Theory of Diffraction, and its Application. 53 



nitely small angles of diffraction ; so that we can write : — 

 a = («! — « ) sin a = (« x — a ) sin et ly 

 £ = (A~ A) sin A= (A~A) sin A, 



7 = (?i— yo) sin 7o = (yi — y ) sin 7i; 

 and 



cos 2 « + cos 2 A + cos 2 7o = 1? 



cos 2 «i + cos 2 A + cos 2 yi= 1. 

 If in the penultimate equation we write 



«o=«i + (*o-«i), A= A + (A— A), 7o=7i + (?o-yi), 



and develop, we get 



cos yi sin 7j_ (70 — 71) = — cos a x sin a x (a — a x ) 



— cos A sin A (A— A) 



cos«i ^cosA cosa ...cos A 



or ry = — a £j = — a p • 



cos 7! cosy! cosy! cosy 



Consequently 7 is a linear function of a and j3; hence, since 

 according to our hypothesis p and pi are constant ; A depends 

 only on « and /3*. We write this, 



A 2 = «K 2 %A- 

 With the chosen position of the system of coordinates, the 

 surface-elements that come into consideration are 



dF-pja^A, y=p;d«„3A)i 



from these we get 



ri 



and the amplitude of the diffracted light is 

 A 2 = K 2 ^ 2 B«i^A^(«^). 



Therefore the kinetic energy of the same light incident upon the 

 element B/, 



CA 2 ^BaoBA=CK 2 ^^B a oBA^(«;/5)^«iBT!. . (0) 



Starting from this expression, we can further develop in 

 two directions: the integration along a Q /3 Q gives the energy 

 incident upon the entire screen, which proceeds from BF; the 

 integration along «iA gives the energy from the whole of the 

 luminous surface F, falling on the element "df Let us carry 

 out both operations. 



1. « A are * ne variables; then 



« = («!— « ) sin «i, /3 = A - /3 , 



