22 DISPLACEMENT INTERFEROMETRY BY 



equation applying throughout the spectrum may therefore, under these 

 simplifying conditions, be written 



(3) nX = 2?M cos r cos 



This changes to (i) or (2) according as ft or r is zero. Moreover, if e 

 and y are the linear coordinates of points of the spectrum and the length of 

 the telescope tube is L, horizontal and vertical angles will be 6 = x/L, /3 = y/L, 

 with the center of ellipses as an origin. 



By the law of refraction (3) may then be changed to 



(4) n w 2 X 2 = 4<? 2 (M 2 - sin 2 i) (^ - sin 2 a) 

 Since a is small, 



\ = D sin 6' = D sin (6 + d)=D (sin 6 -\-9 cos ) 



where 6 is small in comparison with (D being the grating space and 0' = 

 0+0o the angle of diffraction), we may write further 



MW 2 > 2 (sin 0o+0 cos ) 2 +e 2 a 2 /v = e*Kp? 



where K = 4 (/x 2 sin 2 i] . 



This is an ellipse if n at the center corresponds to a maximum value, in 

 terms of the variables and a, so long as K and /* are considered constant. 

 But as n and therefore K vary with X, though slowly, it is true the equation 

 is more complicated. 



When the center of ellipses is not in the field, but passes through the ver- 

 tical plane corresponding to the center of the field of view, the ellipses may 

 soon become appreciably straight lines in their visible contours, and the 

 fringes must rotate in one direction or the other, according as the center is 

 above or below the field. Rotation will be rapid when the vertical axis of 

 the ellipses is relatively long. To bring the center into the field (for a proper 

 value of N), the angle a must be zero, i. e., the two corresponding opaque 

 mirrors which reflect the interfering beams must be rotated on a horizontal 

 axis towards each other, or from each other, until a = o, or the horizontal 

 plane through the field is a plane of symmetry. 



Furthermore, since the fringes necessarily move toward or from the center 

 of ellipses with change of N, the motion of fringes will necessarily be oblique 

 if the center of ellipses is obliquely outside the field of view. In the limit, if 

 the center is in the vertical plane specified, horizontal fringes will rise or fall. 



Finally, if n passes through a minimum instead of a maximum, the fringes 

 will be roughly of the hyperbolic type. 



At the center of ellipses in case of spectrum fringes, n is therefore a maxi- 

 mum relatively to points of the spectrum in the same vertical or transverse 

 line of homogeneous color. This maximum is due to obliquity, the hori- 

 zontal one to change of X. In the case of fringes produced with white 

 light (without dispersion), like the colors of thin plates generally or the achro- 

 matic fringes discussed elsewhere, the center of ellipses (which are now cir- 

 cles) is an absolute maximum, horizontally or vertically, i. e., relative to 

 points in all directions from the center and for each color. The center of 



