482 e. H. SISAM 



principal coincidence of a point-line connex in S,.. Froni this 

 point of view it has been extensively studied (*) in S^ but only 

 slightly treated in space of more than two dimensions (**). It 

 will also be shown that, if m = n, F{xy) --= determines a line 

 complex in S,.. This case has also, therefore, been extensively 

 studied. 



2. Let {xQ...Xr) he the coordinates of a point and (poi,...jV-ir) 

 be the coordinates of a line in 6V. The equation of an algebraic 

 point-line connex in S,. is of the forni 



m — n n 



(2) f{xo . . . X, ; Poi. . .Pr-l r) = , 



where tn >» n. If we impose the further restrictions that the 

 line {p} pasaes through the point (x), we obtain the principal 

 coincidence of the connex. Let {>/) he a point, disti net from {x), 

 on such a line {p). Then 



(3) P2hj = Xii/j — Xj^i i,i = 0,1, .. . r. 



On substituting these values of pi, into equation (2) we 

 obtain an equation of the type (1), that is, the equation of an 

 hyperconical connex (***). 



If, in equation (2), m = n, then f .= is the equation of 

 a line complex. Let (x) and {ij) he two points on a line (j)) of 

 the complex. On substituting the values (3) for the line coordi- 

 nates into the equation of the complex, we obtain the equation 

 of an hyperconical connex in which m = w. 



Conversely, let (1) be the equation of an hyperconical 

 connex. We shall prove that, if m > n : 



m n in — n n 



F{xo . . .Xr-, Vq. . . «/.•) = f[xo r, ; Poi. . . Pr-i ^) 



where {j)oi • • .p>>--ir) are the coordinates of the line joining (x) 



(*) See, e. g., Clebsch, Vorlesungen ìlher Geometrie, Voi. I, Chapter 7. 



(**) Vkneroni, " Memorie Acc. d. Scienze di Torino ,, (2), voi. LI (1901); 

 and " Rendiconti Circolo matematico di Palermo ,, t. 26 (1908). — Kasneu, 

 " Transactions of American Math. Society ,, voi. 4 (1903). 



(***) Equation (2) may be of such a form that it is satisfied by the coor- 

 dinates of every incident point and line. Tn this case F(xy) ^b 0. 



