1878.] Propagation of Plane Waves in a Biaxal Crystal. 499 



The value for the angle between the optic axes, seen in air through 

 a face normal to c is, from these values 



31° 0' 0". 



Kirchhoff found by experiment — 



30° 54'. 



The agreement is fairly close, much closer than that given by the 

 values of the principal indices as determined by Rudberg. 



Having thus found a and c, we proceed to calculate the theoretical 

 values of jjl in different directions and compare with theory, with the 

 following result : — For from about 8° on one side of the axis a to 10° 

 on the other, theory and experiment agree closely. The difference 

 only in two cases amounts to "0001, and is sometimes positive, some- 

 times negative. [The error in experiment is certainly not so great as 

 •00005.] 



But for the next 6° through which the observations extended, the 

 differences continually increase, reaching '00024 for the last observa- 

 tion, the experimental values of fi being uniformly greater thari the 

 theoretical. 



So that the results of observation would be represented by a circle 

 of radius P68125, and an oval curve with the same axes as Fresnel's 

 ellipse, viz., 



a = 1-68580 

 c= 1-53013 



which agrees closely with the ellipse for 10° on either side of the 

 axis <x, and for the rest of the arc observed lies outside it, the difference 

 between the radii vectores of the two curves increasing as we recede 

 from a. 



The differences between experiment and Lord Rayleigh's theory 

 increase much more rapidly, and amount, at the end of the arc of 16° 

 observed, to -00202, or about ten times as much as on Fresnel's theory. 

 Thus Lord Rayleigh's theory differs from the truth by considerably 

 more than Fresnel's. 



The second prism was cut so as to have its edge nearly parallel to 

 OC. 



The parallelism, however, was not sufficiently exact to enable me to 

 treat the principal plane of the prism as coincident with a principal 

 section AOB of the surface of wave slowness. It wa3 necessary, 

 therefore, to determine the values of v x v 2 , "the velocities of normal pro- 

 pagation, from Fresnel's construction. Let the optic axes meet a unit 

 sphere centre at the centre of the surface in OO' respectively ; let P be 

 the part in which any wave normal meets the sphere OP = O'P=0'. 



Then we may show that the values of v x v 2 are given by — 



