50 TRANSACTIONS OF THE [NoV. 16, 



3. Measuremenl of the apparent or of the true angle between 



the op)tic axes. 



In measurements of the angles between optic axes, in general 

 the section must be normal to either the acute or the obtuse 

 bisectrix and an oblique section introduces an error. When the 

 section is surrounded by a liquid which has an index of refrac- 

 tion exactly equal to that of the mean index of the substance 

 under examination, there is no bending of the ray on entrance 

 or emergence, the determination of the angle is independent of 

 the section and all planes of the zone normal to the axial plane 

 act like one plane, that is an entire uncut crystal can be used. 



With a liquid approximately that of the mean index the error 

 introduced is slight. 



For measurement of the apparent angle the axial plane is 

 found as just described, in a liquid of approximately the mean 

 index of the cr^'stal, and the approximate position of the axes 

 noted in parallel light during revolution of A'. The converger is 

 then introduced and placed in horizontal position, the polarizer is 

 then set in the diagonal position, the Berti-and lens inserted 

 with a suitable ocular and a cap anah'zer. The branches of the 

 hyperbola are brought into tangency and the four positions 

 of the emerging axes are read, then the true angles are calcu- 

 lated by the formulas 



„ n sin Ha a • rr '>^ si'^ ^o 



sin Va = ^ — - and sin Vo == — - 



in which 2 H„ and 2 i7o are the apparent angles, n the index of 

 refraction of the liquid, ;? the mean index of the mineral and 

 2 F„ + 2 Vo must equal 180°. 



Direct Determination of the True Angle. 



If n=P then sin Fa^sin H^ and sin Fo=sin Ho and the 

 angle is obtained at once. To practically obtain this, use a 

 liquid with n a little above ft and add diluent drop by drop 

 until the boundaries of the cr3'stal fade. Then for the accurate 

 determination some coarse powder of the substance is placed on 

 an object glass, a drop of the liquid added, a cover glass pressed 

 on and the extinction directions determined in a selected grain. 

 With the condenser lowered and the analyzer out, the two direc- 

 tions of extinction are successively made to coincide with the vi- 

 bration direction of the polarizer and the microscope is focussed 

 sharply on the dark boundary between the liquid and the grain. 

 The objective is then raised and the dark boundary line appears 

 to move towards the substance with the higher index of refrac- 



