SURFACES IN HYPERSPACE. 345 



To determine the outstanding constants X, A, B, so that the indi- 

 catrix takes this form we have merely to take 



B'- = ^, A'-= , ^' „, , \^ ^ 



a 



a(ac — b^) ac — b^ 



and this choice is always possible. Hence we have shown that: The 

 indicatrix may be any ellipse or any segment of a straight line in the 

 normal Sn-i- (We are ordinarily more interested in the domain of 

 reals than in the domain of complex numbers, and this theorem holds 

 for reals.) 



If we are working in the special case of four dimensions we have 

 merely to set / = throughout the work. The results are the same 

 for the special case as for the general case, — the indicatrix may be any 

 ellipse or segment of a line in the normal plane. 



The dyadic $ = (hh — K-M- - 85)a = |yiiy22 + hlfiiyn — 712721, is 



^= {h- - c2)kiki - f k^k. - (.42 + B2)k3k3 

 - /.-I(k2k3 + kak,) - AeOnJn, + kakj) - /c(kik2 + k.ki), (114) 



$S = (? = /i2 _ g2 _ p _ J2 _ 52 



The vertex of Cone II is located at 



„ _ ( |JixS).(tix8xh ) _ B Cfki - rk2) , . 



~ ^ (HLx8xh)2 ~ B fh ' ^^^^ 



and retreats to infinity if h vanishes unless special conditions are 

 fulfilled. If 5 = the vertex is indeterminate. The determinant 

 of $ reduces io ph-B'^ and hence $ becomes singular and Cone II degen- 

 erates when / = or A = or 5 = 0. It is however clear, from the 

 equations (109') of the surface, that li fBh = 0, the surface lies in four 

 dimensions at the point considered. Hence for a true five dimensional 

 surface in the neighborhood of a point, the indicatrLx must be a true 

 ellipse (cannot degenerate to a linear segment) in a plane not contain- 

 ing h, and Cone II cannot degenerate nor its vertex retreat to infinity. 

 Special cases. It remains to discuss the case for a locally four 

 dimensional surface 



zi = h[h(x'~ + y') + e{x' - y')], z^ = hiAix' - f) + 2Bxy]. 

 Here |ax8 = ^eki^kg. This vanishes only when 5 = or e = 0. As 



