208 DOUBLE KEFKACTION AND CIRCULAR POLARIZATION 



In the case of uniaxial crystals, the direction of the optic axis is 

 fixed. We may therefore write 



regarding K and K' as constants. If we had used equation (37), we 

 should have had the factor 7i 4 instead of n n z n L 2 . Since this factor 

 varies but slowly with X, it may be neglected, if its omission is 

 compensated in the values of K and K'. The formula being only 

 approximative, such a simplification will not necessarily render it 

 less accurate. 



20. But without any such assumption as that contained in the 

 last paragraph, we may easily obtain formulae for the experimental 

 determination of 3? and <3?' for the optic axis of a uniaxial crystal. 

 Considerations analogous to those of 13 of the former paper 

 (page 190 of this volume), show that in differentiating equation (39) 

 we may regard <3> and <' as constant, although they may actually 

 vary with X. This equation may be written 



X 2 

 Therefore, 



\X 2 ) 



(42) 



d 



When & has been determined by this equation, <3? may be found 

 from the preceding. 



21. If we wish to represent (p geometrically, like Uj and U 2 , we 

 may construct the surfaces 



Aaj 2 + B2/ 2 + cz 2 + Et/0 + FZX + Gxy = 1, (43) 



the coefficients A, B, etc., being the same by which (j> is expressed in 

 terms of L 2 , M 2 , etc. The numerical value of $, for any direction 

 of the wave-normal, will thus be represented by the square of the 

 reciprocal of the radius vector of the surface drawn in the same 

 direction. The positive or negative character of must be separately 

 indicated. There are here two cases to be distinguished. If the 

 sign of <j> is the same in all directions, the surface will be an 

 ellipsoid, and we have only to know whether all the values of </> are 

 to be taken positively or all negatively. But if <J> is positive for some 

 directions and negative for others, the surface will consist of two 

 conjugate hyperboloids, to one of which the positive, and to the other 

 the negative values belong. 



