MATHEMATICS: G. M. GREEN 
347 
of some of my new results. The details of this discussion will appear with 
some of those of the former note, in extended form, in a single memoir on the 
general theory of surfaces. 
Let P be a regular point of a non-developable surface S, and / a line in the 
tangent plane to 5 at P, but not passing through P. In my preceding note, 
I defined a line V which is in the relation R to the line / with respect to the 
asymptotic net on S. I shall henceforth say that the lines / and V are recip- 
rocal to each other; they are, in fact, reciprocal polars of the osculating quadric 2 
of the surface at P. The line V passes through the point P, and does not lie 
in the tangent plane to S at P. If at each point of S we construct a line V 
protruding from the surface, we obtain a congruence T'; I shall say that the 
congruence V and the congruence T, formed of the lines I which are reciprocal 
to the lines l f , are reciprocal to each other. 
A projective generalization of the definition of geodesies on a surface S 
may easily be formulated, if the congruence of normals of the surface be re- 
placed by a congruence T' composed of lines /'protruding from the surface and 
denned projectively in terms of the surface. There exists, in fact, a two-param- 
eter family of curves on the surface, whose osculating planes contain the 
corresponding lines of the congruence V. I shall, with Miss Pauline Sperry, 3 
call these union curves of the congruence V . In my previous note, I suggested 
that in a projective generalization of metric theorems it is desirable to preserve 
the important property possessed by the congruence of normals: that its 
developables cut the surface in a conjugate net. I also characterized a con- 
gruence T r having this property, 4 and uniquely determined by the surface. We 
may call it the pseudo-normal congruence of the surface, and the lines of which 
it is composed the pseudo-normals of the surface. I have recently succeeded 
in showing that this congruence is the most suitable substitute for the con- 
gruence of normals, in attempting to generalize the surfaces of Voss. A sur- 
face of Voss is characterized by the property, that among its geodesies exist 
two one-parameter families which form a conjugate net; in addition to this 
defining property, however, is the important one, that this conjugate net of 
geodesies has equal tangential invariants. I have proved that this property is 
merely a consequence of the fact that the developables of the congruence of 
normals cut the surface in a conjugate net; that is, if there exist a conjugate 
net formed of union curves of a congruence V ', and if this conjugate net have equal 
tangential invariants, then the developables of T r must cut the surface in a conju- 
gate net. This theorem turns out to be a consequence of the general theory of 
reciprocal congruences. 
The definition of union curves of a congruence T f may be dualized so as to 
yield a new two-parameter family of curves, related to a congruence T, which 
we shall call adjoint union curves. These curves are defined as follows. 3 Let 
C be any curve on the surface; then the tangents to the surface which are con- 
jugate to the tangents of C form a developable, and the edge of regression of 
this developable we may call the first Laplace transform of C. In fact, each 
