ON ALGEBKAIC HYPERCONICAL CONNEXES IN SPACE, ETC. 483 



to («/); and that, if m<^n, F{xij) = (). It will therefore follow 

 that : t/" m >> n, F(xy) = ddermines the principal coincidence of 

 Qfie — and therefare of infinitely mamj — point-Une connexes; and, 

 if m = ia, F(xy) = determines a line complex of degree n. 



To prove the above theorem , suppose first that m = 0. 

 Then F{xy) = is the equation of an hypercone with vertex 

 at any point whatever of S,.. Hence F^O. Suppose, next, 

 that n = 0. Then F{xi/) is independent of (?/o • • • ,«/r) and f{xp} 

 is independent of (poi • • -Pr-ir)- We niay therefore put at once 

 F^f. 



We shall now suppose that w >> and w ;> 0. By Euler's 

 theorem of homogeneous functions 



If the point (x) is fixed, the hypersurface F{yo ... y^) = is an 

 hypercone with vertex at {x). The polar hyperplane of any point («/) 



Xi — ^0. Differ- 



entiate this identity with respect to Xj, multiply the result 

 by i/j, give j the values 0, 1, ... r and add the resulting equa- 

 tions. Then 



(5) "^+2 2-*^-^»- 



From (4) and (5) we obtain 



»■ r 



(6) {m -{-l)nF=^^ (y^Xj — yjX^) ^-^ 



t=0 j=0 



Z4 Zu^'^Xòyiòxi òvjòxi 



1=0 j = .+ l 



where />,-, = yiXj — yjXi , so that ( jt>oi , . . . , pr-\ ,•) are the coordi- 

 nates of the line joining [y] to (.r). 



