Crystals in the Neighbourhood of an Optic Axis. 93 



indicate the fact that the direction A 2 is not capable o£ being 

 represented in the figure itself. The straight line Aj Q 

 further represents a plane through the optic axis A 1 obtained 



as follows : — Let a plane be drawn through the axes A! B l5 and 

 another through the axes A 1 B 2 ; let the angle between them 

 be J\. Then A x Q is the trace of the plane bisecting the 

 angle J\ ; let the angle made by this plane with the direction 

 Ai A 2 be denoted by K. 



6. From the formulae (5) and (6) it follows that the 

 phenomena under consideration in the neighbourhood of the 

 axis A! are symmetrical with respect to the trj system of 

 coordinates (also shown in the figure), the angle made by 

 the — f-axis with the direction A { A 2 being equal to 2K. 



Such being the case, the plane A^ contains two directions, 

 Ci and C/ (see fig. 3), making the same angle 6 with A l5 and 

 which are highly characteristic of the behaviour of the 

 crystal, and may be termed singular axes. The angle 6 is 

 deter mined by the tensors a l5 a 2 . a 3 and Z> l5 b 2 , b-s which are 

 characteristic of the crystal, and is most simply expressed in 

 the form 



e _ (bi-h) sin VuPJiiVm . 



2 V(a 1 — a 2 )(n 2 — a 3 ) 



wherein V n stands for the angle between A : and B 1? and 

 V 21 for that between A Y and B 2 . The angle 6 thus determined 

 is in all practically important cases extraordinarily small. 



7. The four propositions referred to above follow from the 

 formulae (5) and (6), and, using the representation in the 

 %n plane, may be stated as follows : — 



(a) The difference of the squares of the velocities of pro- 

 pagation, w{ 2 — &j 2 2 , of the two waves (corresponding to each 

 direction Z) — and hence also, on account of the small difference 

 between co ] and co 2 in the region considered, the difference 



