24 E. H. Moore, jr.— Theorems of Clifford and Cayley. 
line-geometry ;—(using Clifford’s expressions): On a quadric 4-spread 
Q,; there are two triply infinite systems of planes R,,, R.,,; two 
planes of the same system meet in a point; two of opposite systems 
in general do not intersect at all, if they do, it is in a line. 
There is a similar theorem for quadrics in all flats of odd dimensions. 
1. On a quadric em-spread Qom,oni.1 i (2m+1)flat R,,,., there 
are two 4m(m+1)ply infinite systems of m-flats R,,, .* 
If this holds for R.,,_,, then it will hold for R.,,,,.. For R, (m=1) 
this is the double system of generators R, on a quadric surface. 
Project upon a fixed »,, from a pt. V on the Qon omii- The tan- 
gent R’,,, at Vi cuts Qon omy: 12 & Cone-Qon i om, With vertex at V, 
and it cuts 7, in a fixed 7”,,,, which contains the pr ojection 
through V of the cone-Qy, 4,0 1. €., & fixed @’om_—1), om—1- 
An R,, on the Qon, om41 nt fase tog V) meets the tangent R’,,, at V 
Rn—1 : cag : Ym having an 7-1 
in p aa t and is projected into an rag tye ‘Seiden on the fixed 
q' rae AD Watts 5 
The converse is also true.§ 
The @'sm1),2m1 18 Supposed to have on it two $(m—1)m-ply 
infinite systems of (7—1)flats 7,,_,. In 7,, two 7,, meet in a point; 
hence, there are »” 7, through a fixed 7,_,, i. e., one joining the 
fixed 7,,_, to every pt. of the o” pts. of a random fixed 7,,. These 
7, correspond to R,, on the Q,,, on11- 
Therefore, on the Qs, 41, there are two w”".0 2 = ee 
=}m (m+1)ply infinite system of m-flats R,,. 
2. The intersections of the m-flats. 
(k) m odd. 
Two R,, of the same system (R,,,,, R,,,,) in general do not inter- 
sect, but in special cases they may intersect in R,, or R,, or... 
or Bs. 
Two of opposite systems (R,,,,, R,,,,) in general intersect in a 
point, but may intersect in R,, or R,, or... . or R,,_,. 
* There is on the quadric no R, if s>m; (III, A). 
¢ If an R,, passes through V, it lies completely in the tangent R’o_ at V. 
¢ Thus the intersection of an 7, (the projection of an R,,) with the fixed 7r/o,—1 
lies completely on the fixed g’; and likewise two rm and the fixed 1’, intersect on 
the fixed q’. 
§ A quadric m-spread - passing through V would project into an m-flat 7, having a 
quadric (m—2)spread on q’. 
