49 

 GRATINGS IN THEORY AND PRACTICE l 



[Philosophical Magazine [5], XXXV, 397-419, 1893 ; Astronomy and Astro -Physics, 



XII, 129-149, 1893] 



PART I 1 



It is not my object to treat the theory of diffraction in general but 

 only to apply the simplest ordinary theory to gratings made by ruling 

 grooves with a diamond on glass or metal. This study I at first made 

 with a view of guiding me in the construction of the dividing engine 

 for the manufacture of gratings, and I have given the present theory 

 for years in my lectures. As the subject is not generally understood 

 in all its bearings I have written it for publication. 



Let p be the virtual distance reduced to vacuo through which a ray 

 moves. Then the effect at any point will be found by the summation 

 of the quantity 



A C08&O Vt) + Bsinb(p V) , 



o 

 in which & = ~, I being the wave-length. V is the velocity reduced to 



L 



vacuo, and t is the time. Making 6 = tan" 1 - - we can write this 



sin [0 + b ( p 



The energy or intensity is proportional to (A 2 -f- B 2 ). 

 Taking the expression 



(A +iB)g~ <!-"), 



when i = V 1, its real part will be the previous expression for the 

 displacement. Should we use the exponential expression instead of the 

 circular function in our summation we see that we can always obtain 



1 I am much indebted to Dr. Ames for looking over the proofs of this paper and 

 correcting some errors. In the paper I have, in order to make it complete, given 

 some results obtained previously by others, especially by Lord Rayleigh. The treat- 

 ment is, however, new, as well as many of the results. My object was originally to 

 obtain some guide to the effect of errors in gratings so that in constructing my 

 dividing engine I might prevent their appearance if possible. 



5 [Part II was never written.] 



