22 HE. H. Moore, jr.—Theorems of Clifford and Cayley. 
An m-spread of order P intersects the 2-spread in a curve of order 
mP=2m' P, meeting the lines 7 in u= P points, and the curves v in 
== P pointe; sayin a un) = Cs (aren): 
2t(u—1)=2m' P(P—1)=mP(P—-1) 
lines 7 are tangent to this curve. 
There are mP(P—1) lines 7 lying entirely on the two-spread 
So, m, m1 and tangent to an m-spread S,, p my of order P.* (m odd 
or even ; ef. § 4, b). 
The curves on a full skew two-spread of even order, m=2m', then, 
have an exact correspondence with those on an ordinary quadric, 
m=2; those on a full skew two-spread of odd order, m=2m"+1, 
have almost as exact a correspondence with the curves on a plane, 
m=1; the unique curve O is a singularity, but curves meeting it in 
¢ points play very much the same réle on the two-spread as curves 
having a ¢-ple pt. at an ordinary point of the spread. These close 
correspondences with the hyperboloid and plane curves are the 
marked features of the theory of curves on full skew two-spreads. 
III. Sereaps or Opp ORDERS ON QUADRICS. 
The known theorems, that a curve §, of odd order on an ordinary 
quadricone, cone-Q, ,, passes an odd number of times through the 
vertex V, and that a general quadric 3-spread Q,, contains no 
9-spreads 8, of odd order (ef. Clifford, Mathematical Papers, p. 64), 
may be extended. 
Q,,.4: Will denote a general quadric r-spread in R,,,, and 
cone-Q, ,,, 4 quadric 7-spread in R,,, formed by joining a (general) 
Q,_i,, to a vertex-point V in R,,,. 
The section of Q,,,,, made by a tangent r-flat R, is a cone-Q,_,,,. 
The section of a cone-Q,.,, by an R, through the vertex V is a 
cone-Q,_,,,; but that by an arbitrary R, is a (general) Q,_,,,. 
S, will denote an 7-spread. 
A,. The general Q,,,om ; has on it no spread of odd order 
unless r<cm-+1. 
A,. The cone-Qom, omy: has on it no r-spread of odd order 
ais unless 7<Cm-+1; while 
Avs. m-spreads of .°¢! order pass an 0%} number of 
times through the vertex V. 
B,. The general Qyn so has on it no r-spread of odd order 
unless r<m. s 
——o _ ee 
* Chasles gives this for the case m=2, 
