E.. H. Moore, jr.— Theorems of Clifford and Cayley. 21 
x ? 7 
Xnrps ’ X inits 5) ed OO Cet 0 Rl 
+ 
Ps ’ ee, oA 81k 6) gehen .8 ». +19 Se 
is cut in the curves v by m’-flats, the intersection of corresponding 
R,, in the m'+1 projective pencils of m-flats R,,, 
Meme N= 0, Kg UO Se OR, =O. 
A curve ¢/*" (@'A") corresponds to a curve Ct! (rv) of order wm' +t 
meeting each line 7 in w points, each curve v in ¢ points. 
Since the number of intersections of curves on the two-spread with 
the lines 7, the curves v and with each other, and all intersection- and 
tangency-properties, depend only on the abbildung-curves, it is clear 
that there is a complete correspondence between the curves on a two- 
spread of even order m=2m', and those on an ordinary hyperboloid 
or quadric¢ (m'=1); the two systems of ruling curves, the lines 7 and 
the curves v of order m’, answer to the two systems of generators on 
the quadric (only, in the latter case, the two systems being of the 
same order are indistinguishable). Hence many of Chasles’ results 
(Comptes Rendus, liii, 1861) concerning ‘ Propriétés générales des 
courbes gauches tracées sur Vhyperboloide” apply in this more 
general case; for example: 
A curve O"'*‘(r"v') is determined by ¢u-+-(¢+-~) points. 
Two curves Ort (r*v'), Ct" (rv), meet in tu’+ut' points. 
All curves Cv" (r"v') going through tw+(t+u)—1 fixed points 
form a pencil passing through tu—(¢+u)+1 other fixed points; 
since any two meet in 2¢w points. 
To a curve C’’’ (r“v’) 2¢(w—1) lines 7 are tangent. 
2u(t—1) curves v are tangent. 
2tu curves C”’*} (r'v') of a pencil through a 
pt. P are tangents. 
These numbers are easily derived by considering the correspond- 
ing abbildung-curves. 
The curve C""'*'(r“v’) corresponds to a curve C*t’(z“v‘)'on an 
ordinary quadrie; on the quadric there is no distinction between the 
curves O"+'(z“v'), C“** (z'v"); 1. e., two curves of the same order ¢ +4, 
which meet the generators of one system T in / points, and those of 
the other system v in {, points. If, then, there is a theorem about 
curves of order u,m’+t#, (7=1, 2,.. 8’) meeting the lines 7 in w, pts. 
and the curves v in ¢, pts., the same theorem will be true about cor- 
responding curves of order ¢,7'+-u, meeting the lines 7 in ¢, pts. and 
the curves v in wu, pts. 
' A curve OC’ (r“v") must have P=wm' +4. 
= (0 
