E.. H. Moore, jr.— Theorems of Clifford and Cayley. 15 
2 
the two-spread in curves with Q as (¢+1)ple pt.; these R,,, in the all- 
including flat R,,, meet in an (m+1)—(m—2t+1)=2¢-flat Ry, 
which is the osculating 2¢-flat at Q, which includes the osculating 
t-flat R, at Q to every curve through Q; i. e., which meets the two- 
spread at ¢ consecutive points in every direction from Q; i. e., which 
contains the osculating 2(¢—1)flats KX,,, at all points immediately 
adjacent to Q. 
What is the locus of osculating 2¢-flats at points Q along a genera- 
tor 7 of the two-spread ? 
If a curve ¢” degenerates into @é), say the line 7, taken ¢+1 
times and a supplementary curve ¢”~7 (@ as m—t—2ple pt. and 
through the m—1 pts. 9) the R,, cutting the two-spread in the 
corresponding section will contain the osculating 2¢-flat at every 
point Q along the generator 7. The asyzygetic number for the 
system of supplementary curves @”‘" is (2m—2t—1)—(m—1)= 
m—2t. Hence there are m—2t asyzygetic m-flats R,, meeting the 
two-spread in the generator 7 taken (¢+1) times; these in the R,,,, 
meet in an (m+1)—(m—2t)=(2t+1)flat R,,,,, say the osculating 
R,,,, along t‘*' the (£+1)ple generator z, which is the locus of the 
osculating R,, at the points Q along the generator T. 
The equations of the osculating flats ; the orders of the spreads 
they generate. 
(a2) m even=2m'. Cf. §2, the simplified abbildung-system; the 
equations in ¢. 
The equations correspond to m—2¢ asyzygetic curves @”'*" includ- 
ing the line 7 ¢+1 times; and to the one other asyzygetic curve of 
the system including the line 7 ¢ times and passing through the 
point Q=7rv. 
t=0. RK,, the line z, 
px aX 0; Te ee On 
7X, —X S30); Ti en 40 
> 
TXy— Xin = 0, TM —Xini0 == (5 
2m'(= m) R,, meeting in R,, the line r. 
The additional R,,, vX,—X,,,,.=0, determines the point 7v on the 
line r. 
