E. H. Moore, jr.— Theorems of Clifford and Cayley. 11 
on an r-spread in the (7+1)flat, 8, a ¢epn, 41, the order of which is 
n—-k+r+1, since a (A—r)flat R,_, through the 4-r+1 fixed points 
P meets the 7-spread 8,,,, in 2—A+r-+1 additional points Q which 
correspond to the »—A+r-+1 points Q’ in which the line of inter- 
section of the R,_, with the fixed K,,, meets the projected 7-spread 
2a os ar eee 
Trrorem C. A quadri¢ 7-spread isin an (7-+-1)flat, and is wnricursal, 
its points having (e. g., by projection from a point on it) a one-one cor- 
respondence with those of an r-flat R,. Hence, a fell skew r-spread 
of order 2, 8,2, n4,1, IS Always unicursal, since, by theorem B, it may 
be represented on an 7-spread of order »n—Ak-+-r + 1=2 in an (7+ 1)flat. 
Not only so, but every flat section of a full skew r-spread is étsel/ a 
full skew spread, and, therefore, wnicursal. For an s-flat R, cuts a 
full skew S,, 4.4 @pi2rtn) In an 
Sek, 2,2 =, uu, Which is full skew, 
since A’41l=r' +n’, 
1e@., s+1l—(r+s—/')+n, using the given 
relation, 4-+-1l=r-tn. 
Il. Furi Skew Two-Spreaps. 
1. The abbildung-system. 
The full skew two-spread of order min R,4,, 5 
sal; the curve of intersection with any m-flat R,,, S.,m,m 18 also 
unicursal (I, C); an R,,_,, the intersection of two R,,, and so the axis 
of a pencil of k,,, meets the two-spread in m points (m being the 
order of the spread); further, in the R,,,,, the all-including flat, 
there are o”t! m-flats R,,, i. e., m+2 asyzygetic R,,. Hence, 
there is a representation or Abbildung of the Syn, myi, point for 
point, on a plane y¥,¥¥Y;, Ro; to an K,-section corresponds a wz7- 
cursal curve, say of order n, @"; to the m points of intersection of 
an R,,_, correspond m points of intersection of a pencil of curves 
@"; the Abbildung of the (#-+-1)ply infinite system of K,,-inter- 
sections S;,m,,, is the system of curves &”, (m--1)ply infinite or lin- 
early derivable from-m+2 asyzygetic curves of the system, all of 
which are unicursal and have the equivalent of ’°—m common 
points of intersection, say base-points of the system. A Cremona 
transformation may be found which will change an abbildung-system 
@” of the spread §, jn, m4: into any other abbildung-system @” of the 
same spread, since there is a one-one correspondence between the 
points of the two coincident planes 4, containing the two abpild- 
ima is Uneur= 
