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 4 n — 12, so 
that the curve has in general 2 n — 6 separate stationary tangents. 
I proceed to prove the converse of this, viz. If there are 2 n — 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) ?? =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 such a 
complex contains all the tangents. 
(iii) n = o. 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. 
