37 



but rectangular directions, which may most faciUtate the calculation. 

 For example, when we are considering an extraordinary system of 

 rays in a one-axed crystal, we may take the axis of the crystal for the 

 axis of z, and then the function v will take the form 



« = v' m'y* + n' («4 -f ^'), (M") 



the quantities m, n, being independent of a, /3, y ; and we find by 

 differentiation, 



.77 17^ = '«V +«'^«, -, Sir = '»''/' + n^c^^ , ;;^ ^=m' («« +^'), \ (N") 



v" S'u , v' S^u . v» 3»i! 



values which may be verified by the relations (G), and which give 



we may therefore conclude that whatever be the directions of the 

 rectangular axes of coordinates in an extraordinary system of this 

 kind, the function v" is essentially positive, and is equal to the square 

 of the constant m, multiplied by the fourth power of the constant n, 

 and divided by the fourth power of t; ; t; being the velocity of the 

 extraordinary rays of some given colour, estimated on the hypothesis 

 of molecular emission, and the constants in, n, being the values which 

 V assumes when the ray becomes respectively parallel and perpendi- 

 cular to the optical axis of the crystal. Hence it follows, that in 

 extraordinary systems of this kind, the foci of the osculating systems, 

 considered in the preceding number, are all comprised between the 

 two points in which the given ray touches the two caustic surfaces. 

 It is evident that this result extends to the case of ordinary- systems 



