101 



vol. xvii. and xviii.), and the rule whicli he gives for this 

 purpose affords a remarkable example of the use of the 

 imaginary roots of equations, 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 coordinates x' , y', %', of which %' is 

 perpendicular to the surface of the crystal, and y' to the plane 

 of incidence, while x' lies in the intersection of these two 

 planes, put g/' = in the equation of the index-surface re- 

 ferred to those coordinates, 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' =z sin i, and then :solve it for z'. When i exceeds a 

 certain angle i', the four values of z' will be imaginary, and 

 if they be denoted by 



ti ± V yf^l , u' ± v' V"^, 



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 sin t 



and for the second, 



. , sin i sin 0)' , , , .,„. 



tan (J) — ;-, y = — ^ — r-, r = sv'. (10) 



u sm t 



When i lies between i' and a certain smaller angle i", two 

 of the roots will be real, and two imaginary. The real roots 

 correspond to waves which follow the law of Fresnel ; the 

 imaginary roots give a single wave, following the other laws 

 just mentioned. 



Lastly, when i is less than i", all the roots are real, the 

 refraction is entirely regulated by Fresnel's law, and the 

 reflexion by the laws already discovered and published by 

 the author. 



