DALY. AMPHIBOLES AND PYROXENES. 313 



metry in C E, and B C G a, plane cutting the vertical zone at right 

 angles. Let P represent the vertical crystallographic axis, D the 

 direction of extinction on (010), A (9 and B the optic axes. Let 0' P^ 

 be the rectangular projection of P on B C G, 0' representing the sim- 

 ilar projection of on B G. Further, let O'D' be the direction of 

 extinction on BOG, and A' (/ and B' 0' respectively the lines of inter- 

 section of the planes A 0' and BOO' with the plane B C G. Let 

 the angle B C B' ^ C, ZP A = a, ZP B = /3, ZP'O'A' = a', 

 ZP'd'B' = /3'. We have also ZP'O'B' = 6 and ZPOB^p. 



This construction represents an application of Fresnel's fundamental 

 law that, for a biaxial mineral, the direction of extinction in any section 

 is found by taking the intersection of the plane of the section with that 

 which bisects the angle between the two planes respectively containing 

 an optic axis and the line (ray of light), perpendicular to the section. 



It follows that 



a + ^ a — ft 



and 6 = 



tan a ■= 



tan/3 = 



2 2 



2 

 tan a 

 cos G ' 

 tan^' 



(1) tan 2 p = 



cos O 



tan a — tan /? 

 1 + tan a tan /3 ' 



(2) tan2^=,^^"'"^-/""^>"l^. 



^ ^ 1 + tana tan ^' cos^ O 



Equation (2) is a special case of Michel Levy's general formula on 

 page 65 of the " Mineralogie Micrographique," and also of the new for- 

 mula by Cesaro for expressing the extinction on any face of a zone.* 

 The mathematical treatment of these expressions shows that, as the sec- 

 tion moves continuously from the position (010) to the position (100), 

 it will pass through an intermediate position of maximum extinction if 

 the angle 2 F be less than 90°, or the angle (a + /3) greater than 90°. 



Thus, has a maximum when cos c = .f This is true of 



V tan a tan ft 



* Me'moires de TAcademie Royale des Sciences, etc. de Belgique, 1895, Vol. LIV, 

 p. 26. Kef. in Zeit. fiir Kryst., etc., 1897, p. 181. 



1 The position of tiie section where an extinction may be observed between 



