( 200 ) 



h\jK^liU = () (9) 



tlien 



u = 2 / : (/. + 3) furnishes the pokir conie, 

 ft' ;=_(;. -f- 0) : (4 /. + 12) the satellite eouic of /^. 

 Between the i)amnietei-s jt and ft' exists the bilinear relation 



H — 4 n' := 3. 

 So for n = — i and ji z=i x we find two cnr\'es Z'> for \vhich 

 ])olar conic and satellite coincide. 



In the first case we have / == — i ; so we have the curve P; 

 l)0ssessing in P a node. 



In the second case we find / = — 3, so a curve for wliicli the 

 polar conic is a double right line. 



For P. = — 9 the satellite is indicated bv A'=(). We tlioi have 

 the harmonic curve for which the satellite coincides with the [)olar 

 conic of /l' ; this well-known property indeed, ensues immediately 

 from the definition of A^ 



4. Let us jiow consider I he system of the satellite conies of a 

 given point P witii respect lo the cubic curves of any pencil 



J -f- /J5 = 0. 



By means of a selfevident notation the jusi mejitioned system is 

 represented by the equation 



4 (.i„ + ;. B,) (Ka + ;. Ko) - 3 {La + ;. ur = 0. 



So through each point of the |»lane pass two satellites; the index 

 (.1 is here t/ro. 



The satellite consists of two right lines when P is situïited on 

 the Hessian. Now the Hessians of the pencil e^'idently form a system 

 with index three; the number of pairs of lines d is tlierefore three. 



A double line is found oidy when P lies on the cubic curve; 

 consequently^ for our system i] is equal to 1. 



Betw^een the characteristic numbers of a system of conies exist 

 the wellknown relations 



2 II ^ r -f- ij and 2 r = (i -|- d". 



We lind from the first r =r 3, u being equal to 2 and i] to 1. 

 The second then gives tf=4. From this ensues that the just men- 

 tioned satellite formed of two coinciding right lines must at the 

 same time be regarded as a pair of lines, thus as a figure in \\ liicli 

 the centres of the tw^o pencils of tangents ha\e coincided. 



From the equation 



9 (Z-„-f ;. Ko) (L„+;. Lo)-{A,+?. B^) {A f ;. B)^i) 



