E. H. Moore, jr.— Theorems of Clifford and Cayley. 23 
B,. The cone-Qsn s,m has on it no 7-spread of odd order unless 
By. r<m+1; while 
Bos m-spreads of .°"'torder pass an .{ii}number of times 
through the vertex V. 
Let B, hold in R,,; then will the theorems A hold in R,,,,. 
A, . Q:), 2n+1 
Take arbitrary R,, section of containing an 
ey . cone-Q»,, 2n+1 
r-spread of odd order; there results a (general) ane | with an 
n—1,2 $ 
(r—1)spread of odd order on it; hence, by B,, r—l<n; or r<n+1; 
proving A,, A,,. 
A,,. Notice that the projection in R,,,,, from a pt. P upon an 
R,,, of an r-spread of order s passing ¢ times through P is an 
r-spread of order s-t, i. e., of odd order, unless s-t is even ; i. e., unless 
the r-spread of .°""| order passes an .02)} number of times through P. 
Take arbitrary R,, section of cone-Q,,.,,, containing an n-spread 
5,3 lt isa Q,, ,,., which contains the projection through P of §, ; if 
B, holds, the preceding consideration shows that A,, must also. 
Hence, if the propositions hold for R,, (i. e., the B), they do for 
Ra,+, (i. e. the A). 
If the A (m,=n) hold for R,,,,, the B (m,=n-+-1) will hold for 
Lee 
B,. The Qynsi,om+2 Contains an r-spread 8, of odd order; the tan- 
gent R,,,, at a pt. V (not on the 7-spread 8,), cuts the quad- 
ric in a Cone-Qon on ii, Containing an (7—1)spread of odd. order 
not passing through the vertex V; hence, by A,,, 
r—l<in, r<ntl, r<my,, proving B,. 
B,. An R,,,,-section through the vertex V of the cone-Qoni1,om+2 
shows the dependence of B,,, B,, on the truth of A.,, Ay. 
But A, holds for R,. A curve of 0} order in an ordinary quad- 
even } 
ricone, cone-Q,,, passes through the vertex V an 2%! num- 
ber of times, because its projection through V upon a plane R, 
is (a conic) of even order. 
From A, for R, follow at once the B for R,, and thus the general 
propositions as enunciated for Ry, Ren .. 
IV. Fuats on Quaprics. 
Prof. Cayley, “On the Superlines of a Quadric Surface in 5-dimen- 
sional space” (Quart. J. M., 1872-3, t. xii, p. 176) gives an analyti- 
cal proof of the proposition, suggested by an evident theorem in 
oO 
fo) 
