A Reciprocal Relation in Diffraction. 507 



operation by which W is obtained from V, 



W = DV ) (5) 



or V = DDV. ....... (6) 



These equations hold for light of any colour^ and — with 

 corresponding extension of the definition of D — to any 

 combination of colours. 



The analogy with Fourier's formula is apparent. In fact, 

 the second of equations (5) is, putting n/fa — p, 



V= —win QdZdycospfalj+ytf)— \\ P dgdri sin pfag+ytf)! cos nt 3 



- ^f\\\ QdWv *™ p(?i£ +!/&)+ \\ PdgdqGospfag +y l <n)]s\nnt 2 

 Substituting the values of P and Q from (4), 



y= jp/* i \\ \ dxd y d £ d7 )f(x,y) sin [(«i-«)pf + {yi-y)pi\ 



+ ^A\\\ dxdyd % dr] ^^ cos [(^■" a? )i ? ? + (yi-v)p 7 )} 

 ^ ^\\\\ d ^ d y d ^ d v4>{^y) ^n [Oi-^?+ {yi-y)pv\ 

 + xf lllr^^^^^ cos [(^"^f+Cvi-y)^] 



cos nt 2 



sm nt 2 



The first and third integrals are identically zero, so that 



putting 



n% nrj 



u = 7— and v = — - 



/« /a 



and disregarding the phase difference between t and t 2) we 



have by (1) 



4tt2 .... 



ys <M*i>yi) =1)1) dudvdxdy${x,y) cos [(^—0)11 + (y,-,y)t>j 



This, disregarding the intensity factor / 2 , is the Fourier 

 formula extended to two dimensions. 

 Formulae (5) express the fact that if 



\V is the diffraction image of V, then 

 V is the diffraction image of \\ . 

 In applying the formula? it must be remembered that V 

 and W represent the vibration — not merely the intensity. 



2 L 2 



