346 Prof. D. B. Brace on Achromatic Polarization 



If d 1} d 2 , ... are approximately constant for the entire 

 aperture 



8 (^) , K^) 



8N 



i l — R — + d, R + ....=^-0; . (3) 



and hence 



* &l +d2 8\ + ?•••=*■ • • • (4) 



Equation (3) gives the ratio of the paths in the several 

 plates for achromatism in any part of the spectrum where 

 equation (2) holds. 



Equation (4) gives the length of the path in each crystal 

 necessary for a resultant retardation of N wave-lengths for 

 those rays which can be achromatized. The case of achro- 

 matic l/4\, l/2\, .... wave-plates will illustrate such com- 

 pound achromatic systems. 



If d x and d 2 are not constant over the entire aperture, then 

 for any other thickness we must have as a consequence of (o) 



(St*) . „ . . »(* 



(<*! + &?,) Bx + (d s fta,) gx +• ... =0. (3a) 

 Hence from (3) 



and thus 



Bd, 8 ^^^d 2 B( ^^ + .... = m. . (4 



a 



It follows also that 



Sdi 8(e!— to 1 ) 8^2 8(6 2 -6) 2 ) _ 8N . 



"8/ 8\ + 81 ' 8\ + '- > ' 8/' * * w 



where 81 is the distance between adjacent points in the field 

 of view of a normal spectrum. 



Equation (36) shows that if at any point in the aperture 

 there is achromatism, the same will be true at any other point 

 if the variation in the paths have the same ratio as in (3). 



Equation (4a) gives the variation in paths 8d lf 8d 2 which 

 will produce a variation of 8N in the resultant order. If, 

 for example, the system consists of a series of wedges and 



