Doubly Refracting Plates and Elliptic Analysers 53 



and 



„ . 1 — 2sin 2 2 (1,2) sin'-.7r A A' 



cos 2(0, 2— v )= , = (Q7> 



V 1 — sin-2(i,2) sin^Tr-AA/" 



These give a minimum of I m for any constant value of 

 sin2(i,2)sin27rAA r . The negative sign is used since the positive 

 sign gives a maximum of T m . Substituting these values in 8e a 

 and reducing : 



1 — 1 1— sin 2 2(i,2)sin 2 27rAxV x 



<>e =%. ; — ; rH — tz fU , a ,R,y, . . . . ) (98) 



/4 sin2(i,2)sin27rAA^ J v '" H r ) ^ y * 



This value of le o can be written as a function of I m ,a,/3,y, .... 

 alone and independent of 2^(1,2) and the differential order of the 

 halfshade, AN, for : 



I m -=P o { 1 — 1 1 — sin 2 2 ( 1,2) sin 2 2 7T AN ) 

 from which 



sin2(i,2) sin 2 7r AA f =-p 1 I m (2P — f m ) 

 or 



K=%4^T m f {I ^ y } (99) 



Or approximately, since I m is small in comparison with 2P o : 



*<.=X^jfrfV*<hfl>r> • • • • ) < IO °) 



Therefore, the maximum attainable sensibility of a balanced ellip- 

 tic halfshade of the type discussed is independent of the differ- 

 ence in order of its two halves (AN), provided that the differ- 

 ence is great enough to secure the requisite intensity. 1 



Although the condition above discussed gives theoretically the 

 greatest sensibility, it is more usual and in practice easier to make 



lr rhe conclusion drawn by Zakrzewski (Zakrzewski, M. C, Bull. Int. dc 

 I' Acad, des Set. de Crac, pp. 1016-26, Nov., 1907) that a quarter-wave 

 Bravais biplate (aN= 1 / 4) gives a maximum sensibility is apparently .due 

 to an invalid approximation. 



209 



