754 BELL SYSTEM TECHNICAL JOURNAL 



the plate the value tt must be assigned to v?o; as though the phase were 

 unaltered at the first reflection, but reversed at the second. This is 

 however a point of minor importance. 



The equation (18) is one of the most important in optics; we shall 

 encounter it repeatedly, even so far along as in the X-ray range. 



By comparing (17) and (18) one sees that, if wave-trains of equal 

 intensity fall upon a glass plate from all directions, those which depart 

 in the various directions are not equally intense; their brightnesses 

 depend upon their angle of reflection which is their angle of incidence, 

 i. However they are not separated in space, and hence there are no 

 bands of light and darkness. But if a lens be placed in the region 

 above the plate, it will direct each of the beams to a separate point in 

 its focal plane; and since the illumination at the point where a wave- 

 train converges is proportional to the intensity of that wave-train, 

 there will be fringes in that focal plane. If the lens is set with its 

 axis normal to the planes of the mirrors, as when one looks straight 

 at them with the eye, then the points where the wave-trains reflected 

 at any angle i converge lie all upon a circle, its radius depending on i. 

 The fringes are therefore circular; looking vertically down upon a thin 

 plate, or photographing it with a camera pointed directly towards it, 

 one sees or registers a system of concentric rings, their centre wherever 

 the perpendicular dropped from the lens reaches the plate. These 

 are said to be localized at infinity; but the term is not a good one, for 

 the fringes are not at infinity; they are on the retina or on the camera 

 film, formed by the lens in the focal plane thereof. 



The values of i for which the amplitudes of the reflected beams are 

 least or greatest may be determined by differentiating (17). If we may 

 neglect the variation of A^lAi with i (as usually but not always we 

 may), the result is the expected one: least intensity and blackest point 

 of a fringe corresponds to a phase-difference of tt or an odd-integer- 

 multiple thereof, greatest intensity and brightest point of a fringe 

 occurs with a phase-difference of zero or any even-integer multiple of x. 

 Thus one may compute the actual size of the circular rings to be 

 produced when a given lens sorts out the rays reflected by a given 

 plate, and test the predictions by experiment; or rather, to say what is 

 really done, one may determine the wave-length of the light by measur- 

 ing the diameters of the rings and comparing them with the formuhr. 

 But this is not a customary way of determining wave-lengths. 



At this point it is expedient to remark that the size of the fringes is 

 not affected by the presence of those third, fourth, fifth and indefinitely 

 many reflected beams which I excluded from the computation. For 

 the phase-difference between each of these and the one reflected once- 



