312 FBESNEL ON DOUBLE REFRACTION. 



cities of the two beams, ordinary and extraordinary, is propor- 

 tional to the square of the sine of the angle which the extraor- 

 dinary ray makes with the axis of the crystal. Guided by ana- 

 logy, M. Biot has thought that in bi-axal crystals the same dif- 

 ference ought to be proportional to the product of the sines of 

 the angles which the extraordinary ray makes with each of the 

 optic axes, a product which becomes equal to the square of the 

 sine when these two axes are united into one only, M. Biot has 

 verified this law by numerous experiments, having for their 

 object to determine the angle of divergence of the ordinary and 

 of the extraordinary beam. He has compared these measures 

 with the numbers deduced from the law of the product of the 

 sines by the principle of least action, and has always found a satis- 

 factory accordance between the results of calculation and those of 

 experiment. In transforming the formulae given previously by Sir 

 David Brewster, M. Biot has discovered that the law of the pro- 

 duct of the sines to which he had been led by analogy, was im- 

 plicitly contained in the more complicated formulas deduced by 

 Sir David Brewster from his observations ; hence the experi- 

 ments of the Scotch experimenter, as well as those of M. Biot, 

 establish the accuracy of the law of the product of the sines. 

 In order to translate it into the language of the wave theory, it 

 must be recollected that in it the velocities of the incident and 

 refracted rays are in the inverse ratio of that which they would 

 have in the emission system ; hence, the difference of the squares 

 of the velocities of the ordinary and extraordinary beam, consi- 

 dered under the point of view of this system, answer in that of 

 the wave system to the difference of the quotients of unity divided 

 by the squares of the velocities of the same rays. Now I shall 

 demonstrate that this latter difference must be in reality equal 

 to a constant factor multiplied by the product of the two sines, 

 according to the construction which I have given for determining 

 the velocity of the luminous rays by a normal section made in 

 the ellipsoid constructed on the three axes of elasticity. 



Theoretical Demonstration of the Law q/"MM. Biot and Brewster 

 on the difference of the squares of the velocities. 



Let B B' and C C (fig. 13) be the greatest and least diameters 

 of the ellipsoid : the former I always take for the axis of {x), and 



