16 E. H, Moore, jr—Theorems of Clifford and Cayley. 
t=1. Kk,, the osculating R,,,_,; along z‘*'-*, the locus of tan- 
gent planes of points v along the line T. 
TX, —27X, +X, =0, T Xing 2T Rinses 4 oe 
To OT Ran =} T” Kins 3 — 21 Meta + hoe Oe 
TE" Das BT ta 1 = 0, TK DT ee 
2(m'—1)=m—2 R,, meeting in R,. 
The additional R,, vu(z7X,—X,)—(7Xpi2—Xmii3)=0, determines 
the tangent plane R, at the point tv. 
And so, in general, the law of formation is clear. 
The osculating R,,,, along 7‘*', the locus of the oseulating 
R,, of points v along the line 7. The equations may be written 
Xe) 0, D Aidinel (eee. Gi laser 1) 
Xie x)= Ree Xa 
. 
X"(7—-XYH4=0, X47 X20; 
where, after the binomial expansion and multiplication by the exter- 
nal factor, the exponents of the X are to be exchanged for corres- 
ponding subscripts; X’ is to be changed to X,. 
2(m'—t) =m—2t R,, meeting in R,,,,. 
The additional R,, (vX’—X”"'’) (r—X')‘=0 (the exponents be- 
coming subscripts, as above), determines the osculating 2¢-flat R., at 
the point tv. 
The osculating R,,, along 7’ lies in the osculating R,,,, along 
tt; the two R,,, X'(7—X’)'=0, X™t*(r—X’)'=0, with the m—2¢ 
equas. of the R,,,, written above are easily seen to be equivalent to 
the m—2(¢—1) equations of the R,_,, 
X'(r—X')'=0 X”'*(r—X')'=0 
Kee x)= 0 xX" **(7—X’)'=0. 
Thus the equations show that the singly infinite system of osculat-_ 
ing 2¢-flats R,, at points uv along the generator 7 lie in the oscu- 
lating (2¢+1)flat R.,,,; along 7‘*’; and, at the same time, form a 
pencil of R, in the R,,,, having as an avis the osculating (2¢—1) 
flat R,,_, along 7‘; and the osculating flats of the pencil are homo- 
graphic* with their points of osculation along the generator T. 
Compare the well known theorem: Salmon; Geometry of Three 
* The only equation introducing v is linear in v. 
