306 PROCEEDINGS OF SECTION A. 



Solve now for the ratios X^ ; fx, l v.^ from each of these equations 

 in turn and A^ + /^i + '^2 '■> insert the values of A^, yu.^, v^,, so found in 

 equation f, and we at once obtain for the equations of the four common 

 tangents of Si and Sj the following : — 



t, = aP,^ + /3ao^ + yRo^ = 



t, = aP„2 + /3Qi^ + yR,^ = 



(, = a\\' + /iQ,^ + yRi^ = 



i, = aPi^ + /3Qr + yU;-' = 

 where — 



^i = VJ3^, + V^i» Qi = V^, + V^i, Ph = V^, + V^i^ 



The tangents t^ t.^ t^ t^ will touch Si in the points — 



V P,' Q,' K,/' V F,' Q,' Ri/ 



V Pi' Qo' R,y' V Pi' Qi' Ro/ 



The points of intersection of two conies — 



^1 = /"^ + /^ + /^- = 

 v/ «i v/ /3i V yi 



' v/- «, ^ V i8, / y, 



may be at once found by the above method from the consideration that 

 at a point of intersection we have — 



The co-ordinates required will be — 



(ala,P„^ ^,/3,Qo^ y.y^Ro'), (a.«JV. ^iWi', y.y.Ri'). 

 (ala,Pl^ /i,/3,Q,^ y,y,Ri^), (aia.P;-, /3,/3,al^ yiy,R„^). 

 Hence since the cubics — 



c, = ;/» + y§ + ;/i = o 



t/ a, V /:Ji V yi 



c, = Vi + \IIL + Vi = 



are the envelopes of the axes of points lying on 2, and Sj respectively, we 

 have for the equations of the four common tangents of the two cubics — 



a li y n 



and three others, which may be at once written down. 



