941 



(A'-' + 2 4r ^ + !)■''•' + 2 (A* + V^A— l).^v/-(A' + 

 ys s 



ys ys 



The values of X, other than O or x, for which tliis equation 

 represents a degeneration, viz. a degeneration into two parallel straight 

 lines, are determined by the equation : 



{r + I i/^ A-1)^ + (;.^ 4- 24;. + !)('■= + 2 ?^tJ ;i 4- 1) =: 0. 



Each of the straight lines of a degeneration touches (L') in two 

 points, is therefore a bitangent of {L'). If we write the equation of 

 such a straight line in the shape 



a.v -f- 6v = 1, 

 then we see easily that we have 



a' -f 6' = 1, 

 i.e. the 4 pairs of parallel bitangents touch (C). 



We may observe that the system of conies to which we have 

 arrived is the system of ellipses {A') itself, which is apparent, if we 

 replace the parameter ? by X, in such a way that : 



4,5. + 4j5 + 4,. + t' = t^^^^l- 1 



4jt? \ A 



Let us proceed now to tlie second way of separation. The equation 

 of the second system of inscribed conies and the equation determining 

 the degenerations may be written down. So we come again to 4 

 pairs of parallel bitangents of (L') ; they appear to touch the hyperbola : 



P + 1 . 

 In the same way the third method Of separating leads to 4 paii-s 

 of parallel bitangents of {L'), which touch the hyperbola 



-y = - ^, • 



Hence: 



Of the 28 bitangents which the envelope (/>'), possesses 4 pass 



through 0\ the remaining ones are pairs of parallel lines ; 8 of them 



P 

 touch the circle x' -}- </' = i, 8 the hyperbola x" - -y' = — , 



P + '] 

 and 8 the hyperbola ,vy = . 



We now transfer what we have found to {L) : 



61* 



