PROFESSOR W. K. CLIFFORD ON THE CLASSIFICATION OF LOCI. 
677 
Moreover, we see at once that the theory can be extended to other curves of 
deficiency 1 ; as, for example, the quadri-quadric curve. Starting with any point on 
this curve, we may find the point where the osculating plane at that point meets the 
curve again ; then repeat the process with the point so found, and so on. The plane 
joining any three of these points will meet the curve in another derived point, or else 
touch it at one of the three points. The plane drawn through one derived point to 
touch the curve at another derived point will meet it again in a derived point, or 
touch at the first point, or osculate at the second. The coordinates of any derived 
point are of the order r? in those of the original point, where zL n may be called the 
order of the derived point. In this case the order of no derived point is divisible by 2. 
I was desirous of finding a similar representation of the multiplication of hyper- 
elliptic and Abelian functions ; and therefore sought for cases in which derived 
elements might be found on curves (in the sense explained in the beginning of this 
paper) of deficiency greater than 1. For this purpose I considered scrolls. Taking an 
arbitrary generator on a quartic scroll having two linear directrices, we may draw a 
one-sheeted hyperboloid through three consecutive generators at that place ; this will 
meet the quartic scroll in one other generator, which is thus uniquely derived from 
the given one. Similarly on a quintic scroll contained in a linear complex, the two 
tractors of four consecutive generators meet the scroll in two other points lying on a 
generator. And on a sextic scroll not contained in a linear complex, the linear 
complex having five-line contact at a given generator (containing five consecutive 
generators) will contain one other generator of the scroll. In these three cases, then, 
from any three, four, or five generators we may uniquely derive a fourth, fifth, or sixth 
generator respectively; and the whole theory of derived elements may be applied to 
the generators of these scrolls. 
Unfortunately, however, each of the scrolls considered is at most of deficiency 1, so 
that we merely get more illustrations of the multiplication of elliptic functions. And 
it may be shown, in general, that a curve on which such a theory of derived points is 
possible, is at most of deficiency 1. 
Suppose that it has no singular points, and that Jc — 1 points on it being given, 
there is uniquely determined one other point. If this is effected (as in the above 
examples) by drawing a flat space through the Jc — 1 points, which meets the curve in 
one other point, then it must be of the order Jc. Moreover, it must be in a flat space 
of so many dimensions that the flat of one dimension less is determined by Jc — 1 
points. Now a (Jc — 2) flat is determined by Jc — 1 points; therefore, the curve is in a 
(Jc— l)flat. 
Thus the impossibility of extending the theory of derivation to curves of deficiency 
greater than unity is equivalent to the proposition that a curve of order Pc in Jc — 1 
dimensions is at most of deficiency 1. This failure was the starting point of the 
present paper. 
It remains to explain why, in the group of numbers expressing the orders of the 
