PROFESSOR W. K. CLIFFORD ON THE CLASSIFICATION OF LOCI. 
G75 
In general, a certain number of (n — 2)flats can be drawn through an arbitrary 
(n—k — 1 )flat to have Apointic contact with a given curve ; this number is, in fact, 
the (k— l ) th rank of the curve. If the arbitrary (n—k— l)flat meets the curve at any 
point, then k of these (n — 2)£Lats coincide at that point. For we may project the 
whole figure from an (n — k — 2) flat lying in the (n — k — l)flat on to a k flat. The 
problem is then reduced to drawing (k — 1) flats through a given point to have 
Fpointic contact with a curve in k dimensions. Now we know, from the example of 
the full-skew curve, that, when the point lies on the curve, k of these coincide at the 
point. 
If the arbitrary (n — k — l)flat meet the curve in more points than one, k of the 
osculants will coincide at each of them ; and this result is not affected by the union of 
the points into one. In particular, if it meet the curve in n — k coincident points, the 
number of osculants which there coincide is k(u — k). 
Applying now these general considerations to the elliptic curve, we find at once 
that the (k— l) th rank of it is nk. For we must add to the number k(n— k), just 
obtained, the number, A, given by the theory of elliptic functions for the (n — 2)flats 
drawn through n — k consecutive points of the curve to have Apointic contact else- 
where. In particular, the class of the curve is n(n — 1); we have observed already 
that the number of superosculants is n 3 . 
Thus, a plane cubic is of order 3, class 6, and has 9 inflexions ; a quadri-quadric is 
of order 4, rank 8, class 12, and has 16 superosculant planes. We learn, moreover, 
that a quintic curve in four dimensions, when not unicursal, is of first rank 10, second 
rank 15, class 20, with 25 points of superosculation. Hence a quintic in three 
dimensions, with five apparent dps., is of rank 10, class 15, and has 20 superosculant 
planes ; this follows by projection from the former case. 
A curve of this kind, viz., an elliptic curve of order n in an (n — l)flat has its 
coordinates x 1; x 2 . . . x /t determined by the equations 
x x , X 2 , . . . £C„=1, t, t' , tx, tt' . . . 
(the last term on the right being t lhl ~ l) t' or else t hl , according as n is odd or even), 
where Z=sn 3 (u-f-iK') and 
i'—~r = 2sn(u-F'iK / )cn(».-l-iK')dn(H + ?'K / )== s /2t( L — /!)(l — krt). 
Cb'lO 
Pf* is even, we may write £ = sn 3 w instead of sn a (it+iK').] The condition for n points 
u 1} u 2 • . . u n to be co-flat is then 
u l J r u i + . . . +m„= 0. 
See Lindemann ; Clebsch’s ‘ Lectures on Geometry,’ vol. ii. 
