PROFESSOR W. K. CLIFFORD ON THE CLASSIFICATION OF LOCI. 
671 
Since there are n — 2 identical relations between n linear qualities, the n — 2 equations 
of the unicursal curve may be written in the form 
1 
1 
i 
X n 
A 1 5 
x 3 * . 
Y » 
• • Y /t 
cq, 
cq, . 
■ ■ cq 
/^i, 
A, • 
• • A 
it is evident that the equations X,, X 2 , * ’ ' X„= 0 represent stationary osculant 
(n — 2)flats, that is to say, ( n — 2)flats which pass through n consecutive points of the 
curve. 
The properties of this curve may be very conveniently studied by regarding it as a 
projection of the curve considered in the last section. If all the points of that curve 
be joined to a point O, not situated upon it, the joining lines will form a cone of 
order n ; and on cutting this cone by an (n — l)flat we shall obtain the curve now 
under discussion. 
The n points of superosculation, whose existence has just been proved, are then 
clearly the projections of the points of contact of osculant (n — l)flats to the full-skew 
curve drawn through the point O. It follows that when n is odd, these n points of 
superosculation are on the same (n — 2) flat; but when n is even this is not the case, 
unless the point 0 lies on the quadric locus associated with the full-skew curve, in 
which case we have a special variety of the projection. Thus the three points of 
inflexion of a nodal cubic are in one straight line ; but a unicursal skew quartic in 
ordinary space has not in general the property that the points of contact of its four 
stationary osculating planes are in one plane. The property established above for the 
full-skew curve shows that this will be the case if the four points form an equian- 
harmonic system, or if the quadrin variant of the quartic which determines them is 
equal to zero. And generally when n is even, the n points of superosculation will be 
co-flat if, and only if, the quantic in t which determines them has its quadrinvariant 
zero. 
By using the values of the coordinates of a variable point of the curve expressed in 
terms of a parameter t, we may obtain an expression of this quadrinvariant and also of 
its product by the discriminant in terms of the roots of the quantic. Let cq, cq, . . . a n 
be the values of t which belong to the points of superosculation, and aq, x. 2 , . . . x n the 
coordinates of a variable point on the curve. Then we may write 
x,— (t — af l , i=l,2,...n, 
and the coordinates of the point a; are (a, — cq)", (a, — cq)", . . . (a, — «„)". If for short- 
ness we write (/< k) instead of cq — a/, then the condition that the n points shall be 
co-flat is 
4 R 
MDCCCLXXVIII. 
