<正> 1.引言近年来,嘉当的外形式法被应用到几何学的各分支里,起着相当大的作用.特別地要提起,苏联几何学家菲尼可夫运用这个有效的方法,在普通空间线汇论中作出了系统的研究.对高维射影空间线汇论的应用仅仅是开始的阶段.捷赫讨论了普通和高维射影空间线汇的可展变换和射影变形之后,希伟茨进一步发展后面的问题并做出总结.

The object of the present paper is to develop the theory of conjugate nets in ann-dimensional projective space S_n by utilizing Cartan's method of exterior forms.HsiungChun-Chih has demonstrated that the two tangents at a generic point of a conjugate netN_x intersect a fixed hyperplane at two points which describe in turn two conjugate netsand stand for Laplace transforms to each other.A generalized theorem with a simpleproof has been obtained by P.O.Bell,but there is no improvement concerning the secondtheorem of Hsiung:The point of intersection of the tangent plane at a generic point ofN_x with any fixed subspace S_n—2 of n—2 dimensions describes a conjugate net in S_n-2.In the present paper we merely consider the general ease where the associate Laplacesequence of the conjugate net N~x is neither periodic nor degenerate,so that a movingframe{A_1 A_2…A_(n+1)}can be attached to a generic point A_1 of the net N_x(A_1),suchthat…,A_5 A_3, A_1,A_2,A4,…constitute the Laplace sequence.Suppose that n≥2k≥4,and S′_kand S″_k denote the k-dimensional osculating spaces of the corresponding net curvesat A_3 and A_1 respectively.If we take two points X and Y respectively,in S′_k and S″_kin such a way that the tangent plane of the surface(X)[(Y)]at X [Y] passes throughY [X],then X and Y must describe two conjugate nets which are Laplace transformsto each other,and the determination of such points X and Y depends upon 2k arbitraryfunctions of one argument.The above result not only furnishes a natural generalization of Bell's theorem,butalso contains a special case where X and Y are respectively the points of intersection of S′_kand S″_k with any fixed space S_(n-k)of n—k dimensions.Obviously,even this parti-cular case may be seen as an extension of the second theorem of Hsiung.Moreover,the:last part of our theorem also gives a generalization of a former result of the presentauthor,namely,when k=1,the determination of Levy transforms of a conjugate netdepends upon two arbitrary functions of one argument.The above consideration leads us to generalize the notion of the conjugate as well asharmonic relation between a conjugate net and a rectilinear congruence.If we take apoint X in the osculating space S′_k,for example,of the curve u at the point A_3,suchthat X describes a conjugate net X(u,u),then the net X(u,v)is said to be conjugateof the k th species to the congruence T_(A3A1).According to this definition the ordinaryconjugate relation is of the first species,since the point X lies on the corresponding rayof the congruence.In the last case it is known that the Laplace transform Y of the netX must lie on the corresponding Laplace transform of the congruence,and in consequence,the congruence T_(XY)is harmonic to a conjugate net.We can now extend this fact to thecase of conjugate relations of the k th species and show,in fact,that the Laplace trans-form Y of X(u,v)along the direction u must lie in,the osculating space S″_k of thecurve u at the point A_1.Thus we reach the general notion of the harmonic relation ofthe kth species between the conjugate net(A_1)and the rectilinear congruence T_(XY).