944 



If we allow .V to change tVoui x into 0, b passes through an 

 equal change of shape. If we consider a and b, however, together, 

 then the general and special values of s, for which two asympto- 

 tical directions coincide, and those for which points of osculation 

 occur, will not be the same for a and b. 



If we take into consideration what has been observed in ^ 5 with 

 respect to the asymptotical directions of (L') and its points of inter- 

 section with (C), it is evident that we have to distinguish the fol- 

 lowing cases, which are represented in fig. 1 (with the exception 

 of the 3'd): 



J. a and b both cut (C) ; they have each two intersecting 

 asymptotes. 



2. a touches (C). b cuts {C}; a has two intersecting, /; two paral- 

 lel asymptotes. 



3. a lies outside (C), b cuts {C) ; a has two intersecting asymp- 

 totes, b consists of closed branches. 



4. a lies outside (C), b touches (C); a has two parallel asymp- 

 totes, b consists of closed branches. 



5. a and b lie both outside (C); both consist of closed branches. 



In this we have not yet paid attention to the presence or absence 

 of the points of intlexion in the closed branches; the number of 

 cases would be increased by this. 



It is evident that a value of .v exists, below which points of inflexion 

 occur both in the closed branches a and />. In that case all the 28 

 bitangents of {L') are real. 



We have now allotted to ^' all positive values, for negative values 

 of s {L') lies in the other four angles. If we revolve the system of 

 axes 45°, we shall get the same cases again. 



The value of /) determines the situation of the bitangents drawn 

 from 0. For increasing values of p they move towards the axes, 

 for decreasing values of p towards the lines that bisect the axes- 

 angles. We shall have to consider the limit-cases separately. 



^ 7. 2". p <i^, consequently the bitangents from are imaginary. 

 For a very great value of s (which we have always to take posi- 

 tive here) {L') consists of a small closed branch, given by 



x' — l (2p + 1) .ef ^ y' = -^, 

 symmetrical with regard to the axes and the bisectrices. It possesses 



