of Fresnel's Optical Theory of Crystals. 79 



Then *^ = i- nearly, and r = ^y^ -f- 2:"= a:, 

 or, if we change the origin to the other extremity of the prime ray, 



y J 



*/=y r^b-x, 

 so that the =" becomes 



-F-i<-*±'V('-S)(5-) 



Hence at each singular point the surface is touched by a cone, the 

 =" to the generating line of which is given by the above, the ex- 

 treme angle between it and the prime ray being 



"-' ■ {(v/'-5)(l-)} 



When 6= a 4' always = — and the cone returns into a plane. 



Again, let us suppose that the position of any perpendicular 

 from the centre is given, and that of the corresponding radius vec- 

 tor required. 



Let O A, O B* denote what we have termed the optic axes, but 

 which it will be more agreeable to analogy to term the prime per- 

 pendiculars from centre, and let O P be the given normal. Take 

 O Q, O R contiguous perpendiculars from centre in planes P O Q, 

 R O Q, perpendicular to P O A, P O B respectively, then the incli- 

 nation of the two former will be the same as that of the two latter, 

 and may be termed ju,. 



Let i^ 1^^ now denote the angles P O A, P O B respectively, then 

 QOA = i^ QOB = »^^ + ^lyy 



R O A = i^ + J ^ R O B = <,^ 



The ray will be found by joining O with the intersection of three 

 planes drawn at P, Q, R, perpendicular to O P, O Q, O R, respect- 

 ively. 



Now from Prop. 9 it appears that 



using only one sign for the sake of simplicity, which we may do by 

 throwing the ambiguity upon the way in which i^ or i^^ is measured, 

 also 



0Q = 0P + ^^-2Z^i,, 



OR = OP-fl:^l« 



d 



• O A, O B are not expressed in the figure, 



