ox SOME NEW IIEOREMS ON CUEVES OF DOUBLE CDRVATUEE. 441 



third with one of the second ordei-, to which curve six chords proceed 

 from an arbitrary point, all being situated on a cone of the second degree.* 

 Hence it follows, that, if li < h^, a plane curve of the ni\\ order with 

 h double points is not necessarily the perspective of a twisted curve of 

 this order with /; apparent doable points — a result with which M. 

 Halphen also appears to be acquainted. 



3. If, farther, two cones of the orders m, in' be drawn through the 

 Ji chords their mvi' — h remaining edges of intersection will be always 

 situated on at least one cone of the order v = m + in' — (n — 1). If 

 7i > /i4, this is self-evident and without interest ; it is not so in other 

 cases, however, and I have found interesting theorems relating to the 

 mutual dependence of these remaining edges of intersection and to the 

 manifoldness of the cones of the order v which pass through them. I 

 must not, however, attempt to communicate these theorems on this 

 occasion. 



4. "We have seen that, if h > Jt^, the Ji chords are independent of one 

 another, so far as the conducting of cones through them is concerned. 

 They cannot all be given arbitrarily, however ; no more than can, in 

 general, the double points of a plane curve when their number surpasses 

 a certain limit. But I must not enter into this matter. 



I will give a construction when h ^lij^, of such a system of /trays issu- 

 ing from a given point, as are formed by the li chords of a curve [«, /i]. 

 n and h being given, seek, first, that value of j which satisfies the condition 



hj > h > hj + 1 ; 



then construct an arbitrary cone of the order n — 2j + 1, with its vertex 

 at the given point; take, next, any (n—jy — h edges of this cone, and 

 pass through them two cones of the order n —j ; their remaining h edges 

 of intersection will be rays of the required kind. The orders 

 n~-2j + 1 and n —j and the number (n —j)'^ — A are so chosen, that, in 

 general, neither of the last two cones breaks up, and that these cones have 

 no farther intersections on the first. I do not say that there always 

 exists a curve, for which the /; rays thus constructed are chords, or a 

 cone, for which they are double edges ; I say merely that the h rays 

 in question have the same properties, with respect to cones passing 

 through them, as have the h chords of a curve \_n, A]. The degree of 

 manifoldness of such a system of h rays issuing from a given point is 

 3/i, _ (Ji^ + 1) ; h being ^ h^. 



5. But I was more interested in the degree of manifoldness of the 

 curve [n, 7i] itself in space, or — to use a phrase of Schubert's- — the 

 number of its constants, or, in other words, the number of simple con- 

 ditions to which it can be submitted. I have, however, not yet suc- 

 ceeded with the general problem. I have once more found the result, 

 for h > 7^4, which Halphen has communicated, without demonstration, in 

 the above-quoted paper in the ' Comptes Rendus ' ; viz., that if 7; > 7t4, 

 the number of conditions to which a curve [_n, 7i] can be subjected is 

 always 4?i.2 Bi-ill and Noether have given ^ a lower limit for 7;-. I have 

 not yet found a connection between their reasoning and mine. 



' Kohn seems to have given similar theorems ; but an accurate comparison 

 will soon show that his theorems are not only different from those above com- 

 municated, but for the most part selP-evident and therefore without interest. 



- Cf. also my remark in a paper published in Crelle's JoHrnal, vol. lxxx-\-iii. 



' Math. Ann., vol. vii. 



