262 
[675 
675. 
ON THE FLEFLECNODAL PLANES OF A SURFACE. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878), 
pp. 49—51.] 
If at a node (or double point) of a plane curve- there is on one of the branches 
an inflexion, (that is, if the tangent has a 3-pointic intersection with the branch), 
the node is said to be a flecnode; and if there is on each of the branches an 
inflexion, then the node is said to be a fleflecnode. The tangent plane of a surface 
intersects the surface in a plane curve having at the point of contact a node; if 
this is a flecnode or a fleflecnode, the tangent plane is said to be a flecnodal or a 
fleflecnodal plane accordingly. For a quadric surface each tangent plane is fleflecnodal; 
this is obvious geometrically (since the section is a pair of lines), and it will 
presently appear that the analytical condition for such a plane is satisfied. In fact, 
if the origin be taken at a point of a surface, so that z = 0 shall be the equation 
of the tangent plane, then in the neighbourhood of the point we have 
z — (x, y) 2 + {x, y) 3 + &c.; 
and the condition for a fleflecnodal plane is that the term (x, y) 2 shall be a factor of the 
succeeding term (x, yf. Now for a quadric surface the equation is 
z = \ [ax 3 + '2hxy + by 2 + 2 (fy + gx) z + cz 2 }; 
that is, 
z (1 — fy — gx — \cz) = ^ (ax 2 + 2lixy + by 2 ), 
or developing as far as the third order in (x, y), we have 
z = \ (ax 2 + 2hxy + by 2 ) (1 +fy + gx), 
so that the condition in question is satisfied.