E. H. Moore, jr.— Theorems of Clifford and Cayley. iW 
Dimensions, § 459; “the anharmonic ratio of four tangent planes 
passing through a generator of a ruled surface is equal to that of 
their four points of contact.” 
(6) m odd = 2m" +1. 
There is one more equation in each case from the first group of 
coordinates, X, . - Xjw42, than from the second group, Xan. - - 
X40; the v equation is of exactly the same nature; the conclusions 
stated above hold equally for m odd or even. 
As the line 7 generates the two-spread §,,,,,.4, the osculating 
R.,,, along z*’ generates a (2¢+2)spread of order (¢+1) (m—2t) 
Sats, ((4-1)(m—2t), m-+1 * 
Let the m—2t asyzygetic R,, determining Rk,,,, in terms of 7 be 
Pees 45... :- Agi; the A involve 7 to: the, power ¢-+-1 ; -let 
Ans» Am—i,0) - + + Asii,s, be what the A become when the coordi- 
nates of a point, P, in the (m-+1)flat are substituted for the current 
coordinates. The (2¢+2)spread is met by an (m—2¢—1)flat in say 
x points P,; any point in the (m—2t—1)flat may be given in terms 
of m—2t asyzygetic points P,,.. . . Pozi; 
(ge 7 cet La ya ap 
8 é., XA Xs, mt AX, ma SUC G Nog Xr, opt » ete. 
Substituting the coordinates of P, in the m—2¢ A,, and eliminating 
from the m—2¢ A,,, the m—2¢ 1 which enter homogeneously in 
the first degree, we have the determinant of the order m—2t 
A,,, my ys Nee m—1 9 CEO sae %t+1 =0, 
JA m9 Bey m—1l) * ° Js is +1 
Puoraa: a's Pier 9 oo Aoees, 241 
an equation of degree (¢+-1)(m—2t¢) in r; for each value of 1, there 
is one set of values of the A, one point P,. Hence the order of the 
(2¢+2)spread, the locus of osculating Ry4,, is (¢+1)(m—2¢), as 
stated. Through a point P, an osculating R,,, along 7‘t’ may be 
drawn; this contains the osculating Tas) along zt‘; the R,, joining 
the point P, with this R,,_, is the osculating R,, at some point v of 
the line 7, This may be expressed thus; through an R,,_,_, may 
be passed (¢+1)(m—2t) m-flats R,, which meet the spread in a 
(t+1)ple line 7; and (¢+1)(m—2t) (m—1)flats R,,_, which meet 
the spread in a (¢+1)ple point rv. 
TRANS. Conn. AcAp., Vou. VII. 3 Sept., 1885. 
