20 EF. H. Moore, jr.— Theorems of Clifford and Cayley. 
44s(s+3)—z¢(¢+1)} points. Hence two curves C (O'r**) through 
${s(s+3)—¢(¢+1)}—1 points determine a pencil of such curves 
through these and ${s(s—8)—¢(¢~-1)}-+-1 additional points. 
(a) meven=2m’'. The simplified abbildung-system, @”'*’ having 
@ as m’-ple point and 4 as an ordinary point. 
Curves @?*' (@,4’) of order p+1 having @ as p-ple pt., through 
A, correspond to full skew curves of order m'+-p; these are the only 
curves on the spread in a flat of less than m-+1 dimensions. The 
proof is like that of (b) for m odd. 
In particular:—A line y,—Ty,=0 through @ corresponds to a 
line z on the spread (§ 2); two lines z do not meet. A line v through 
A corresponds to a full skew curve v of order m’ on the spread ; two 
curves uv do not meet. Through every point on the spread pass one 
line 7 and one curve O”’, uv; a line 7 and a curve vu meet in one point. 
@ corresponds to a curve O of order m’', meeting every line Tt. 
A corresponds to a line A of order m’, meeting every curve v. 
The curve O meets the line A in a point OA. (§ 2.) 
In fact, the curve O, the line A, the point OA in no way differ from 
an ordinary curve v, line 7, point tu of the spread. 
Observe that a curve 6’ (@'A") of order s (having O a ¢ple and A 
an w-ple point), corresponds to a curve C&%™*6™ (7*“*u) of order 
(s—t)m'+-(s—w) meeting every line 7 in s—¢ points and every curve 
v in s—w points: it passes s—t—w times through the point OA and 
meets the line A elsewhere in uw points, the curve O elsewhere in ¢ 
points; (.°. in all, it meets the line A in s—¢ points, the curve O in 
3—w points). 
A curve G°=6 OE) (OE MA"—E™) with an (s—t—u)}ple point at 
say @ corresponds to a curve CO™t¢—) (ru) with an (s—t—w)- 
ple point at Q. Thus the order and character of intersection with 
the lines 7 and curves v and the (s—¢t—w)ple point are exactly the 
same. The asyzygetic numbers are equal; as shown by the follow- 
ing equality (where v=s—t—uw, s’=s+-v=2s—t—u,t’=s—u,w'=s—), 
(s+1)(s +2)—¢ (¢ +1) —u (uw +1) 
@' a. 
=(s'+1)(s'+2)—7’(¢'+ 1)—u'(u' +1)—v(v+1) 
oO" a Qe 
The statement above is justified, and it is therefore proper to con- 
sider only curves which have no especial relation to the point OA; 
i. e., in the abbildung-plane, only curves @° (©'A") where s=t+u. 
The spread is ruled with the lines z (§2); and also with the curves 
v, O™, The curves v correspond to y,—vys=0; the (m'+2)spread 
