1 1 6 On the Laws of Crystalline 



substituted in equation (40), which will then become 



\ /A f\ t\\ i 2 M\ 2 / 8m M cotan 



cos (! + 1 2 ) (tan + cotan 0) - (a 2 - o 2 ) sin 2 <i sm w( : 



\ sm i<i 



= 0, (46) 



sm 



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

 and write cos 2< 2 instead of sin (<i-/ 2 ). Multiplying all the 

 terms of (46) by sin B cos 9, we find 



/ 2 7*\ 'a fl /sinwcos0 cosw \ 



= ('-6'i sm tism w cos0 : 1 - . (47) 



\ sm 2/2 cos 2t 3 / 



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 



cos o) = cos q cos ( p - 1 2 ) , 



sin <u cos ti = cos q sin (p- 1 2 ), 



aiJorded by the right-angled triangle -47fo, the equation (47) 

 will take the form 



, \ ( 2 - & 2 ) sin 2 *! ... . / \ f N 

 cos (t + iz) = - ^r cos 2 ^ sm ( p - i t ) sm (^> + 1 2 ), (49) 

 sin 2t 2 cos /ct2 



or 



cos (i 4 i 2 ) = Kco$?q (sin z p - sin 2 i 2 ), (50) 



where 



(a 2 - y) (1 + ^ 2 ) . 



26(l-6 2 )~' 



this value of K being found by assuming tan i 2 = cotan ii = b, 

 which is accurate enough for the purpose. 



Thus we have obtained ii + i z , 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 < 2 = b sin ti. In this way, if we use OTI instead of ti to dis- 

 tinguish the polarizing angle from other angles of incidence, 

 and if we put 



I- K *- (52) 



~ 1 + V ~ 26(1 - If)' 



