PROCEEDINGS OF SECTION A. 205 



By forming the conies whose equations are — 



(T.A,) + (T.B.) + (T.CO - "' 



and (T.Ai)=' + (T.B,)' + (T.Ci)' = 0, 

 we find that the tangents to S at any related triad of points on S form 

 triangles whose vertices lie on the conic — 



V + s = 0, 



Avhile such triangles are self -conjugate with respect to the conic — 



Lq^ — S = 0. 

 If Li L2 L3 be the three chords of 2 formed by joining a triad of 

 points on that conic, then from the equations — 



vl, + vl; + ve; = 



Li^ + W + L3^ = 0, 

 we learn that the triangle LiLgLj envelopes the conic — 



5V-16S = 0, Xf\^\^^ 



and is self -con jugate with respect to the conic — ^\J^ " --•' — "~^ Z'' 



3L^2 - 8S = 0. y^ #*'"*"* ^<fi;\f 



S 6. On the cotnmon tanqents of St and So. j"^ \^. 



R Lj t ' 'S R A * Y'sal 



LetS, = ^ + ^§ + 7.= „^ — J^ 



s. = »^ + § + ^ = o.W-;-\-^' 



Then the lines — 



a > ^ /3 2 17 2 n 

 t =z ~ A, + ^ /x/ + iL 1/3- = 



«2 p.^ y^ 



will touch S, and Sj respectively. 



If ti and to represent a common tangent of Sj and Sj, then 



a, ~ " ttj ' fii ~ " pr yi ~ " y,' 

 Hence A^ = ± A, /.-^, /., = ± /., M , '-i = ± »'2 /'cl', 

 and therefore since A, -f- /*i + I'l = 0, we obtain — 



± K /"^ ± /^2 /^' ± "o /^ = 0, 



V ttj V /?2 V 72 



which represents only four distinct equations of condition. 



Writing a, 6, c for , " ' , /— ' /'' respectively, these equations 

 V «2 V 13/ V 72 

 are — 



Ci ^ Xod + /X2^ ~l" "^2^ = 



C2 r^ A2O — fJiJ> — J'2C ::= 



C3 ^ — AoW 4* H-i^ — "2^ = 



Ci — Ajtt {J-2^ -\- v^c =1 



V 



