of Minerals in the Thin Section. 359 



Fedorow* has also shown how it is possible to measure the 

 birefringence 7 — /3 and /3 — a b y use of the universal stage 

 and the Fedorow mica-comparator and thus to ascertain the optic 



axial angle from the approximate formula cos 2 —- = - — , either 



by graphical means or by calculation. 



Lanef has also used the birefringence of different sections 

 as a rough measure for the optic axial angle, but his methqds 

 are even less exact than those of Fedorow and can only give 

 first approximations to the true optic axial angle of a given 

 mineral. In cases of parallel intergrowths of different amphi- 

 boles and pyroxenes they have, nevertheless, rendered valuable 

 service. Both his methods and the one of Fedorow will not, 

 however, be discussed further in this paper. 



Extinction angles of faces in zones whose axes lie in the plane 

 of the optic binormals. 



This method is particularly adapted to monoclinic minerals, 

 as amphiboles and pyroxenes, and may be of service to secure 

 a rough estimation of the optic axial angle of such a mineral. 

 The underlying principle of this method is again the rule of 

 Biot-Fresnel (page 322), and mathematical formulae suitable for 

 its solution have been developed by Michel -Levy ,J Cesaro,§ 

 Harker,| Lane,^[ Daly,** and others. These formulae show that 

 for the exact determination of the optic axial angle, the method 

 of extinction angles on different faces in the same zone is not 

 well adapted to optic axial determinations, especially when the 

 optic axial angle of the mineral is small. In certain cases, it 

 is possible to express this relation, as Lane has shown, in a 

 slightly different form w T hich is better adapted for measure- 

 ments. Lane's method, as applied to the pyroxenes and amphi- 

 boles, consists in measuring the angle between the clinopinacoid 

 and that face of the prism zone which shows the same extinction 

 angle. For this case, in which the plane of the optic binormals 

 contains the zonal axis, the formulae of Cesaro and Michel-Levy 

 reduce to the form, 



( tan A + tan u ) cos v . , 



tan 2 x = ± ^- ^ J— (1) 



1 — tan A tan /*. cos v 



\ and fju being angles between the zonal axis and the two optic 



*Feodorow, E. von, Zeitschr. f. Kryst., xxv, 349-356, 1896. 



f A. C. Lane, this Journal (3), xliii, 79, 1892. 



X Michel-Levy et Fouquee, Mineralogie Micrographique, p. 368. 



§ A. Cesaro, Mem. de l'Acad. Roy. d. Sci. d. Belg., liv, 26, 1895. 



i| Harker, A., Miner. Magazine, x, 239, 1894. 

 ^[ See Daly, Proc. Amer. Acad. Arts and Sci., xxxiv, 314, 1899. 

 **Daly, R. A., Proc. Amer. Acad. Arts and Sci., xxxiv, 314-323, 1899. 



