On the Laws of the Double Refraction of Quartz. 69 



The magnitude of k varies considerably with the colour of 

 the light, increasing from the red to the violet, while the coeffi- 

 cient of sin 2 0./.-, in formula (13) diminishes. If we take the 

 violet ray H, for example, this coefficient will he 159. But it 

 would be useless to make any more calculations, as we have no 

 experiments with which they might be compared. 



The figure of the wave surface yet remains to be examined. 



Eliminating k between formulae (5) and (6), we obtain the 

 equation 



(' - A) ( - B] = 4** ^, (14) 



which expresses the nature of the surface, s being a perpendi- 

 cular from the origin on a tangent plane. From this equation 

 it follows that the two values of s can never become equal in 

 quartz, as they do in other crystals ; for if we solve the equation 

 for s~, and put the radical equal to zero, we shall get the condi- 

 tion 



(A - BY + 16n- 2 ~ = 0, 



which cannot be fulfilled, since the quantity which ought to 

 vanish is the sum of two squares. The two sheets, or nappes, of 

 the wave surface are therefore absolutely separated. 



It is commonly assumed that one of the rays is refracted 

 according to the ordinary law ; but this is not the case, since 

 neither of the values of s is constant. However, the ray which 

 has the greater velocity (a being greater than b] may still, for 

 convenience, be called the ordinary ray. Of the two roots of 

 equation (8) , the one, A' , whose numerical value (supposing not 

 to vanish) is less than unity, corresponds to this ray. When C 

 is positive, k is negative ; and when C is negative, k is positive : 

 therefore in both kinds of quartz, by formulae (5) and (6), we 

 have s 2 > A, and s e * < B ; denoting by s and s e the respective 

 velocities of propagation of the ordinary and extraordinary waves. 

 Hence, if we conceive a sphere of the radius a, with its centre at 

 the origin, and a concentric prolate spheroid, whose seiniaxis of 



