04 <; 



2 of the bitangents from O. Of llie 4 bitangents 0, 1 or 2 are 

 consequently isolated. 



Fio-. 3 i-efers to the case that two of (lie bitangents are isolated. 

 For a few positive and negative values of s, {L') has been drawn. 



Fig. 4 refers to the case that 1 bitangent is isolated. 



Fig. 5 to the case that none of the bitangents is isolated; {L') 

 therefore touches the 4 bitangents drawn from in real points. 



2". /^ <^ 0, so the bitangents from are imaginary. 



Fig. 6 gives a representation of this (p is supposed > — h). 



(In the figures {K) and [C) are represented as intersecting; this 

 is indeed always the case if {K) is an hyperbola). 



{K) is a (le(jent'raüon. 



As p =1= is supposed, we have only to consider the case of 

 degeneration in two parallel lines that touch {C). Generally speaking 

 we can say that substantially everything is as when (/v) is an 

 hyperbola. If the bitangents are real they will generally touch {L') 

 in real points. 



§ 9. Special cases p = and p =: cc These cases had to be 

 considered separately (§ 1). 



For p ^0 and q =\= the tirst equation which w^e ha\'e found 

 in § 1 for {L') passes into : 



+ (4/- + I') iy' — '^y- -f ^' a^' y' + h (1 + % - y") iy" — '^') = ^^ 



If we write -. 



4r + /- _ 



4q ~~ 

 then the equation becomes: 



I i,.' + Ivy —{t+l)y' + l \ iy"- - x') = qa^y, 

 {L') has now a node in 0. For the rest various cases may occur 

 also here, which we are not going to consider separately. 



If p = and besides ^ = 0, then we have to consider the problem 

 separately (cf. note p. 143). It is evident then that (L) consists of two 

 rectangles ^). 



For p = oc and q =|= oo, the first equation of (L') found in § 1 

 represents two hyperbolae, intersecting in the points (0, + 1) ; p = x 

 involves, according to the relation between g and cp (§ 1), ? = 0. 

 There is therefore no question of an envelope (L'). For p = oc 

 and at the same time ^=00 the envelope must be found again. 

 It appears that (L) consists of 2 rectangles '). 



1) Phil. Mag. p. 315. 



2) Phil. Mag. p. 315. 



