38 



of rays, to which the expressions (M"), (N"), fort;, and for its partial 

 differentials, may be adapted by making n = to, a change which gives, 

 hy(0"),v" = m\ 



Principal Foci and Principal Rays. 



X4. Another important property of the function v'\ is that when, 

 by the nature of the light and of the medium, this function is essen- 

 tially greater than zero, (which we have seen to be the case for all 

 ordinary systems of rays, and for the extraordinary systems produced 

 by one-axed crystals,) the intersection of the two caustic surfaces 

 reduces itself in general to a finite number of isolated points. To 

 prove this theorem, let us resume the formulae of the twelfth number, 

 and let us suppose that the ray which coincides with the axis of z, 

 passes through a point of intersection of the caustic surfaces, so that 

 -the two roots of the quadratic (C") are equal ; then the two values of 

 tan. ^, deduced from the quadratic (-E"), will be equal also ; and if 

 we put this quadratic under the form 



E (tan. <p)» + £" tan. (p + E" = , ■ (P") 



in w 



hich 



§»« yw Vv vw 



we must have 



(Q") 



Now the coefficients E, E', E", are connected by the following 

 relation: 



