( 199 ) 



ill (lieii- points of iiilcrsectioii willi (lie polar line />', lliorcloro (lie 

 polar line of P willi res[»cc( to all the cni'ves /t''; of this [leneil. 



For the cni'\e /^ ])assinj2; through P ensues from this that it 

 niiist have a node in P. 



Evidently the e(jiiation of this eurve is 



((■^jl>'\/ — (t".y 'ty /:,■ /''^/ = *), • • ... (5) 



\vhilst its polar coiiie is indicated l)y 



or by 



//',/ A'— /." =z 0, 



from wliieli is evident that it is composed of the tangents thi-ongh 

 P to the polar conic P with respect to /j\ 



For / = — 3 we find a //* with the i)olar conic L' = 0. So it 

 possesses three intlectional tangents meeting in P. 



3. The satellite conic of /-* with respect to /•- (that is the conic 

 through the points where /•'' is intersected by the tangents drawn 

 out of P) has for equation ^) 



4 a'^j: «y h^,j — 3 cix a',, b^- Ir ,j = , (G) 



or 



4^=^^ /v — 3L^ = (7) 



To determine the satellite conic for the curve h^y we put 



^'It = «'li- ^'^y + ^- «^l■ «.y l>x ^'^ir 



Then we find 



3 I'j; I,, = {I -f- 3) a\, a,j //"'^ + 2 P. a,, «',/ }>, Ir,^ ; 

 G /, l-'y = 2 (;. + 3) a,,. a\, h\j + 2 ;. (a-\ h, li^ ^ + a, a^, Z/'y), 

 or 



Z.i I-, I ==:(;. -f 1) «, rt^y //y ; 



So according to (6) the equation of the satellite of Z^ is 

 4[(;.+3) a\^yb\,,i-2)a,a%jb^0'>f] (A+ 1) c=',/Z^_y - 9 (;.+ l)"-' a,«y',^c,(;V^\y = 0, 

 or as a, fj, c and d are equivalent symbols, 



(4 A -I- 12) a\r a,, h\, - (;. -f 9) a, a^,^ A, Ir ,, = 0, 

 or 



(4 ;. + 12) b\i K - {). + 9) /y^ = U (8) 



From this ensues thai the satellite conies and the jioiar conies of 

 P w ilh respect to the curves k'^, belong to the same [)eiicil. If \vc 

 re[)resent this by the equation 



TliL' doduclioii of this lmiluiIiou is tuiiiul in Salmon "Higher plane curves"" 

 A slereoinelrifal treatment of the satellite curves is found in the ahove-nientioned 

 (iitisertaliou of Dr. H. uk Vkies, p. 18, 19 etc. 



