with respect to Three Lines and a Line. 421 



g v 2(v-/i) =0 



z y x + y + z ' * 



*±.*£=A =0 , 



x z x + y + z 



^__P + 2(/*--\) =0 



y a? # + ?/ + .? ' 



x y z 



and each of these is the equation of a conic passing through the 

 three cubic centres. 



If two of the three centres coincide, then the conies all touch 

 at the coincident centres. Consider the first and second conies : 

 these intersect at the point z=0, x + y + z = 0; and the line 

 x + y + z=2kz, if k be properly determined, or what is the same 



thing, the line x\-y+z= -^—^ — -z, if is properly determined, 



will be a line passing through the last-mentioned point and one 

 of the other points of intersection : k or will of course be de- 

 termined by a cubic equation ; and if this has a pair of equal 

 roots, the conies will touch. But the equation of the line, com- 

 bined with those of the two conies, gives 



_i i i 



and substituting these values in the equation of the line, we 

 have 



1 + JL+_L_ 3 =0 



+ \ 6 + iJu + v 



which is (as it should be) a cubic equation in 0, 

 If the equation in has equal roots, then 



1 1 12 



"*~ tQ i ..\2' I A i ,.\2 /32 — "> 



(0 + \)*^(0 + p)* ' (0 + V? 



and putting in these two equations, 



_ m m m 



we have 



+ V y-$ + p' + v' 



2m n 

 x + y + z- -g-=0, 



x* + y* + z*- 2 -^=0-, 



or eliminating m, 



(* + y-M) 2 =2(* 2 -ry+3 2 ); 



