14 E.. H. Moore, jr.—Theorems of Clifford and Cayley. 
A point corresponds to the dine OF, viz: the intersection of the 
full-skew curve O and the line A. y,=0; or w= m, t=0; 
Xe XS= teense Se = 0, D.C Trane. 5.0 — ae, Se = X= 0: 
3. Tangent and osculating flats. 
On the two-spread let Q be a point and C a curve passing 
through Q; there are at Q a tangent line R,, and .osculating 
2-flat R,, osculating R,... R, to the curve C, determined by 2, 
3... (¢+1) consecutive points (Q included) of the curve; the oscu- 
lating ¢-flat R, meets the curve C in ¢+1 points at P. The tangent 
plane R, at Q to the two-spread is the locus of tangent lines R, to 
all curves C through Q; every m-flat R,, through this tangent plane 
R, meets the spread in a curve having a.double-pt. at Q. 
Suppose there were an s-flat R, such that every R,, through it met 
the two-spread in a curve having a (¢+1)ple pt. atQ. Take any 
curve C through the point P; every R,, through the R, meets the 
curve C in ¢+1 points at Q, and, therefore, contains the osculating 
R, to C at Q; hence the R, contains the osculating ¢-flats R to all 
curves C through Q, or say it meets the two-spread at ¢ consecutive 
points in every direction from Q. . 
In the case of this full skew two-spread it will be shown that at 
every pt. Q there is an osculating R.,., containing the osculating 
R, to every curve through Q; but probably s=2¢ for the general 
two-spread in any number of dimensions, R,; (2¢£4, of course). 
Considerations from the abbildung-system. 
The general abbildung-system (cf. §1) of curves @” has as base-pts. 
the (m—1)ple pt. ©, and the m—1 pts. 9; there are m+2 asyzyge- 
tic curves of the system. Consider those curves of the system which 
have a (£+1)ple pt. at @, say &” (@‘t’); these correspond to the 
R,,-sections of the two-spread which have a (¢+1)ple pt. at Q. Such 
a curve @” (@*') includes the line @@ ¢ times, and, besides, a sup- 
plementary curve ¢”~ (@), passing through @ and the m—1 pts. 9 
and having an (m—t—1)ple pt. at ©. For say the curve &” (@") 
includes the line ©@ « times, and, therefore, also a supplementary 
curve &”~ (Q*) with (¢+1—z2)ple pt. at @ and (m—2x#—1)ple 
point at @; the line ©@ meets the supplementary curves in 
(m—a—1)+(¢+1—x)=m+t—2x pts. and will be again thrown off, 
if m—x7<m+t—2z, i.e, if «<t. The asyzygetic number for the 
system of supplementary curves ¢”~ (@) is (Q2m—2+1—m—1—1=) 
m—2t+1. Hence there are m—2¢+1 asyzygetic m-flats R,, meeting 
