1888.] Theorems in Analytical Geometry. 389 



AV + B"xz + C"x 2 = p, 



D"z 2 + E"xz + F"a 2 = q, 



GV + U"xz + K"x 2 = r. 



Hence the equation to the envelope is — 



Q _ J p(EK - HF) + g(HC - BK) + r(BF - EC) V 

 P 1 A(EA - HF) + D(HC - BK) + G(BF - EC) J 



/ pCE'K' - H'F') + g(H / C / - B'K!) + r(B'F - E'C) \ * 

 q I A'(E'K' - H'F') + D\HC - B'K') + G'(B'F' - E'C) J 



J p(E"K" - H"F") + g (H"C" - B"K") + r(B"F" - E"C") \ * 

 I A"(E"K" - EPF 7 ) + D"(H"C" - B"K") + G"(B"F"~ WW) J ' 



When the curve F is of the third order the first polar becomes a 

 curve of the second order, which is called the polar conic. Let us see 

 what curve the pole must move on for the polar conic to break up into 

 two straight lines. Let — 



£ O 2 + 2myz) + r] (y° + 2mxz) + V (z 2 + 2mxy) = 0, 



or the equation to the polar conic. Then 



x* + 2mx (| + |) + (*Ly* + 2myz + |z 2 ) = 0, 

 and that this equation may break up into factors 



fe|) 2 -(f ^ + 



must be a square ; or 

 must be a square ; or 



(^-p)(^-|l) = ^(f -O 3 . 



or _ m 2(£3 + ^3 + r 3) + (1 + 2 W 3)^r = 0, 



the equation to the Hessian. 



Hence the equation to the straight lines is of the form — 



+ - (J + |) = ± mQ, 



