304 PROCEEDINGS OF SECTION A. 



Among other relations which connect the quartric with 2 is the 

 following : — 



The equation of the line fi,?! is — 



"L/XI.(/X - ,.) + ^^ uX{u - A) + '^- A/X(A - fM) = 0, 



flo /^o 7o 



which may be written L' — L" = 0, 



where L' = ": [jl',' + f^u'X + "^AV = 



"o /5o 7o 



L" = " ixi>' + ^ i/A^ + I A/x^ = 



"o Po 7o 



1/ is satisfied by R' (^^ ^ ^,) and L" is satisfied by K" (^. ^, --;) . 

 Kj R' R" form the related triad f p ^ — ^ j on the quartic — 

 L' + L"^'^ +^ +7_ = L,. 



"o Ho Jo 



Hence OjPi, iljR', fiiR" form with Lq an harmonic pencil. 



The lines AjRi BiR' CjR" intersect L^ in the points ^i ^^ ^i respec- 

 tively. 



8 5. On some sjjecial relations of triads. 



Let (X.12,) (X.0,3) touch 2 in the points P" (/>t% ./^ A^) P'" (i.\ A'% /x^). 

 Then the equation of P"P"' is— 



^ (X'- + /xz.) + ^ (/x^ + ,.A) + ^ (^^ + A/x) = 0, 



tto Po 7o 



which may be written — 



(T.A,) + A/.i.(X.i^,) = 0. 



"VVe have also — 



eL, = (T.AO-A/..(X.120 = 0. 



Hence P'P'", L^, (T.Aj), (X.Oj) form an harmonic pencil whose 

 vertex is at the point 12". 



The lines— (T.A,), AiO, (X.fi,) (X.li^) : 

 AAi, AB„ ACi, AA2 

 AAi, AA2, AO, Ai2"' 

 all form harmonic pencils, while the four points— 



I2i 12, 123 12' 

 constitute an harmonic range. 



The conic whose equation is — 



(X.12,f + (X.i2,)2 + (X.123)^ = 

 reduces to — 



^ + ^ + tI = 0, 



«o' Po 7o 



showing that the axes of any triad on L^ form a triangle self-conjugate 

 with respect to the above conic. 



