440 EEPORT — 1881. 



0)1 some New Theorems on Curves of double Curvature. 

 By Professor Sturm. 



[A communication ordered by tlie General Committee to be printed in extenso 



among the Reports.] 



The following theorems on curves of double curvature are the result of 

 my sti^dy of the very interesting pajDers on twisted curves of the fifth 

 order, which Professor Cayley has published in the ' Comptes Rendus,' 

 vol. liv. and Iviii. (1862, 1864). Cayley there considers monoids of 

 the order vi with a given vertex or summit, and passing through a 

 given curve of double carvatui'e ; that is to say, surfaces of the order 

 m, passing through the curve, and having, at a given point O, a multiple 

 point of the degree m — 1 of multiplicity. The cone of the order m — 1, 

 generated by the right lines which have, in common with the surface at 

 this point, m coincident points, contains all lines passing through 0, 

 and meeting the curve twice. With reference to these lines or chords, 

 and to their mutual dependence, the consideration of the above-men- 

 tioned cone has led me to some new results. 



1. Consider a curve of the 7ith order, without singular points, to which 

 7t chords proceed from a given point : let us term it a curve [_n, K\. The 

 maximum value of li is known to be \{ii — 1) (« — 2). Halphen has given,* 

 without proof, the minimum value of A. Kohn, however, has published, 

 in the ' Sitzungsberichte der Wiener Akademie,' - a demonstration of the 

 same result, and last winter I also found this minimum value, before 

 seeing the papers of Halphen and Kohn. This minimum value is 



the greatest integer contamed in ( — -— J therefore t(t — 1), or f^, 



according as n is equal to 2^ or to 2i -f 1. Curves with this minimum 

 always exist ; they are each situated on a surface of the second degree, and 

 form, in the former case, its complete intersection with a surface of the 

 order t, and, in the latter, when completed by a right line, its complete 

 intersection with a surface of the order t + \. 



2. Cayley has found that through every curve of the %th order a 

 monoid of the {n — l)th order, with a given vertex, can be passed. Hence 

 we infer that through the // chords which proceed from the given point, 

 a cone of the order n — 2 always passes. And if h has its maximum 

 value, no cone of an inferior order can possibly contain all the chords ; 

 but if h diminishes, such cones become possible. There are certain 

 values of h to be distinguished. I shall denote them by 7;.^, and define 

 this symbol by the equation : 



where ^ + 1 ^J ^ 3, and N{i) has the usual signification h'{v -f- 3). 

 The most important of these values seems to be h^ = ^(w — 2)(7i — 3), 

 a fact with which M. Halphen, I understand, is well acquainted. We 

 have, moreover, h^ = -i-(M — l)(/i — 2) — 1. 



When h becomes ^ Jij, cones of the order n —j through the h chords 

 are possible, and the degree of their manifoldness is always hj — 7i; j 

 being > 3, the chords are so related to each other that every cone of the 

 orders— y, passing through any Ti — \(J— 2)(y — 3) of them, contains 

 also the remaining ones. I was surprised when I first detected this 

 relation in a special case, viz. the known intersection of a surface of the 



' Com2)tes Rendus, vol. Ixx. (1870). « October, 1880. 



