Mr Grace , A Theorem on Curves in a Linear Complex. 133 
For n> 5 a different method is necessary, and it applies 
equally well to n = 3, 4, or 5. 
Suppose the curve is defined by 
*=/.W, y =f* <». *=/»(*•)> w =/«(*-)» 
where X is a variable parameter and the / s are rational integral 
functions of order 
Then the coordinates of the tangent at any point of the curve 
are the six Jacobians of the f’s taken in pairs, i.e. 
^ 23 ) ^ 31 > Jl2> Jli) *^ 24 ) J 34 ) 
and each of these is of order 2 (n — 1), so that the rank of the 
curve is 2 ( n — 1). 
Now consider the points of the curve at which a linear com- 
plex contains six consecutive tangents ; it is easy to see that their 
parameters are given by 
J 23) 
dJ 23 
~dX 
Jn, J 12) J 14) *L 24) J & 
d“J 23 
dX? 
d 3 J 23 
dX 3 
= 0 
d*J 23 
^23 
dX 5 
(A), 
and that the order of this equation in X is 6 (m — 5) where m is 
the order of each /. (Of course the easiest way to see the last 
fact is to make the J 1 s homogeneous for a moment by the addition 
of a variable fu.) Since m — 2n — 2 we infer that there are in 
general 12^—42 such points on a rational curve of order n. If 
there are more than this number the equation (A) is an identity 
and then all the tangents belong to a linear complex. 
Now in our case there are 2n — 6 stationary tangents and each 
of these 1 counts for six of the points defined by (A); hence the 
equation (A) has 6 (2 n — 6) = 12n — 36 roots and therefore it is an 
identity. Consequently when there are 2 n — 6 stationary tangents 
all the tangents belong to the same linear complex. 
1 Camb. Phil. Soc. Proc. xi. 28. 
