MR. J. J. WALKER ON THK DIAMETERS OF A PLANE CUBIC. 197 



the envelope of which as x'y describes the line L or 2 = 0, is 



w = ajz 8 + 8& 3 7/z = 0, ........ ( H8 ) 



another conic having double contact with * and v at the same points. 



80. Lastly, the chord through a point (x"y"z") on the polar line of x'y, any point 

 on L which has the former as its CoTES-point, but which is not one of the pencil of 

 chords through (x'y'z'}, is (53) when 2' = 0, and L is 2 = 



^' {,/z" (zx* - *-) + x'z" (z,f - yz") } + 2 y'z" (z,j" - yz") - 0. 



But here 



9V' 



whence the chord in question is 



-y' y "z"x-(x'y"+1y'x")z"y + (x'y" + 3y'x")y"z = 0. . (119) 

 81. The polar line of x'y'Q being 



the envelope of the chord (119) above, as x'y'z" describes the polar line just 

 mentioned, determined as the condition that 



y a! " + ajy + b&'*z" = 

 shall touch 



x' 2 y" 2 - (y'x + x'y)y"z" - 2yyz"x" + 3y'zx"y" = 0, 



is dividing out the factor i/* 



(2a. f r'x -f Sfejy'z) 8 + 8a. 2 b s x t3 yz = 0, ...... (120) 



or, as it may be otherwise written, 



'x - btf'z)* + 86, (Za^y'x + a^ + b^z) 2 = 0, (121) 



a conic inscribed in the triangle formed by the Hues y = 0, L or 2 = 0, and the polar 

 line of the point x'y on L. 



