JOUENAL 



OF THE 



EOYAL MICROSCOPICAL SOCIETY. 



AUGUST 1904. 



TRANSACTIONS OF THE SOCIETY. 



IX. — A Direct Proof of Abbes Theorems on the Microscopic 



Resolution of Gratings. 



By Prof. J. D. Everett, F.R.S. 



(Bead June 15th, 1904.) 



Suppose the Microscope to be adjusted for viewing a transmission 

 grating laid on its stage, illuminated by a source so placed that 

 the incident light may be regarded as consisting of a single train 

 of plane waves travelling in a direction perpendicular to the 

 rulings. Diffraction spectra will be formed in the focal plane of 

 the objective, and will consist, for light of given wave-length X, 

 of bright points, all of them lying in the plane drawn through the 

 axis perpendicular to the rulings. The optical paths which we 

 shall have to discuss lie in this plane. 



Let the spectrum of zero order be called C, the two spectra of 

 the first order A x ~B 1} the two of the second order A 2 B 2 , and so on. 

 It will be necessary for us to consider the optical paths from an 

 incident wave-front (in a fixed position) to points midway between 

 successive bars of the grating and thence to the spectra, These 

 paths increase or diminish in arithmetical progression as we pass 

 from each middle point to the next, increasing for the spectra on 

 one side of C, and diminishing for those on the other side. The 

 common difference of the progression is zero for C, + A for A x 

 and B l5 + 2 \ for A 2 and B 2 , and + n \ for the two spectra of 

 order n. 



The optical disturbance at a fixed point P, in the image-plane 

 conjugate to the plane of the grating with respect to the objective, 

 is the sum of the disturbances due to the different spectra. All 

 these disturbances have the periodic time T characteristic of light 



Aug. 17th, 1904 2 E 



