132 Mr Grace, A Theorem on Curves in a Linear Complex. 



A Theorem on Curves in a Linear Complex. By J. H. 

 Grace, M.A., Peterhouse. 



[Received 4 March 1901.] 



When all the tangents of a twisted curve belong to the same 

 linear complex, the curve cannot have a proper point of super- 

 osculation ; for at such a point the osculating plane contains three 

 consecutive tangents not in general meeting in a point, which is 

 contrary to the supposition that they belong to a linear complex. 

 In fact two of the tangents must coincide in order that the three 

 may be concurrent, i.e. the curve must have a stationary tangent. . 

 The point of contact of a stationary tangent may be regarded as 

 the coalescence of two points of superosculation, and hence we 

 may say that : — 



The points of superosculation of such a curve coincide in pairs, 

 and each point of coalescence is the point of contact of a stationary 

 tangent. 



A rigorous algebraic proof of this proposition for the case of 

 rational curves has been given by Picard — the method incidentally 

 shews that the equation giving the parameters of the points of 

 superosculation is of the form 



T=0 

 where T is rational. 



Suppose the curve is of order n, then it is easy to see that the 

 equation for the parameters just mentioned is of order 4m — 12, so 

 that the curve has in general 2n — 6 separate stationary tangents. 

 I proceed to prove the converse of this, viz. If there are 2n — 6 

 stationary tangents to a rational curve of order n, then all the 

 tangents to the curve belong to the same linear complex. 



The proof is very simple for n = 3, 4, or 5, and we therefore 

 consider these cases separately. 



(i) n = 3. Here there are no stationary tangents and the 

 rank is 4, so that a linear complex drawn to contain five of the 

 tangents contains them all. 



(ii) n = 4. Here there are two stationary tangents and the 

 rank is 6. Obviously therefore a linear complex containing more 

 than six of the tangents contains them all. But the linear com- 

 plex containing the two stationary tangents and any other three 

 tangents has seven lines in common with the tangent developable 

 because each stationary tangent counts for two. Hence snch a 

 complex contains all the tangents. 



(iii) n = 5. Here the rank is 8, and a linear complex drawn 

 to contain the four stationary tangents and one other tangent 

 contains in reality nine tangents and therefore contains all the 

 tangents. 



