E. H. Moore, jr.— Theorems of Clifford and Cayley. 25 
(2) m even. 
Two R,, of opposite systems in general do not intersect, but may 
intersect in R,, or R;, or. . . or R,_;. 
Two R,, of the same system in general intersect in a point, but may 
intersect in an R,, or Ry, or... or R,,5.. 
This may be expressed (R,= a point, 0 =no intersection), begin- 
ning with the most infrequent cases: 
Two R,, of *s =n} system intersect in 
am Te A 
in general 
m even m odd 
alee aries. OF... | OF kinis: 20 
Pee Oe bees sOLt).<... ORS Umea ig 
(1). A few considerations to be used in the proof are given here. 
In 7,,, two 7,, meet in a pt. 7; if in another pt., then in a line 7, , 
if two 7,, have an 7, common, and also another pt., then they have an 
7,,, 19 common. 
Two 7, intersecting on qg’ in r, may intersect in another point ; 
and thus in 7,,,; but they do not intersect in éwo other asyzygetic 
points, for then the intersection with the fixed 7’,,,_, would not lie 
entirely on the g’. (Cf. foot-note {, §1.) 
If two 7, intersect only in an 7, lying on the q’, the two correspond- 
ing R,, meet the R,,, joining this common 7, to the fixed point V in 
two R, (which were both projected into the common 7,) which in the 
R,,, intersect in an R,,. Asa special case, if two 7, intersect in a 
point r, on gq’, the two corresponding R,, [intersect the line R, , join- 
ing 7”, to V, in two points R, and] do not intersect. 
(2) meven. The7r,, on q' intersect according to (k), m—1 being 
odd. 
On q@ ?ma,2 2nd 7, 4,, in general do not intersect, but may in- 
tersect In 7,7; .... OF Tas. Twor, through them must intersect 
in a point, at least. Hence, in general, m being even, two R,, of the 
same system intersect in a point. In the particular cases, if the two 
7, mMtersect entirely on g’, the two R,, intersect in R,, R, ... or 
R,,4; but if the two 7,, intersect also in a point not on q’, the two 
im intersect in K,, RR, . 2.. or R,3: 
On q’' 7,,4,, and 7,,,,, intersect in a point 7; but may intersect 
IN %,%... OF M». Two r, through them would in general in- 
tersect in no external point; hence, in general, m being even, two 
R,, of opposite systems do not intersect. In the particular cases, if 
the two 7,, intersect entirely on gq’, the two R,, intersect in R,, R, 
... or R,,; but if the 7, intersect also in a point not on q’, the 
two R,, intersect in R,, R, ... or R,,_,. 
Trans. Conn. AcaD., Vou. VII. 4 SEpPt., 1885, 
