^' r 



!>51 



7('7-l)( 1 --^^ 



4 y\ 



— - — . The ('iisps of Fi"-. 18 have 



{q-2f V (?-2) 



coincided in pair«. 



^ ^ (^-2)-^ \ 4 ^ ^ ^ (^ - 2)'' / 



/ l \ 1 



s = 0. { ?' = q j Fig. 20. Degeneration in the circle .r'-f-y^rzr— . 



1 

 We iiave now supposed, that (/ lies between and — . It a lies 



between — and — , we have a little chanee. Then the 8 cusps of 

 4 2 ^ ' 



Fig. 14 would already have disappeared for s = <j. 



For q between — and 2 the forms of the envelope, indicated by 



z 



Fig 16, do not exist. 



For q positive and ^ 1 no figures have been drawn for reasons 

 stated already. 



§ 12. Let us now consider the shape of (L) in general, first in 

 case {K) is an ellipsis. 



The symmetry with regard to the axes and the bisectrices does 

 not exist anymore now. The nodes, which for / = lie on the 

 axes, lie for positive values of / in the second and the fourth 

 quadrant (^ 3) ; those which lie for p -{- q = on the bisectrices 



have been removed for positive values of into the direction of 



P 

 the P-axis (§ 3\ The changes in form which (A) undergoes in con- 

 sequence of this are easily understood. 



Other forms of (L) are, however, possible. 



Let us first suppose p ^ 0. We have to start now from the 5 

 cases mentioned in § 6. 



In case i, (L) has mainly the shape which has been represented 

 in Fig. 9, in which we have to take into consideration the observa- 

 tions just mentioned. 



In case 5, (L) has, with due observation of these remarks, the 

 general shape of Fig. 11, or of Fig. 12, or it is a combination of 



I) For q = 1 (S) consists of two circles; we have then the well known case 

 of the conical pendulum. 



