faces ; but when it is required to determine also the two series of 

 caustic curves contained on these two surfaces, or the two series of 

 developable pencils composed bj the tangents to these curves, we 

 must then have recourse to integration. The differential equation in 

 X, y, z, which determines the developable pencils, may be found by 

 eliminating ^ between the formulae marked (E'), and may be put 

 under any one of the three following forms : 



VL 



%x ' 



rL') 



in which a, j8, 7, are considered as given functions of x, y, z, deduced 

 from the equations (C) . The developable pencils having been thus 

 determined, by integrating the equations [L'), the caustic curves will 

 be known, because they are the aretes de rebroussement of those pen- 

 cils ; the caustic curves may also be found by the condition of being 

 contained at once on the developable pencils and on the caustic sur- 

 faces ; or, finally, we may find the differential equations of these 

 curves in a/', y", z'', without reference to the developable pencils, by 

 combining with the formulae {F') the differential relation between 

 «» /3, y, which results from the equations {B') and admits of being 

 put under any one of the three following forms : 



S'^J^. S^=5>. s|., V (M') 



0/ 



oy dy 



•a. 



