IN CRYSTALLIZED MEDIA. 345 



v- = a" (sin' ^ cos' e + cos' m siir e cos' ^ + ^ sin 2;u sin Se cos x//) 



+ i' (sin- M sin' e + cosV cos' e cos' >// - ^sin 2/usin Secos v^) 

 + c^ cos^ fx sin \//^. 



Hence «>' + v'- = «' (cos' e + sin' e cos' f) + ¥ (sin' e + cos' e cos' \//) 



+ c'sin'x// (1); 



4)'^ _ v"- = (a- - b-) {sin 2m sin 26 cos v/. - cos 2m (cos' e - sin' e cos' \|/) | 

 — {Ir — c') sin' i// cos 2 m (2). 



21. Now in tlie equation of condition (^9) since 



M=~-'l- N=^-^- p-^.t. 

 3 6 ' ^^ 3 6 ' ^ - 3 6 ' 



__{(l-tr) ,. p_ A'-C. 



■•■ M- N = - '- ^ ' N-P = - 



6 ' 6 ' 



by substituting these values, that equation becomes 



sin 2m { («^ — f>^) (cos' e — sin' e cos' \//) + (i' — c') sin' \|/} 

 + cos 2m («■ - b') sin 2 £ cos \// = ; 

 .-. («' - ¥) (cos'e- - sin' £ cos^ >//) + (i' - c')sin'>// 

 = - cot2M(«' - i') sin 2m cosx// (a) ; 



hence »" - «' = («' - Z»') sin 2m sin 2e cos x// + (a' - 6') 52!_*^ sjn 2e cos x|/ 



sm 2m 



, 2 ,,, sin2ecos\i/ 



= (a - o') ^ — - — J- . 



^ ' sin 2m 



If -f = |, it appears from the above equation for cot2M(«), that 



M = 0. or -, and 



sin 2m = 0. 



This vahie therefore does not make tlie expression vanisli, as would 

 appear at first sight ; it can vanish only, when 



sin 26 = 0; 



