PROFESSOR W. K. CLIFFORD OR THE CLASSIFICATION OF LOCI. 
67 4 
must suppose the n points to become identical, or the t, still satisfying this equation, 
to become equal. We thus obtain 
nt=c-\-aoj-\-bo) 
c a , b , 
t — 1 — — I — Ct> , 
n n n 
and values of t, representing distinct points, will he got by giving to the numbers a, b 
the values 0, 1, . . . n — 1 independently. Hence there are n 3 points of superosculation. 
Thus a plane cubic has nine inflexional tangents, and a quadri- quadric curve has 
sixteen superosculant planes. 
Propositions hold good in general in regard to the grouping of these points, which 
are analogous to those which relate to the inflexions of a cubic. Thus, an (n — 2) fiat 
drawn through n — 1 of them will always pass either through another besides, or through 
the tangent line at one of the n — 1. This is obvious from the values already given for 
the parameters of points of superosculation. 
Through any point of the curve can be drawn [n — l) 3 osculant (n — 2)flats. This is 
proved in the same way as the preceding proposition, which is, in fact, the projection 
of it ; for if through the given point we draw a cone containing the curve, and cut it 
by an (n — 2) flat, the section will be an elliptic curve of order n — 1 in n — 2 dimen- 
sions, and the projections of the points whose osculant (n — 2) flats pass through the 
given point will be points of superosculation on the projected curve. Hence, also, 
the lines joining the given point to the points of contact are grouped in respect of co- 
flatness in the same way as the points of superosculation in the curve of next lower 
order. 
More generally, through k given points of the curve there can be drawn (n — kfi 
(n — 2)flats which have ( n — Jc ) pointic contact with the curve. If u x , u 2 , . . . U/ c are the 
parameters of the k given points, those of the required points are given by 
-fyi + Uo+ . . . U/-\- aoj-\-bco ), 
where the integers a, b may take independently the values 0, 1, . . . n — k — 1. 
From these results we may now determine the various ranks and the class of the 
curve. Suppose that we know the number of (n — 2)flats which can be drawn through 
7i — 2 arbitrary points in space — or, which is the same thing, through an arbitrary 
(n — 3)flat P — to touch a certain curve. Then, if the arbitrary (n — 3)flat meets the 
curve in any point, two of these will coincide at that point. For taking an (n — 4)flat 
in the (n — 3)flat, and joining it to all the points of the curve by (n — 3)flats, we 
may cut this figure by a plane or 2 flat. Every (n — 3)flat will cut this plane in a 
single point. The problem is then reduced to drawing tangents from a point (viz. , the 
intersection of P by the plane) to a plane curve ; and we know that when this point 
lies on the curve, two of the tangents coincide at it. 
