348 
MATHEMATICS: G. M. GREEN 
point of this edge of regression I shall call the first Laplace transform of the 
corresponding point of C. Let T be a congruence composed of lines /, one in 
each tangent plane of the surface. Then the curve C is said to be an adjoint 
union curve of the congruence V if the first Laplace transform of every point 
P of C lies on the corresponding line / of T. 
A study of the relations between the union curves of a congruence T' and 
the adjoint union curves of the reciprocal congruence T leads to many inter- 
esting results in the general theory of surfaces and rectilinear congruences. In 
particular, light is thrown on a number of theorems which first appeared in 
my second memoir on conjugate nets. 5 In the terminology of that paper, 
combined with the concepts of the present note, if T' be the axis congruence of 
a conjugate net, then the reciprocal congruence V is the ray congruence of the asso- 
ciate conjugate net. That is, the first-mentioned conjugate net is formed of 
union curves of the congruence r', and its associate conjugate net consists of 
adjoint union curves of the congruence V which is reciprocal to T f . This 
theorem is fundamental in the study of surfaces of Voss and their projective 
generalization. 
In general, the union curves of a congruence V do not coincide with the 
adjoint union curves of the reciprocal congruence T; in fact, this can happen 
only on a quadric, and always does on such a surface. But on any non-ruled 
surface S, there are particular curves, which are union curves of certain con- 
gruences T', and at the same time adjoint union curves of the reciprocal con- 
gruences T. These curves are very important in several connections, and 
the property just described serves as a simple geometric characterization of 
them. They were first defined, in a different way, by Segre, 6 and I shall there- 
fore call them the curves of Segre; they are, however, merely curves which are 
conjugate to certain similar curves which were previously introduced by Dar- 
boux 7 in a memoir on the contact of curves and surfaces. 
The curves of Darboux were called by him lines of quadric osculation. He 
defined them as follows. Each member of the three-parameter family of 
quadrics which have contact of the second order with the non-ruled surface S 
at a regular point P cuts the surface in a curve which has a triple point at 
P. The three tangents at such a triple point may coincide, but, if they do, 
it must be in one of three directions, which Darboux calls tangents of quadric 
osculation. Associated with each of these three tangents of quadric osculation 
is a one-parameter family of quadrics having second-order contact with the 
surface at P, and cutting the surface in a curve having a triple point with coin- 
cident tangents in that particular direction of quadric osculation. There are, 
therefore, three one-parameter families of curves, the curves of Darboux, whose 
tangents are tangents of quadric osculation. The curve of Segre may likewise 
be assembled into three one-parameter families, each family being conjugate 
to a family of curves of Darboux. 
Both Segre's and Darboux's treatments depend essentially on the notion of 
order of contact. It has always seemed to me to be desirable, whenever pos- 
