1 1 6 On the Laws of Crystalline 



substituted in equation (40), which will then become 



sin <> cotan 



/ QII 



cos (/! + 2 ) (tan 9 + cotan 0) - ( 2 - & 2 ) sin 2 ^ sin wf - 



sin 2/2 



+ ^2^--> ^ 



if, denoting the arc Ao by w, we confound </ with w, 0' with 0, 

 and write cos 2/ 2 instead of sin (ii 1'). Multiplying all the 

 terms of (46) by sin cos 0, we find 



( 2 - tf) sin 2 t ! sin w cos 



~ 



sm 2 it cos 2t 2 



From A draw the arc AR meeting the arc iY at right angles 

 in the point R, and put RY=p, AR - q. Then by means of the 

 values 



2 , 

 sin w cos 9 = cos # sin (p - 1%) , 



afforded by the right-angled triangle ARo, the equation (47) 

 will take the form 



/ x i . , v / \ /^nx 



cos (i + 2 ) = -^^ IT- cos 2 ^ sm (p - 1) sm (;j + 1), (49) 

 cos /c/2 



or 



cos (i 4 2 ) = Kvvtfq (sin 2 /? - sin 2 t 2 ), (50) 



where 



jr=(a> 2fta-V)- ; (51) 



this value of JT being found by assuming tan 2 = cotan f! = b, 

 which is accurate enough for the purpose. 



Thus we have obtained ii + i 2 , or the sum of the polarizing 

 angle and the angle of ordinary refraction. The former angle 

 itself may be inferred from formula (50) by help of the relation 

 sin i 2 = b sin i r . In this way, if we use zs l instead of ci to dis- 

 tinguish the polarizing angle from other angles of incidence, 

 and if we put 



*-r^'w^j' ( 52 > 



