PROCEEDINGS OF SECTION A. 307 



Also the common points (other than the vertices of the triangle of 

 reference) of the two quartics — 



Q,= /^ + /§ + /Zl= 



V a V /3 V y 



Q,= /'^ + /^' + /^ = 



V a V (3 V y 



are the four points whose axes are the common tangents of Si and Sj. 

 The co-ordinates of these points of intersection are at once seen to be — 



(iV'u^'V/' (lV'Q?'R?)' VW'Q?'R?)' (iY'Q?'^V* 



§ 7. On certain special triads. 



The lines AO, BO, CO will meet Lq in points which determine on 

 that line the triad ( — 2 1 1) •: the axes of this triad form a triangle 

 whose vertices are at the triad ( — |- 11) on S and whose sides touch 2 

 in the triad (4 1 1). 



Calling the points (4, 1, 1) (1,1,4) (1, 4, 1) pi po and pj respec- 

 tively, the lines Ap^, Apa : Bpa Bpi . Cpi Cp2 determine on L^, six points 

 constitutingthe two triads ( — 5 4 1) ( — 5 1 4) ; the axes of these triads 

 determine on S the triads (25 IG 1) (25 1 16). 



If these six points on 2 be joined to A B and C we obtain the follow- 

 ing triads : — 



( — 17 16 1) ( — 17 I 16) ( — 26 25 1) ( — 26 1 25) 

 ( — 41 16 25) ( — 41 25 16) 

 lying on L^. 



This process can be continued ad infinitum to the determination of 

 triads on L^ S and 2. 



§ 8. On some particular constructions. 



If with the conditions given in the following cases, the point O 

 (uq, ^q, -/q) can be found, the construction required can be effected. 



It is to be noticed chat the tangent to S at the vertex A of the triangle 

 of reference makes with AB, AC and AO an harmonic pencil. 



Case 1. — To construct conic through five given points and to find 

 tangents to conic at such points. 



Taking A B C as the vertices of a triangle of reference, the axes of 

 the two remaining points D and E with respect to the triangle ABC will 

 intersect in O. Join AO and construct ( T-A) making with AB AC and 

 AO an harmonic pencil : similarly construct (T.B) and (T.C). To find 

 tangent at D, take the triangle BCD. Draw BO' so that BC, BD, (T.B) 

 and BO' form an harmonic pencil. Draw CO' so that CB. CO', CD, 

 (T.C) form an harmonic pencil ; then the fourth harmonic to DC, DB, 

 DO' will be (T.D). 



