943 



15' cos^ a -\- 2pl cos a sin n — (2p -\- q) sbi^ a -{- 2p\' =z ^ 



=r 6' [cos^ a — 2 {2p -\- 1) cos'' a sin'' a + si)i'^ «}. 

 If we write this in the form : 



1^ sin^ a — 2pl cos a sin a — (2p + 9) <'<'•?'' f(\^ == 

 =. s \sin* a — 2 (2p + 1) «^'os' « «<«'' « + <^os^ «j 

 theji it is evident, that 



y =^ — X cotg a 



in an asymptotical direction of {L^}. 



If (L') touches (C), two asymptotical directions coincide, they are 

 perpendicular to the line that connects with the points of contact. 



^ 6. (K) is- an ellipsis. 



1°. p ^ 0, consequently the bitangents from are real. They 

 cut {K) in real points, in which points they touch {L'). 

 The bitangents divide the plane into 8 angles, in which 



n, = x'—2{2p-{- 1) X' ƒ + y' 

 is alternately positive and negative. (L') lies for positive values of 



s = q^-p (4r + I') 



in the angles, where H^ is positive. 



Let us call the branches of (L'), which are lying in the one pair 

 of opposite angles, a, those which are situated in the other pair, b. 



Let us begin by giving positive values to s and let us first consider 

 a exclusively. 



For 5 =: GO degeneration in two bitangents. For large values of 's, 

 a consists of one branch with two asymptotes and four points of 

 inflexion. For decreasing values of s the angle between the asymp- 

 totes becomes smaller, the apices are removed from each other and 

 the points of inflexion move towards infinity. For a definite value of 5 

 the asymptotes are parallel. If there is a further decrease in s, a will 

 consist of two closed branches in which for another special value 

 of s points of osculation occur in the sides turned towards 0. Then 

 two points of inflexion appear in each branch and the branches 

 contract, till we have for s =: degeneration in the ellipsis {K) ^). 



•) The case S = q^ — p '4r + /-) = must be inquired into separately. For s = 

 is the condition that in the second part of the relation between ^ and c (p. 939) 

 the root may be drawn. In this case (A') represents two jjencils of ellipses. 

 Consequently the required envelope (L) has now degenerated into 8 straight lines, 

 which are the polar lines of the base points of those pencils, and in (K'), which 

 is the polar curve of (X). 



