186 



MR. EOBEET FINLAY ON 



{{Bo: + E)2/ + Cx'+ ¥x + H}' = {Cx'' + Tx + H)' 



-- (Bx + E) (Da;" + Gx' + Ka; + L) (4), 



{{Bx + E)y + Cx'-hFx+Iiy = — {Cx'-\- ¥x + liy 



+ {Bx+ E)(Dx»+ Ga;^ + Kar + L) (5); 



from which we see (III) that each of the supplementary 

 curves touches the four straight lines represented by the 

 equation 



(C*— BDX+(2CF— BG— DE)x' + (F'+2CH— BK-GE)a?» 



+ (2FH — BL— KE)a. + H* — LE = o (6) 



at the eight points in which these lines are cut by the 

 hyperbola (3). 



(c) By adding together equations (4) and (5), we get 



{{Bx + E)y + Cx' +Fx + Uy = o, 



from which it is evident (III) that the two supplementary 

 curves touch each other at eight points, which lie on the 

 hyperbola (3). 



{d) Let a straight line be drawn from the origin A to any 

 point P (t, y) on the curve (4), cutting the hyperbola (3) in 

 the points Q. and Q.^, and meeting the four straight lines (6) 

 in the points R^, R^, R,, R,; then, (V), 



PQ,.PQ, ^ (Bx + E)y -4- Ca:'+ Far + H 

 AQ,.AQ» ~ H 



PR,.PR,.PR,.PR, ^ (C'— BD)a:* + &c +(H'— LE) 



AR..AR,.AR,.AR, H*— LE ' 



and in virtue of these equation (4) becomes 



/ PQ^.Pg x', 3^^.PjR,,PR..PR..PR, 



*VAa,.AQ/ ^ ^AR,.AR,.AR3.AR/"^ ''• 



If the same straight line meet the curve (5) in a real point 

 P, we should find, in a similar manner 



„^ /PQ,.VQ,y_ PR^.PR^.PR^.PR. .^. 



^ • VaqTaQ ) -(^E-H ) Xr AR.AR.AR-^^^- 



