( 373 ) 



cof it (.v/'/;' V +w),v' Vro.^'' ./•) —2 rot il sin" Vtln .v cot 27/'-|-(c-ox' .v—kIh" F) = O 

 sm^ V sin .V cot 2y' 



cot a = 



sin'' V^cos^ Vcos^ w 



sin^ V sin ,v cot 2?/' 



cos x — sin 



V 



sin^ V-^cos^ Vcos^ .v.J \sin^ V-\-cos^ V cos'' o;^ 

 As long as the second term reiiiaiiis smaller than the first, the 



:i 



condition for which being cosu:^shi V, or ,?,' <^ [', this equa- 



tion will yield two positive roots, and accordingly two values of « 



between and — will satisfv it at a given valne of 2?/ smaller 



2 ' • 



than the maximnni. The extinction will have reached the maximum, 

 when the t\\'o roots are eqnal, so if 



(sin'' V sin w cot 2v/)^ =r {sin' V -\- cos^ V cos^ x) {cos^ x — sm^ V) 

 or: 



sin V tq V 



(7) 



sin 2limax -=■ 



COS X cot ,r, 



whilst the corresponding value of a ^) is found from 



sin'' V sin x cot ^y'max 

 <'0t Umax = — 



. - (8) 



sin^ V -{- cos^ Vcos^ x 



The })lane ON^, in which lies the corresponding pole, then makes 

 witli the plane OP an angle 



If we take for olivine the value 2F=87''^), this gives according 

 to the aboxe foi'mulas the following figures : 



TABLE I. 



^) ci{y' — mux) is denoted by z,«„.r, wlierevei- it coald ii^iL give rise to ambiguity 

 2) Min. d. Roches, p. 248. 



