Crystalline Reflexion and Refraction. 191 



fracted wave ; and a normal, drawn to it through the origin, 

 lies in the plane of incidence, making with a perpendicular to 

 the face of the crystal an angle o>, which may be called the angle 

 of refraction ; so that, if i be the angle of incidence, we have 



sin o> = s sin i, 



the velocity of propagation in the fluid being regarded as unity. 

 To each refracted wave, or system of vibration, corresponds 

 a particular system of values for r, s, o>. These the author shows 

 how to determine by means of the index-surface (the reciprocal of 

 Fresnel's wave-surface), which he has employed on other occa- 

 sions,* and the rule which he gives for this purpose affords a 

 remarkable example of the use of the imaginary roots of equa- 

 tions, without the theory of which, indeed, it would have been 

 difficult to prove, in the present instance, that there are two, and 

 only two, refracted waves. Taking a new system of co-ordi- 

 nates #', i/ ', s', of which z' is perpendicular to the surface of the 

 crystal, and y' to the plane of incidence, while of lies in the in- 

 tersection of these two planes ; put tf = in the equation of the 

 index-surface referred to those co-ordinates, the origin being at 

 its centre ; we shall then have an equation of the fourth degree 

 between x' and z', which will be the equation of the section made 

 in the index-surface by the plane of incidence. In this equation 

 put x f = sin i, and then solve it for z'. "When i exceeds a certain 

 angle /, the four values of z' will be imaginary ; and if they be 

 denoted by 



u v <, u f 



each pair will correspond to a refracted system, and we shall 

 have, for the first, 



sin i sin w 



tan o> = - , s = r, r = sv ; (9) 



u smz 



and for the second, 



, sinz sin a/ , , , 



tan u/ = -, s' = r , r = s'v . (10) 



u sin i 



* Transactions of the Academy, VOLS. xvn. and XYIII. (supra, pp. 36, 96). 



