940 



where 



s^q^ — p [ir -\- P). 



§ 2. It is evident from the equation jnsi found that out of the 

 origin 4 bitangents may be drawn to {L'}, given by 

 x^ -2{2p-{- l),v"-y'-\ry' = 0. 



The 8 points of contact are lying on the conic 

 (K) qx'- + 2plxy - (2p + q)y'^ + 2p = 0. 



The bitangents are real or imaginary, according to p being positive 

 or negative. They form two pairs of perpendicular lines, lying sym- 

 metrically with regard to the axes and with regard to the straight 

 lines that bisect the angles of the axes. 



If {K) has its axes along the axes of coordinates or along the 

 bisectrices, then {L') and consequently {L) as well will have those 

 lines as lines of symmetry. The first occurs for 1=0, the second 

 for p -\- q z= 0. These two suppositions consequently give rise to the 

 same simplification in the shape of (L). In the formerly amply dis- 

 cussed case that /=i:0 as well as p-\-q=:0, {K) becomes a circle 

 with 1/2 as radius. 



§ 3. ^Jode.'i of (L). Let us write the equation of {L') found in 

 § 1 in tiie shape 



in which 



^^_ qx' 4 2pla;y - (2p + q)y' + 2p 

 ~ )/s 



and M and JSf are expressions of the second order, obtained by 

 separation of the expression x' —'2[2p -\- lyv^i/' -\- y\ then we see, 

 that {L') is touched in 4 points by each conic of the system, 



):-M ^2).U + N=zO, 

 in which k represents a parameter. 



The separation of the expression mentioned, may be executed in 

 the following ways : 

 x'-2{2p + l).^•y -f y' = {x' + 2\/p xy-y'){x'-2]/p xy-y') 



= {x' -f 2\/p~+lxy + y-){,^--2y^j^lxy + y"-) 

 = y-{\/2f^\-2V^p{p + \))y^\ \x-^ — 

 -(1/2^1^1 + 2 Vp.p -t- l))y'\. 

 The first way of separation leads to the following system of inscribed 

 conies 



