300 PROCEEDINGS OF SECTION A. 



This line is satisfied by the co-ordinates of Qj : hence P3 is the point 

 of intersection of the lines (X.Q") and QiPi- 



The equation of P2P3 is — 



": ^ , ^ /^ , 7 ^_ _ : 



«o /A — // ■" /^Q 1/ — A, 7q \ — /t 



hence the points P2 ^rid. P3 are collinear with the point — 



The line A1OA2 is the axis of a point A3, lying on S, whose co-ordi- 

 nates are — 



r^ 1 ^1 



\\ (yH. 1') f.1 {\> A,) V {\, ^t)J 



The equation of (T.A3) is — 



1 ^- (^, _ ,)'^ + ^^ ^^ (, -icf^\ V- (X - ^Cf = 



= G \a^ >"' + ~o^'" "^ 7^ " J "~ ^^^"' ^. ^ + ^o ^* "^ 7° " J ~ ^ 



= e (T.AO -3A/n.(X.A0 



Er (X' + /r + V' — /(»/ — ,.\. - A;0 (T.A2) 



— 3 (^ — v){v — \){X — fi) (X.A2) 



Hence (T.A3) is determined as the line joininp: the intersection of 

 (T.A,) with (X.Ai) to the intersection of (T.A3) with (X.Aa). 



A1OA2 will meet Lq in the point Q'" (0i 02 ©3), where 

 e, = X' + 2/41^, 02 = ytr + 2,A, 03 ^ 1'' + 2\/n. 

 The equation of OA3 is — 



~ Hf^ - 01 + 1 K^ - X) 02 + ^ <x - /O 03 = 0. 



It is at once verified that OA3 and S are both satisfied by the point 

 A T- - — 



' V0, 02 03 



To determine the position of A^. Ave have — 



(x.A,) =^-e.-h ^^ e. + 1 03 = 0. 



= (T.AO + 2Vu.(X.Q0- 



Hence (X.A,) is the line joining to the intersection of (T.A,) and 

 (X.i2i), and therefore A4 can be constructed geometrically. 



(X.i2"') will touch S at the point P^ (0i- 0.^ 03^) and the equation 

 of OP4 will be— 



^ (02^ - ©3^) + f^- (03^ - 0r) + '- (01^ - 02^) = 0, 



"■o '-^O 10 



which easily reduces to — 



^ X(;t - 01 + 1 /<" - ^) 02 + ^ "(^ - /') ©3 = 0- 



Hence OP4 is identical with OA3, that is to say, P4 is the point of 

 intersection of OA3 with (X.12'"). 



