39 



and it results from this relation, that if 



the condition (Q") cannot be satisfied without supposing separately 



E = o. t^ =0, E" = 0. (S") 



We may therefore put 



yjv_ yv_ ^2K.~ 111. yw _ rv_ 



f/> being a quantity which can be determined by substituting these 

 values in the quadratic (C") ; for this substitution gives, 



R being the common value of the two equal roots. Hence it follows, 

 that when R is made equal to this value in the equation (A") for the 

 focus of an osculating system, that is, when we place this focus at the 

 intersection of the caustic surfaces, the coefficients of Ja', 23«5/3, 5j8*, 

 namely, ' 



become separately = ; and it is easy to prove that in like manner 

 the coefficients of 



must separately vanish, in the more general equation (V) of the ele- 

 venth number; we have therefore generally, for the intersection of 

 the caustic surfaces, when the function v" is essentially > 0, the fol- 

 lowing equations : 



