the wave-length, 

 could not be optically distinguished. 



It is worthy of notice that while Frauenhofer speaks of the 

 whole wave-length, modern investigation fixes the half wave- 

 length rather, as the limit of microscopic vision. It seems, 

 however, that, on his own principles, Frauenhofer should have 

 arrived at the latter result ; for a grating whose period 

 is equal to the wave-length can show colours when sufficiently 

 inclined f . It is easy to see that when the angular aperture 

 of a microscope is nearly 180°, a displacement of the radiant 

 point amounting to half a wave-length, perpendicularly to the 

 line of vision, will entail on one side of the pencil an acceler- 

 ation and on the other a retardation of that amount, so that 

 at the original focal point upon the retina the phases will now 

 range just over a complete period. The displacement of half 

 a wave-length corresponds therefore to something rather less 

 than the half width of the best possible image of a mathema- 

 tical point. 



A definite limit to an operation such as visual resolution 



* JEnc. Met., ' Light/ § 758, 1830. 



t Frauenhofer seems to have gone wrong in his formula for the case of 

 oblique incidence. 



