1)47 



Various ska pes of tlo' envelope (L). 



§ 10. The mimber of various shapes which (/>') and consequently 

 also (L) may assume is, as we have deduced in what precedes, 

 very great. In order to facilitate the survey of those various forms, 

 we shall begin with the case that p-\-q=:{\ and at the same time 

 /=iO. The equation of (/>') runs: 



The equations of the 4^'' order in I as mentioned in ^ 3 are now 

 of a quadratic form. The situation of the double points of iL) may 

 therefore be determined by means of quadratic equations; of the 

 double points 8 are lying on the axes, 8 on the bisectrices. The 

 cases g' := and q=z oc hiive been considered separately (^ 9). 



For an arbitrary value of q we have besides the values 5 = 

 and s = 00, for which {L') degenerates, two more special values of 

 s, viz. a value for which the asymptotical directions coincide in 

 pairs and one for which the points of inflexion coincide in pairs. 



The asymptotical directions are determined by : 



{q'-s){x^—yy + mq-s)x'rf = 0. 



They are real if q^ — s and q {q — s) have different signs. 

 They coincide in pairs : 

 for s = q^[7'^0) with the directions of the axes, 



for s = qlr = — {l — q) j with the directions of the bisectrices. 

 For s = q^ the asymptotes are removed at a distance I /" 



*^ 1 — q 

 1 I /^^ 



from O, tor s = q at a distance ~'\/ — -. 

 ^ 2y l-q 



For s = q- (L') touches (6') in 4 points, lying oji the axes, for 

 s = q in 4 points on the bisectrices {^ 5). 



If the points of inflexion coincide in pairs those points are 

 situated either on the axes or on the bisectrices. 



If they are lying on the axes at a distance a from 0, then the 

 equation should run : 



(.r* -f y' -a-y = s' {.c'-a') {y''—a'). 



From this we deduce : 



(1-2^)" ' V '^ (1-2^)7 ■ 7-1 



The points of inflexion coincide in pairs on the bisectrices for: 



