MATHEMATICS: G. M. GREEN 
591 
The corresponding line / joins the points 
The following geometric characterization of these points p and <t will 
of course afford a characterization of the line y^. 
Let / be any line in the tangent line, and I' the corresponding line 
through the point y. Project the asymptotic curves on the tangent 
plane, from any point on V , and let Ci and C2 be the conies which oscu- 
late these projections at y. There exists but one pair of lines /, V such 
that the interesections of / with the asymptotic tangents are the double 
points of the involution determined by the two pairs of points in which 
/ cuts the conies Ci and C2. Let R and S be these double points; they 
/are given by the expressions 
R-yu + \\y, 5 = % + i^y- (15) 
h a 
Recalling that the directrix of the first kind intersects the asymptotic 
tangents in the points r and ^ defined by equations (11), one finds that 
the point p is the harmonic conjugate of r with respect to y and and 
0" is the harmonic conjugate of 5 with respect to y and 5. This com- 
pletes the required characterization. 
The points R and S, whose coordinates are given by equations (15), 
are of importance for still another reason. Darboux has shown^ that in 
terms of the non-homogeneous coordinates of a regular point of a sur- 
face the equation of the surface may be written in essentially either of 
the following two forms, provided a local tetrahedron of reference be 
properly chosen : 
2 = xy + i {x^^y^) 4- i-^xy {Px^ -F jy) -f . . . . 
Darboux did not characterize either tetrahedron. The tetrahedron 
which gives rise to the first expansion was completely characterized 
by Wilczynski.^ To obtain the second expansion, three of the vertices 
of the tetrahedron of reference must be taken at the points y, R, S, and 
the fourth at the intersection of the canonical quadric with the line corre- 
sponding to RS in the relation R. 
The congruence of lines y^, since its developables cut the surface in 
a conjugate net, would very naturally take the place of the congruence 
of normals to a surface in projective generalizations of metric theorems. 
The said conjugate net would then play the part of the lines of curva- 
