SURFACES IN HYPERSPACE. 



339 



f.f' = i, f.g = o, g-g=], g-g' = o, f'.r = o, 



in*n = 0, q-n = 0, n- =1, a = m^; 



yu = a-^ 



m n p 

 m^ m • p 

 1 n-p 



= a-^{— iri'pm + m'p — m^ii'pn), 



yi2 = a-i 



m n q 

 m- ni'q 



1 



= a-^(m-q — m-qm), 



y22 = 0; 



^ = 2(yiiy22 + y22yii) - yi2yi2 



= — a~-(m-q — m^qin) (m-q — m-qm). 



Moreover, a^") = 1/a, a^^^) = 0, a^^-'^ = 1, 



2h = a-"(— ixi-pm + m-p — m-ii'pn), 



$/a = hh — |X|x — 55, |i|JL + 55 = hh — ^/a. 



p,|i -[- 55 = ja-^(— in'pm+ m-p — m-'n-pn) ( — m'pm + m-p — 



m-n • pn) 



+ a~^(m-q — rn-qm) (m-q — rn^qm). 



|A^5 = — - (— m'pm + m-p — m-n-pn)- (m-q — m«qm). 

 2m,' 



Hence j^^S contains h and the indicatrix lies in a plane with the 

 surface point, no matter how high the dimensionality of the space 

 in which the surface lies. Hence: A ruled surface is at each point 

 of the four dimensional type and never of the general ti/pe, i. e., a ruled 

 surface is made up of planar points, in Levi's nomenclature. 



The formulas will serve to investigate the whole surface. If we 

 are interested only in the neighborhood of some ordinary point we 

 may assume that the point lies on the trajectory y = i(u), that is, 

 V = 0. The formulas then simplify further; for 



m=r, p = r, m-p=f'.f" = 0, ^, = m-=l; 



2h = f" - g'f'g, $ = - (g' - f'g'i') ig' - f'.g'n. 



