PROFESSOR W. K. CLIFFORD ON THE CLASSIFICATION OF LOCI. 669 
A%=wB', (7) 
A.%=±n(n— 1)C', 
AX=L' 
so that, when n is even the equation of the polar is — 
0 = AL' + A'L - n(BK' + B'K) + {n - 1 ) (C'H + C'H) - etc (8) 
that is, it is simply the polar of the point in regard to the quadric (6). 
It is to be observed that the quadric is completely determined when the curve is 
given. I reserve the question of the conditions to which the curve is subject when 
the quadric locus is given, or, say, the discussion of the problem to represent the 
relation of poles and polars in regard to a quadric locus (in space of an even number 
of dimensions) by means of a unicursal curve. 
But when n is odd, the last term of equation (4) is negative, and the equation of 
the polar is — 
0 = AL' — AX — ?i(BK' — B'K) +bi(n-l)(CH'- C'H) — etc. . . . (9) 
that is, it is skew symmetrical, and every point lies upon its polar. It is convenient 
to use the term co-flat for n -f- 1 points, which are in the same (n — l)flat ; with this 
nomenclature we may say that when n is odd every point is co-fiat with the n points 
of contact of the osculant (n — 1) fiats, which can he drawn through it. This will be 
recognised as an extension of the property of a skew cubic, that every point in space 
is co-planar with the points of contact of the three osculating planes which can be 
drawn through it. 
A case of this skew symmetrical relation is given by any arbitrary state of motion of 
the whole space as a rigid body, the relation between two points being that the line 
joining them moves perpendicularly to itself. The polar of any point is an (n — l)flat 
drawn through it perpendicular to the direction of its motion. When n is even 
there is always one point which remains at rest, and all the polars pass through this 
point. Thus the general motion of a solid in an even number of dimensions always 
depends in this simple way on the motion in one dimension less. In an odd number 
of dimensions, however, every point moves in the general case ; but if any point is at 
rest, then all the points in a certain straight line are at rest. 
Besides its order and class, a curve has, in general, characteristic numbers inter- 
mediate to these, which may be called its first rank, second rank, etc. The first 
rank is the order of the locus traced out by straight lines through two consecutive 
points of the curve ; the second rank, of that traced out by planes through three 
consecutive points ; and generally the k ih rank is the order of the (k J r l)wide locus 
traced out by A flats through k -\- 1 consecutive points. For the curve just considered 
these numbers are 2 (n — 1), 3 (n — 2), . . . (k-\- 1) (n — k) ; it is convenient to derive them 
