SURFACES IN HYPERSPACE. 



341 



If we use Ti, Si, ft, for the second derivatives of zi in accord with the 

 usual notation, the quantities p, q, r, are 



p = Sr.ki', q = ^Siki , r = 2f ik, . 



With these values, yu, yi2, y22, etc. may be calculated. 

 We shall at this point merely calculate, from (106'), 



(107") 



$c = Ga 



— ^v'iti Si") 



1 + Xpi^ XpiQi ^piti 



^PiQi 1 + S^i^ ^qiti 



'LpiTi ^qiTi 'Z/Titi 



1 + ^pc ^piqx "^piSi 



l^PiQi 1 + Sgi^ XqiSi 



ZipiSi t^qiSi -^Sf 



1 + 2^r '^piqi 



^Piqi l+S^i^ 



■ pm + ^q^) 



— 9r(l + '^pi") + 2piqi^piqi 



+ '^ijitiVj — SiSj) [— piPiil + 2gr) + piqj^Piqi 

 + qiPi^ptqi - qiqiiX + ^Pv^)]- 



(108) 



In the particular case n = 4 where i and j run over the indices 1 

 and 2, the formula becomes, 



Ga = {rA - ^r) (1 + p^j + q-r) + {r4. - 6v) (l+i^r + qi') 



— (tir2 + Vih — 2^152) (^12^2 + giQ'2). (108') 



The case 7i> 4 is much more complicated but consists of a sum of 

 terms rt — s'-^, with coefficients, and some supplementary terms. 



If the surface is a twisted curve surface its rulings will project into 

 lines and hence each of its projections Zi must be a twisted curve 

 surface and the terms rt — s^ vanish; but as 6' = there are sup- 

 plementary conditions to be satisfied, the condition in case ?i = 4 

 being 



(^i'-2 + rofi — 2*152) (2^12^2 + 9192) = 0. 



But the surface may be developable, that is, G = 0, without making 

 the individual terms rt — s- vanish. Indeed if in four dimensions 



