240 Mr. A. A. Michelson on Interference Phenomena 



If, on a plane perpendicular to EE' at distance D from E, 

 we call x distances parallel to P'C, and y distances parallel to 

 P'D, counted from the projection of E on this plane, then, 

 putting tan<£ = K and D + P=S, we have for the equation to 

 the curves, 



DT+BK* 

 Vt> 2 + tf 2 + / 

 or 



Ay = (4S 2 K 2 -A 2 > 2 + 8TSKD^ + (4T 2 -A 2 )D 2 . . (6) 



If, numerically, 



A < 2SK, the curve is a hyperbola. 

 A=2SK, „ „ parabola. 

 A>2SK, „ „ ellipse. 

 K=0, „ „ circle. 



A = 0, „ „ straight line. 



All the deductions from equations (4) and (6) have been ap- 

 proximately verified by experiment. 



It is to be noticed that in the most important case, and that 

 most likely to occur in practice, namely that of the central 

 fringe in white light, we have A = 0, and therefore also t = 0; 

 and in this case the central fringe is a straight line formed on 

 the surface of the mirrors. Practically, however, it is impos- 

 sible to obtain a perfectly straight line ; for the surface of the 

 mirrors is never perfect. 



It is also to be noticed that the central fringe is black ; for 

 one of the pencils has experienced an external, the other an 

 internal reflection from the surface of b (fig. 1). This will 

 not, however, be true unless the plate g (which is employed to 

 compensate the effect of the plate b) is of exactly the same 

 thickness as b and placed parallel with b. When these con- 

 ditions are not fulfilled, the true result is masked by the effect 

 of " achromatism " investigated by Cornu (Comptes Rendns, 

 t. xciii. Nov. 21, 1881). This remark leads naturally to the 

 investigation of the effect of a plate of glass with plane parallel 

 surfaces, interposed in the path of one of the pencils. 



The effect is independent of the position of the glass plate, 

 provided its surface is kept parallel with the corresponding 

 mirror. Suppose, therefore, that it is in contact with the 

 latter, and let c d (fig. 4) represent the common surface. Let 

 t = hi= thickness of the glass, ?'= angle of incidence, r— angle 

 of refraction, n = index of refraction, X = wave-length of 

 light. Let ef represent the image of the other mirror, and 



hk 



put ?j. = — • 



