12 E. H. Moore, jr.— Theorems of Clifford and Cayley. 
ung-systems. There is, therefore, no loss of generality in assuming 
as the abbildung-system, the system of curves @” having as base- 
points an (m—1)ple pt. @ and m—1 other points, ~,,.. +. Pmt 
the conditions given above are satisfied by this system ; for the curves 
are unicursal; the asyzygetic number is 2m+1—m—1=m4+23 the 
base-points are equivalent to 
(m—1)?+(m—1)=m(m— 1) =n’ —m=n*?—m 
common points of intersection. 
This abbildung-system may be simplified. Transform the plane 
%, by a quadric transformation, having ©, Qj, and }, », as fun- 
damental points. 
A curve @”, of order m, passing through the two fundamental 
points B15 Dm—2 and having an (m—1)ple pt. at @, the third fun- 
damental point, and passing through m—3 fixed points $)..- Pn—s, 
is transformed into a curve @’ of order m—1=2.m—(m—1+1+41), 
having an (m—2)ple pt. at @, (since the line Dn—1 WBn—2 Meets the ¢” 
in m—2 additional pts.), passing through the m—38 fixed points 
Pi. + - Dns corresponding to the pts. Q,. .. $s, and not pass- 
ing through Pm [2,2], (since the line OB, [OBn—] does not 
meet the @” except in @ and 9,,» [© and 9,,_,]). 
Thus such a quadric transformation reduces by unity the order of 
the curves of the abbildung-system, and deletes two of the base- 
points. By 7 such quadric transformations, the abbildung-system of 
curves @”, with @, an (m—1)ple point, and m—1 other points @, as 
base-points, is changed into a system of curves @”” with @, an 
(m—r—1)ple point, and m—1—2r other points #, as base-points. 
The simplest abbildung-systems are, evidently, 
(a) m even=2m',r=m'—1. — &""*! vith @ as m’ple point 
and through one pt. &. 
(6) m odd=2m"+1,r=m". &”"* with @ as m"ple point. 
2. The canonical form of the equations of the 85m, m4i+ 
Let the X, (s=1,...m-+3) be the homogeneous coordinates in 
R,,41 and the y, (s=1, 2, 3) be the homogeneous coordinates in §,. 
(a) meven=2m'’. @i8 ¥1, Yr, Y¥z=0, 0,1; A is Yi, Yo, Ys—1, O, 0. 
By the simplified abbildung-system set 
». CF xs tp. Ge Dia On etree 1e, te - 8 8 oleh oe, be 2. Oe Mees DOT 
Xe ree . oe Xeai ft le a ‘eee ee ee ee et Yee | 5 D. Carve Koma 
m-+1 m-+2 
Yi rey: Y page Yj mi—2 Ayer ee na. | ae eee Vie oar Yi Yor uid Yr mirth. 4 
a as. Re Ue eid oo + + Sy ee ag aes Fa Pinar Feel b 
