10 E.. H. Moore, jr.—Theorems of Clifford and Cayley. 
Tueorem A, Hvery proper r-spread of the n" 
space of n+r—1 dimensions or less. 
For through x+1 points of the 7-spread we can draw an »-flat, 
order is in a flat 
R,; this meets the 7-spread 8, in a number of points greater than 
its order, and, therefore, contains a curve, or 1-spread 8, of the 
r-spread §,. 
An (n+1)flat-R,,; drawn through this -flat R, and an external 
point of the spread §,, for a similar reason, contains a 2-spread Ss, 
of the r-spread §,. 
Thus, finally, there is reached an (n+r—1)flat RW, which 
completely contains the r-spread 5$,. 
An r-spread of order », say S,,,;, may lie in a flat space of k 
dimensions, where k= n+r—1; when A=n+r—1, the S,,,, .4,1 may 
be called a full skew r-spread of order 7. . 
Turorem B. An r-spread of order nin a flat space of k dimen- 
sions (and no less), say 8,,1,x, may be represented, point for point, on 
an y-spread of order n—k-+-1-+1 in an (vr-+1)flat, 8, . arti, rpi- 
Join P, an arbitrary fixed point, to Q, a variable point, both being on 
the 7-spread; the resulting (7-++1)spread 8,4, is of the order »—1; 
for a (k—r)flat R,_, through P meets the 7-spread 8, elsewhere in 
n—1 points Q, and, therefore, meets the S,,, in the »—1 lines PQ, 
i. e.,in a curve of ordern—1. Each line PQ meets a fixed (4—1)flat 
R,_, in a point Q’ corresponding to Q; the (r+1)spread §,,, meets 
the fixed flat R,_, in an 7 spread of order n—1. The r-spread of the 
n order in k-flat S,,,, is thus projected into one of order n—1 in 
a (k—1)flat, 8, 4,21. A second projection from an arbitrary 
point upon a fixed (4—2)flat R,_, gives an r-spread of order n—2 
in a (kK—2)flat, 8, , 9,4 -9- 
Thus, finally, after #—r-+1 successive projections the original 
Sn, 18 represented, point for point, on an spread of order — 
n—k+r+1 in an (7+1)flat, 8, cp -ti, pas 
But the result may be reached at once. Through any k-r+1 
fixed points P of the r-spread and a variable point Q pass a 
(k—r+1)flat R,=;, cutting a fixed (r41)flat R,,,, in a point Q’ 
corresponding to Q. Thus the §,,,,, is represented, point for point, 
space of k dimensions. We shall in certain cases go further, and speak of an h-spread 
or h-way locus, viz: a locus determined by the whole or an algebraically separate 
portion of the system of solutions of /—h equations among & coordinates; if none of 
these equations are linear, the h-way locus will be said to be in & dimensions.” 
A proper curve or spread is one which does not break up into two or more alge- 
braically distinct parts. 
