162 



LIPKA. 



(23) 



2a 



2i3 = 



dx dx^ 



dy d^y 

 dx dx^ 



(25) 



With the aid of (22) we can express these coordinates in terms of t', 

 -v", v'". We are considering the coi trajectories with the fixed initial 

 ■element (o, o, v') but for which v" and v'" are variables. Equations 



(23) and the equation of the trajectories (11) furnish us tlii'ee equations 

 from which we may elimmate v" and v'", and thus get the equation of 

 the focal locus. After considerable reductions we find for the equa- 

 tion of this locus 



(24) /U ia' + ^'Y + A, (a/- /3) (a2+ ^2) + {av' - 13) (A^a + ^3^) = 0, 



where Ao, Ai, Ai, Az are functions of u, v, v', the coordinates of the 

 fixed element, only, and may thus be considered as constants. We 

 find 



j ^2 = - 3 .T„2 (1 + v' 2) i^v' -+ 2cf>v'- ^P), 



Us = - 3 xu' (1 + v' -) (cPv' '- 24^v'- 4>). 



Now (24) is the equation of a bicircular quartic (a quartic possessing 

 a pair of nodes at the circular points at infinity). The form of the 

 equation indicates that the bicircular quartic (24) has the following 

 properties : 



ii) Its tliird double point is at the origin or initial point and is a 

 node, having two real and distinct tangents at 0, viz., 



(26) a/- i3= and A^a ^ A^^ = Q, 



the first of which has the direction of the initial element v' . 

 (ii) The inverse of the quartic with respect to is 



(27) (av'- iS) (.42a + As^ + ^li) +^0=0 



a hyperbola with ai)'— /S = 0, the initial element, as asymptote.^ 



5 For a study of the bicircular quartic, see Basset, Elementary treatise on 

 cubic and quartic curves, chapt. IX; also Loria, Specielle algebraische und 

 transscendente ebene Kurven, pp. 102-108. It may easily be shown that prop- 

 erty (ii) may be replaced by the following: the fundamental point of the 

 bicircular quartic lies on a Une through perpendicular to the initial element. 



