68 NOTES ON THE RESOLVING POWER OF 



ill consequence no exact statements can be made. All we can 

 get is rough estimations by which the quality of oj^tical instruments 

 can be compared. The conventional limit probably supplies this 

 desideratum as well as any other, and since it possesses greater 

 convenience, it may continue to be adopted, except, perhaps, in 

 special problems. 



The problem of the microscope has been studied specifically bv 

 Helmholtz (1874), Abbe (1873), and by the late Lord Rayleigh. 

 The name of the late Lord Rayleigh may be repeated, because he 

 has dealt with the whole problem in all its ramifications in a way 

 which no other investigator has done. In particular may be men- 

 tioned the following pajDers by him: '' On the Diffraction of Object- 

 glasses," Coll. Papers, Vol. I., 163 (1872), " Investigations in 

 Optics," I., 415 (1879-1880), '' Resolving Power of Telescopes," 

 I., 488 (1880), " Wave Theory Light," III., 47-187 (1888), " The 

 Theory of Optical Images, with special reference to the Microscope," 

 IV., 235 (1896), Ditto (supplementary paper), V., 118 (1903). 



The microscope problem possesses several peculiarities which are 

 not met with in stellar observation. In the first place the object is 

 never self-luminous like a star, and much depends upon the char- 

 acter of the light transmitted through the object when it is semi- 

 transparent or reflected from it when it is opaque. Again, the 

 object seldom consists of points (which would be imaged as dif- 

 fraction discs and rings), but may be isolated or series lines or may 

 be of any other shape; and in each case may be either bright or 

 dark compared with the " background "; each case requires specific 

 consideration. No one has worked these cases out in full except 

 Lord Rayleigh, and reference must be made to his papers cited 

 above for the full investigations. We can deal here only with some 

 general considerations. 



In the first place the essential difference in detail between a 

 telescope and microscope arises from the object being near the 

 objective. It becomes convenient to refer to the semi-anr/Ie that 

 the objective subtends at the object and the distance, e (instead 

 of the angle) between the two sources which are here separated. 



Now for two independent points the distance e for which resolu- 

 tion will occur is for a rectangular opening 



\ 



2 n sm a 



where n is the refractive index between the object and objective and 

 X is the wave-length of the light employed. On the other hand, 

 Abbe, by considering a series of linear openings as object, found 

 if the phase of the light passing through each opening is the same 

 for all 



X 



6 = — . ; 



n sm a 

 which is twice as great as before. The quantity n sin (t he called 

 the Numerical Aperture, and the reciprocal of e the resolving- 

 power. These two examples bring out a necessary condition for 

 securing fine resolution. The value of e is half as great as when the 

 lights from different points of the object are independent, as when 

 they are isophasal. Now this independence can be fairly secured 

 by focussing a source of light by means of a condenser upon the 



