64 On the Laws of the Double Refraction of Quartz. 



the equations are reduced to the ordinary form, in which state 

 they ought to agree with the common equations for uniaxal 

 crystals. Hence, putting a for the reciprocal of the ordinary 

 index, b for the reciprocal of the extraordinary, and for the angle 

 made by the axis of z with the axis of the crystal, we must have 



A = a\ B = a z - (a* - b 2 ) sin 2 0, (3) 



supposing the velocity of propagation in air to be unity. 



Now, from the nature of equations (1) and (2), the vibra- 

 tions must be elliptical. In fact, if we put 



-=- (st - 2) , r] = q sin \-f (st - s) (4) 



= cos 



where p, q, s, I are constant quantities, the differential equations 

 will be satisfied by assigning proper values to s and to the ratio 



-. For, after substituting in equations (1) and (2) the values 



of the partial differential coefficients obtained by differentiat- 

 ing formulae (4), we shall find that every term of each equa- 

 tion will have the same sine or cosine for a factor : omitting 



therefore, the common factors, and making - = A*, we shall get 

 the following equations of condition : 



s 2 = A - Ck, (5) 



Subtracting these, we have 



which, by formulae (3), becomes 



(8) 



Let us now interpret these results. It is obvious, from for- 



