American Journal of Mathematics, Vol. 45, No. 1 (January 1923), 25-41

Introduction

It has been shown by Professor Wilczynski that a non-developable analytic surface S may be regarded as an integrating surface of a non-involutory, completely integrable system of partial differential equations of the form

(1)

y_{uu} + 2by_{v} + fy = 0

y_{vv} + 2a'y_{u} + gy = 0

y

where the subscripts denote partial differentiation, and where the coefficients, which are seminvariants, are analytic functions of u and v satisfying the integrability conditions

(2)

a'_{uu} + g_{u} + 2ba'_{v} + 4a'b_{v} = 0

b_{vv} + f_{v} + 2a'b_{u} + 4ba'_{u} = 0

g_{uu} – f_{vv} – 4fa'_{u} – 2a'f_{u} + 4gb_{v} + 2bg_{v} = 0

b

g

Then the curves u = const., v = const. form an asymptotic net on the surface S. We shall assume that in general a' ≠ 0, b ≠ 0, thus excluding ruled surfaces from our discussion.

Under the above conditions (1) has exactly four linearly independent solutions

(3)

y^{(k)} = f^{(k)}(u,v) (k = 1,2,3,4),

which are interpreted as the homogeneous coordinates of a point y on the surface S. The semicovariants of (1) are

(4)

y, y_{u}, y_{v}, y_{uv}

Substituting the values (3) for y in (4) we obtain four points
y, y_{u}, y_{v}, y_{uv}
which are not coplanar since no relations of the form

αy^{(k)} + βy_{u}^{(k)} + γy_{v}^{(k)} + δy_{uv}^{(k)} = 0

can exist among them. For, otherwise (1) could have at most three linearly independent solutions. Hence these points may be used as the vertices of a local tetrahedron of reference for the purpose of studying S in the neighborhood of the point y. An expression of the form

τ = a_{1}y + a_{2}y_{u} + a_{3}y_{v} + a_{4}y_{uv}

where a_{1}, a_{2}, a_{3}, a_{4} are analytic functions of u and v, assumes four values τ^{(1)}, τ^{(2)}, τ^{(3)}, τ^{(4)} corresponding to the four values of y. Hence τ determines a point whose local coordinates may be defined by writing

x_{1} = a_{1}, x_{2} = a_{2}, x_{3} = a_{3}, x_{4} = a_{4}

Consider the case when two such points

(5)

τ_{1} = a_{1}y + a_{2}y_{u} + a_{3}y_{v} + a_{4}y_{uv}

τ_{2} = b_{1}y +b_{2}y_{u} + b_{3}y_{v} + b_{4}y_{uv}

τ

are given for every point y of the surface S. If we associate the line L, determined by τ_{1} and τ_{2}, with each point of S, these lines form a congruence. Wilczynski and Green have considered such congruences in cases where the lines L pass through the point y or lie in the corresponding tangent plane. In this paper we shall consider the more general problems connected with the congruences determined by the lines L when L has an arbitrary position relative to S. General formulas will be obtained for the torsal curves and the guide curves (to be defined later), and for the focal points on L. These general formulas will then be applied to certain certain special congruences in connection with which various configurations of the lines themselves will be studied.

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