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MATHEMATICS: G. M. GREEN 
In the general case, i.e., when the parametric net is not necessarily 
asymptotic, the developables of the congruence r correspond to a con- 
jugate net on S if and only ii jiu — vv = 0. Borrowing a locution of 
Guichard's, used by him in a quite different connection, I shall say that 
a congruence V is harmonic to a surface S if its developables correspond 
to a conjugate net on S. If a congruence V is harmonic to a surface S, 
or if its developables correspond to the asymptotic curves of S, and then a 
line I of V is met in its focal points by the tangents to the curves correspond- 
ing to the developables of V, and conversely. This theorem is a general- 
ization of a corresponding theorem for the case in which the parametric 
net is conjugate and the congruence V is the ray congruence, i.e., the 
congruence of lines joining the first and minus first Laplace trans- 
forms of the parametric conjugate net. The theorem affords a new geo- 
metric characterization of conjugate nets with equal Laplace-Darboux 
invariants.^ 
A geometric characterization of isothermal nets may be given in 
terms of the relation R. Let the surface be referred to an orthogonal 
net, and let the congruence consist of the normals to the surface. 
Then the parametric net is isothermal if and only if the developables 
of the related congruence V correspond to a conjugate net on S. This 
characterization, together with another, is soon to appear in the Trans- 
actions of the American Mathematical Society. 
In what follows, I shall always suppose that the asymptotic net is 
parametric. Again borrowing a terminology used in a different sense 
by Guichard, I shall say that a congruence is conjugate to a surface S 
if its developables cut the surface in a conjugate net. Then if the 
congruence V' is conjugate to the surface S, the related congruence V is 
harmonic to S, and conversely. The two conjugate nets can coincide 
only when T and are the directrix congruences. 
An important question naturally arises concerning the existence of 
congruences conjugate to a given surface and uniquely determined 
thereby. The surface normals form such a congruence, so that a pro- 
jective generalization of metric theorems would demand the existence 
of a congruence projectively determined by the surface and conjugate 
to it. I have found that such a congruence is generated by the lines 
/' joining the point y with the point 
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