PROCEEDINGS OF SECTION A. 299 



The equation of A^O is satisfied by the point Ai ( - - - j which 



clearly lies on the conic S. To determine the position of Aj, we may- 

 proceed as follows : — 



The equation of AA^ may be written — 



The equation — 



^'^{l + £) - ^{k + 1) = <^ 



reduces to — 



Let Lq and (X.Q,) meet a = in H and K respectively : Then 

 AA2, AH, AK and the line whose equation is — 



form an harmonic pencil. 



This last line determines the point in wdiich 



(X.AO = ^^x + |^;. + ;-. = 



meets « =: 0. Therefore since (X.A,) passes through O, two points in 

 the line are known. Hence (X.A,) is found, and consequently Ai can be 

 constructed geometrically. 



(X.Ai) will meet L^ in the point Q' whose co-ordinates are 

 (/Li — V V — \ \ — /i). The equation of (X.Q') is — 



/^ . ^/ ^ = : 



+ :r-h=-T-> + 



«o(/' — ") /^V»' — ^) 7o^^ — /O 



this line will touch 2 in the point Pj R/t — i')' (j' — ^Y (^ — /O J > 

 it may be at once verified that P^ lies on the straight line A, O Pi Aj. 

 The equation of (T.Aj) is 



— /V + — a -f- - V — U. 



«o ^o 7o _ 



Since A," — /ti^ = /r — v\ =. v'- — \/j =: 0, this equation may be 

 written — 



Q U + 3; + y + V"' (^. + .^ + ^j = 0- 



Hence (T.A,) passes through the intersection of L^ and (X.Qi) : (T.Aj) 

 can therefore be constructed. 



(T.Ai) will meet L^ in the point Q" [x (/t — v) u (v — X) v (X — /«)], 



therefore, since L^ is the polar of O witl\ respect to S, the line A^Q" is 

 the tangent to S at the point A^. 



The axis of Q" will touch S in the point — 



P3 [X- {^t - vf fi^ {v - X)- V- (X - i.Cf\. 

 The equation of P1P3 is — 



I- A"' (/t — v) -\- p~ v\ {v — X) -\- ~ X/t (X — /t) = 0. 



