298 PROCEEDINGS OF SECTION A. 



The axes of all points of S with respect to tiie triangle AjBiCi pass 

 through O, while the axe*; of all points on Lq with respect to the same 

 triangle envelope the conic 



V/(X.Q,) + y/{X.Q,) + l/(X.Q3) = 0, 

 which on expansion reduces to the conic 2. 



In order to establish a triad on the line L^, draw any straight line 

 thi'ough O, and find the point H, of which this line is the axis. Since 

 H must lie on the conic S, we may assume its co-ordinates to be 



(?' ^^ 7) "'hen X + ^, + „ = 0. 



AH, BH, CH will meet L^ in the points H, (X, v, /li), H.,(i/, /n, \), 

 H3 {juL, \, v), i.e., H, H, H3 constitute the triad (X j/ /<) on L^. 



(X.H,) (X.H3) will meet in A'f ^, 7, ^\ : (X.H3)(X.H,) will meet 



X 



^^^'{l^i \) : (X.H,)(X.HO will meet in C'(^,i, 7). 



Let AA', BA', CA' ; AB', BB', CB' ; AC, EC, CC meet L^ in the 

 points Di E, F, ; D.^ K, F.^ ; D3 E3 F3 respectively. For the co-ordinates 

 of these nine points we obtain — 



{K /', I') (/*, I', ^) (j'. X., yW,) 



(l'. X, /<) (\, /<, v) (/<, V, X) 



i^i, V, \) (^, X, /O (A.. /<, v) 

 showing that AA^', BB^', CC^ meet in Di, that BA', CB', AC meet in E,, 

 that CA', AB', BC meet in Fi, and that the points Di Ei Fi establish the 

 triad (X /i j'). 



If the pcint Q, (\, /t, v) be given on L^, we may find the two other 



points of the triad as follows: — Determine the point A, ( - , -, - j — 



for the construction of this point see § 2 — ; Draw HA; CAj meeting L^ 

 in H2 {v /(, X) and H3 {jn X v) '. (X.Ho) (X.H3) will meet on S in 



A" (1 , J , ^) : AA" BB" f:C" will meet L^ in (X /t v) {/a v X) {v X /t). 



The axes of the triad (\ /<. j^ will touch 2 in the points Pi (X^ fj^ v''), 

 P2 (;tt* v^ A,'^), P3 (j'^ A^ /i')- If now lines be drawn from the vertices of the 

 triangle ABC to Pi P2 P,, we obtain nine points on L^, which may be 

 arranged to form three distinct triads. 



§ 2. On the concurrent determination of points on mid tangents to the 

 associated cotiics S and 2. 



On L° let any point O; (\ /t j') be taken. The equation of the line 

 joining Qi and O («o^o7o) "''^^ ^^ 



i (/^ - ") + I ('^ - ^) + 7 (^ - /O = «• 



This line is the axis of a point A. ( _ ^ ;^n x - ^) b'i"g O" S. 

 The equation of A2O is — 



^^ K (;t - .) + |;. (. -\) + 1^.{X - ^0 = 0. 



This line will meet (X.Qi) in the point Pj (X^ /t^ v"^) ; this point will lie 

 on the conic 2, as may be seen from inspection. 



