26 E.. H. Moore, jr.— Theorems of Clifford and Cayley. 
Thus, the (4) holding for 7—1 odd, the (7) hold for m even. 
(3) m odd. By similar considerations it is shown that, the (?) 
holding for m even, the (4) hold for m+1 odd. But the (#) hold 
for m=1 on the quadrie Q,., in R,; (as might be shown immediately 
by the projection from V on a plane); and, therefore, (4) and (/) are 
true in general. 
3. One m-flat of each system passes through every R,, of the 
Q,-. The Ry, projects into an vr” having an 7,,_, on g’, through 
which (suppose) there pass an 7, ,, and an7,,,,,; the original 7,,_, 
and the 7,,_,,,, intersecting in 7,., determine an 7,, the projection 
of an R,,, through the R,,_,. This is true on the Q, ,; therefore, in 
general. ‘ 
Every pt. in R,,,,,; has a polar R,,, with reference to the Q,,,; the- 
polar R,,, of every pt. in an R, passes through a certain R,,,_, which 
is called the polar of the R,. (Clifford.) 
If the s-flat R, lie on the Q,,,, the polar h,,, of every point in it, 
i. e., the tangent R,,, at that point, passes through the s-flat itself; 
hence the polar P,,,_, must include the original s-flat R,. 
The m-flats R,, are self-polar. 
The R,,, and R,,, through an R,_, taken together lie in and 
determine the polar R,,,, of the R,,_,. 
More generally, two R,, intersecting in an R, lie in and determine 
the polar R,,,_, of the R,. 
