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XLVI. A Reciprocal Relation in Diffraction. 

 By A. A. Michelson *. 



SUPPOSE the vibration at the surface A of a sphere 

 whose centre is at to be a known function of x and y, 

 the origin being a point P on the sphere, 



V = (f> cos nt + t|t sin nt (1) 



Then the vibration t on a sphere B passing through 

 whose centre is at P will be 



W= — ^~A\ ^dxdysinnit- t) — r-^,11 ^frdxdy cos n(t — t) (2) 



Putting 



a\ f 2 ) a fa ' fa 



W= — —A 11 cf>dxdy cos(ux + vy) — \\ ijrdxdy sin(ux + vy) sin nt j 



— z-> 1 1 </> dx dy sin {ux + w/) -t- \ I ^ dx dy cos {ux f u#) cos n£ ' 



or W= Psinw^ + Qcosw^ (4) 



If now a spherical mirror be made to coincide with the 

 sphere B an image of the source will be formed at A. 



This image may also be considered as the resultant of the 

 vibrations at B. Hence, if we designate by DW the 



* Communicated by the Author. 



+ Scientific Papers of Lord Rayleigh, vol. iii. p. 80. The results given 

 by Lord Rayleigh for the intensity of the diffraction figure in the focal 

 plane do not apply to the phase of the vibration. This restriction is 

 removed if the surface considered be the sphere B ; for the distance 

 between two points on the spheres is 



orif / 2 =* 2 +2/ 2 +* 2 =£ 2 +»? 2 +(t-/) 2 



p*=r~-M-tyrj-2z£+2f£. 



If | and r) are small, £ will be of the second order, and so is f—z, so 

 that {z—f)£ is of the fourth order and may be neglected. Hence 



or approximately p =/ J . 



