948 



From what was observed in § 7 follows that we have to consider 

 for q negative values only, and positive, ones smaller than unity. 



The asymptotes, parallel to the axes, are real for all these 

 values of q. 



The asymptotes, parallel to the bisectrices, are real for negative 

 values of q, imaginary for positive ones, smaller than unity. 



The points where the points of inflexion coincide on the bisec- 

 trices, are always real. 



The points where the points of inflexion coincide on the axes are 



real for all negative values of q, and further for positive values, 



1 1 



of q, smaller than -. For values of q between — and 1 they are 



imaginary. Further we observe that the value of s, for which these 



1 1 



points occur, is between oo and q, it q lies between —and — ; 5 lies 



1 

 between q and q'', if q lies between and -. 



After the deductions made in § 6 and § 7 and this § it will be 

 superfluous to give an explanation of fig. 7, where {L') is repre- 

 sented for a negative value of q and some various values of s, and 



fig. 8, where {L') is represented for a positive value of q\<C.~]- 



^ '11. From the shape of (/>') that of (L) as reciprocal polar 



curve may be at once deduced. 



Let in the first place q be negative. There are 4 pairs of parallel 



1 

 asymptotes, touching at the circle x^-\-y'' =—. They are parallel 



with the bitangents of {L'), passing through 0. Let us now^ consider 

 various values of s. 



s'^q''. (r<^0). Fig. 9. Besides the 8 asymptotes just mentioned 

 there are 4 more, which pass through 0. The entire curve {L) lies 

 outside (C) and can therefore not be of any consequence as an 

 envelope. For on (C) the velocity of the moving point is 0; outside 

 (C) the vis viva would be negative. In fact q^ is the greatest value 

 that s can have in the dynamical problem. 



.s^ = ^*. (r =: 0). Fig. to. The cusps have coincided in pairs in 



