E.. H. Moore, jr.— Theorems of Clifford and Cayley. 19 
If s>t, p2=m"+1; which shows that the curve O of order m” is 
unique, since it is the only curve of so low an order on the two-spread 
(the lines 7 excepted). 
Curves on the two-spread and in flats of less than m+1=2im"+2 
dimensions are full skew curves. Such a curve is an R,,-intersection 
or a part of an R,,-intersection ; therefore, its abbildung is a curve 
6° (@") ; any m”4+1-s lines 7 belong to the supplementary system, 
of which the asyzygetic number is m’+2-—s; therefore, m”+42—s 
asyzygetic R,, meet in an R,,,,,, in which the curve O%"t)—G-Dmttemut 
lies. A curve C”** in R,,,,,,. is a full skew curve. (I; theorem C.) 
The general plane curve @’ (¢=0; not through ©) corresponds to 
a Ot) which does not meet the unique curve. The plane is a full 
skew 2-spread 8,5 (m=2m"4+1=1, m’=0). All the geometry of 
plane curves depending upon intersections and tangencies and the 
order of curves is immediately applicable to the general full skew 
two-spread of odd order m=2m"-+-1, the curves C*”"’* on the two- 
spread corresponding completely to the curves @° of the plane. A 
curve Cw"'t)“"" (O*) meeting the unique curve O ¢ times is a par- 
ticular case of C%”"t», and in fact plays the same role as a Ot) 
having a ¢é-ple point. A few examples are given. 
There is a double infinity of curves C”"*’; two meet in one point; 
one is determined by two points; they correspond to the lines ¢' of 
the plane. A line z together with the uniqte curve O is a special 
case of a curve C”""'. Five points determine a curve Ct); which 
corresponds to a conic ©? of the plane. 
Pascal’s theorem becomes : 
If six points P’.. P® lie on a curve C*"t” the three points of 
intersection of the two curves Ot’ joining P'P’, P*P®; P’P®, 
P>P*®; P*P", P’ P*, respectively, lie on another curve O?"", 
Mo acurve Ctr) (O’r**) there: are .s(s— 1) —2 (¢-+ 1) = 
(s+-t)(s—t—1) tangent lines 7, and s(s - 1)—¢(¢—1)=(s—t) (s+¢—1) 
tangent curves C”’'' in a pencil through a point P.* 
An m-spread of order P meets the 2-spread in a curve C”” meeting 
the unique curve O m’P times, and every line 7 in P points; 
ee (ONT a Seay +l), tm’ PP, , To an m-spread 
of order P in Row,» my: there are mP(P—1) tangent lines lying 
entirely on the Sy m9, mit 
Two curves C (O'r*“), C (O’r""), meet in ss’ —¢¢’ points; in particu- 
lar, two curves C (O'7*”) meet in s’*—@ points; one is determined by 
* These formule are similar to some given by Chasles, Comptes Rendus, 1861; cf. 
the following (a). 
