CLASSICAL THEORY OF LIGHT 757 



of the reflectors through a chosen distance and counting the fringes 



which are born or consumed during the motion, one may evaluate the 



distance in terms of the wave-length of the light, or the wave-length 



in terms of the chosen distance, according as the one or the other is 



independently known. For, returning to equation (18) and putting 



/x = 1 (since the two surfaces of the "thin plate" are separated only 



by air) and r = (since the light is normally incident), we see that cp 



is changed by 27r when the spacing between the reflectors is changed 



by ^X; but when <p is changed by lir, a single ring is added to or 



subtracted from the system of annular fringes; hence the total number 



of rings created or destroyed during the motion of the mirror is equal 



to twice the number of wave-lengths comprised in the distance which 



it traverses. In this way Michelson counted, as the first step in his 



determination of the length of the standard metre, the waves of the 



red cadmium line covering the distance between the two ends of an 



"intermediate standard" or etalon, about half a millimetre long. 



In practice the real and the virtual reflector are frequently not 



quite parallel with one another; and this is sometimes a convenience. 



If they intersect (another of the things which are not possible with a 



pair of real reflectors) and the lens of eye or camera is located vertically 



above the line of intersection, this line stands forth embodied as a 



fine straight black fringe — the central fringe — companioned on either 



side by a multitude of others.^" If now either of the reflectors be set 



'" Imagine a pair of mirrors inclined to one another at a very small angle <p. 

 Establish a coordinate-frame such that the z-axis is the line of intersection of the 

 two mirrors, the x-axis lies in the plane of either. Locate the lens of the eye or the 

 camera at any point, say P; drop the perpendicular from P to the z.v-plane at Po; 

 let R stand for its length, and to for the distance between the mirrors at its foot. 

 Consider the pair of reflected wave-trains arriving at P from any direction, making 

 an angle i with the aforesaid perpendicular. Denote by a and /3 the projections of i 

 upon the xy-p\a.ne and the jz-plane respectively. Assume all these angles to be small. 

 The pair of reflected wave-trains arise from the reflection, at first and second mirrors 

 respectively, of a single primary wave-train which fell at the same angle i upon the 

 mirrors at a point where the distance / between them is equal to {to + R- tan a ■ tan (p) ; 

 or, to first approximation, t = to -{- Raip. The phase-difference between them, by 

 equation (3), is equal to (27r/X)2^-cos i. To first approximation we have 



cos i = 1 - iz2 = 1 - l(a:2 _^ ^2)_ 



Hence to this approximation we have for the phase-difi'erence: 



5 =^{to+ Ra.p){l - W+^'). 



Rearranging, and dropping the terms in a-/3 and a/3-, we get 



«2 + ^2 _ 2iR<p/to)oi - 2 + (X5/4;r)(2//o) = 0. 



The loci of constant phase-difference, and hence the fringes, are circles centred upon 

 the line inclined at angle ao = R<plto to the perpendicular dropped from the lens to 

 the mirrors. If this perpendicular meets the mirrors along their line of intersection, 

 as assumed in the text, we have to = 0; the centre of the circles is infinitely remote, 

 the fringes are straight. If then either the lens or the line of intersection is shifted 

 sidewise, the fringes march sidewise and acquire a curvature. — It should be realized 

 that if the wave-fronts are wide and the mirrors not perfectly parallel, there are 

 fluctuations of intensity along each wave-front; it may be necessary to narrow the 

 aperture of the lens in order to avoid confusion due to these. 



