18 EE. H. Moore, jr.—Theorems of Clifford and Cayley. 
(a) meven=2m'. The simplified abbildung-system; § 1. 
g”’*" may degenerate into the line 7 taken m’ times, and a line 
through 4; for this the limiting case t=m’'—1. 
The (2¢+2)spread, locus of osculating R,,,, along zt! is of order 
(t-+1)(m—2t). But the (m—-2z¢)spread, locus of osculating R,,», 
along t”’~ is also of order 
(m!' —t) (m—2m'-—t—1)=(t+1) (m= 20). 
For instance, (=0; the two-spread locus of lines R, (i. e., the orig- 
inal two-spread) is of order m; and, also, the m-spread locus of oscu- 
lating R,,_, along 7” is of order m. 
Thus, when m is even, the orders of the (2¢+2)spread for t=#,, 
t=t,, are equal, if ¢,+-¢,=m’—1. 
(b) m odd = 2m"-+-1. 
t=m,; since the curve ¢”"*' in the limiting case degenerates into 
the line r taken ¢+1—=m"-+1 times. 
For t=m’" thé order of the (2¢4+2) = 2m" +2 —=(m-+1)spread is 
(m”+1)(m—2m")=m"+1; i. e@, through every point R,, +41» of 
Ry may be passed m’-+1 R,, meeting the 2-spread in an (m” + 1)ple 
line r. 
- There is no symmetry analogous to that for m even. 
4. Curves on the two-spread. 
(b) m odd = 2m"-+-1. 
The abbildung-system, @”""' having @ as m’-ple pt. 
Let us denote the wniqgue curve of order m” in an m’-flat corre- 
sponding to @® by O; and the right lines of the spread by 7. A 
curve on the spread of order p, meeting the unique curve O in g 
points and every line 7 in r points may be written C? (O%7’). 
A curve on the abbildung plane @°(@') of the order s with a éple 
point at @ is met by a curve of the system @”""' in s(m”+1j—tm" 
points and by any line 7 in s—¢ points; therefore, it corresponds to a 
curve on the spread of order s(m"-+-1)—tm", which meets the unique 
curve O in ¢ points, and every line z in (s—¢) points, say 
Cea er Ores: 
So a curve @* (@‘7°“) transforms into O°™"'t)—™" (Of7**), 
A curve C? (O°), of order p, meeting the curve O in g pts., must have 
p>g and p=q (mod. m”+1); say p==s(m" +1) -tin’, 
qt. 
s(m"+1)—tm"=p. (s=?%). 
s=1,t=1,p=1. A liner on X, corresponds to a line 7 on the spread. 
ss, t=s, p=s. s lines 7 correspond to s lines r. 
