Doubly Refracting Plates and Elliptic Analyzers 55 



Differentiating with respect to o and introducing the condition for 

 a perfect match, 2^(0,1 — o )=o, gives: 



3 [A/1 2sin2 (i,2)sin2TrA / 'sin27r AN 



<^o lA™ J I+COS2(l,2) 



or 



c/i / I+COS2(l,2) rtr H \ f n 



/+ S1112(l,2)Sin27rA r Sin 27T AiV 7 



This will be a minimum for constant I m when sin2mVsin27rA.Y is 

 a maximum, i.e. N=% and AN=%. This is the condition in 

 the Laurent saccharimeter halfshade. Then : 



s/i ,I+COS2(l,2) ' r „ . , N 



M -=- * lin ? (,,,) /^-"-fty > < I0 3) 



Substituting 



^=I+COS2(l,2) 



* 



M *-X\jjf^f'V~+P>* • • • • ) ( IQ 4) 



or approximately 



*K=X\{pJV~*>to* ■ ■ ■ • ) 0°5) 



A comparison of the formulae for the maximum sensibility of 

 these three halfshades — balanced elliptic halfshade, Laurent 

 saccharimeter halfshade, and Lippich halfnicol — is of interest : 



Lippich: 86=^^1^/ (/ m ,a,(3,y, ...'.) (94) 



Laurent: 80=^yj^p-f(/ 7ii> a,/3,y, . . . . ) (105) 



Elliptic: *e.=X^rp-f(lJ*P,y, • • • • ) (100) 



Whatever difference, then, exists between these sensibilities in 



practice arises from the constants, a, /3, y, depending 



on the sharpness of the dividing line, the parallelism and homo- 

 geneity of the light, and other incidental factors. Since these 

 factors have great influence on the value of f(/ M ,a,(3,y, . . . ) 



211 



