MB. J. J. WALKER ON THE DIAMKTERS OF A PLANE CUBIC. 173 



will be proportional to 



n (x"z - z"x) - m (y"x - x"y) . . . 

 or 



Lx" - L"x, Ly" - L"y, Lz" - L"z, 



and the substitution of these coordinates in 



, 



+. .. = 



gives the equation of the two chords in question, in the form 

 . (Lx" - L"*)* * + . . . + 2(L/' - L"y) (Lz" - L"z) ,, + . . . = 0, 



wherein L stands for Ix + my + nz, L" for Ix" + my" + nz". 

 The equation of that one of the pair which cuts L in the point (x'y'z) being 



(y'z" - z'y") (Lx" - L"x) + (z'x" - x'z") (Ly" - L"y) + (x'y" - y'x")(Lz" - L"z) = 0, 

 that of the other will be either 



+ {(x'z" - z'x") ^ + 2 (y'z" - z'y") ^} (Ly" -L"y) 



(Lz''-L"z) = Q, . . (53) 



or one of the two analogous forms obtained by interchanging x", y" or *", z". 

 41. The reciprocal of the equation (52) beiug, to a factor, 



ra^av' /avMfi J^L.^L ** w \ 



I a/' a*"* ' \$y" w)\ l * \$# &<' &> sy" ~ w a/' a^y 



" u> 



if x"y"z" should be a point on the poloid *, the two chords having it as their COTES- 

 point will coincide as would otherwise be inferred from the consideration that any' 

 point on s might be regarded as the point of intersection of two coincident tan en ts 

 become a " double chord," and its equation will be either 



"-L-,)=0, . (54) 



or one of two analogous forms resulting from the interchange of x", y' or x", z". 

 Otherwise, the equation of the double chord through (x"y"z"), a point on * is 



