945 



8 points of iiitlexiou or none, according to p being <^ or 



>-\ 



We shall suppose p^ — 1. This is siifHicient, for it is easy to 

 prove that {L') for a value of p<^ — 1 by revolving the system of 

 axes 45° passes into a curve answering to a value of p^ — 1. 



\ï s decreases, the closed branch will increase while the symmetry 

 is lost. For a certain value of a' it touches (C) in two points. Then 

 it cuts (C) in four points, in consequence of which according to the 

 observations made in § 5, infinite branches occur. For a smaller 

 value of s the closed branch which we shall call a, again touches 

 (C) internally in two points. Then a cuts (C) in 8 points while new 

 infinite branches appear. If .s- decreases further, then a touches (6^) 

 externally in two points; two asynijitotes of b become parallel. 

 Further a cuts (C) moreover in 4 points while two asymptotes of 

 b have become imaginary. After this external touching occurs again, 

 after which a has quite passed outside (C). At the same time b has 

 become a closed branch. All the time a has remained inside {K), b 

 outside {K), for {L') cannot cut {K) now as H^ cannot become 

 zero. It is evident, that, if {L' \ has assumed the form of a ring, a 

 must have lost its points of inflexion if it possessed them. They will 

 have disappeared with four at a time. After the falling together of 

 two asymptotical directions, points of inflexion will occur in b so 

 that the closed branch b may possess 8 points of inflexion. On further 

 decrease of s these points of inflexion will disappear by four at a 

 time, while the branches a and b approach each other, in order to 

 coincide with (A') for 5 = 0. 



In Fig. 2 {L') is represented for a certain value of p<^0 (viz. 

 <^ — \) for some values of s. 



From the equation of {L') appears at once that ïovp=: — 1,{L') 

 has degenerated into two conies; at the same time (L) has degener- 

 ated into two conies. 



In the figures [K) and (C) have not been drawn as intersecting; 

 it is easily shown that they cannot intersect each other if {K) is 

 an elHpsis. 



^8. {K) is an hyperbola. 

 1°. p^i), so the bitangents from are real. 

 From the equation of [K) we deduce easily that the angle of the 

 asymptotes is always greater than 90^. Hence {K) will cut at least 



