SURFACES IN HYPERSPACE. 



341 



If we use Ti, Si, ti, for the second derivatives of Zi in accord with the 

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



p = S/iki , q = S^iki , r 



i/Cikt . 



(107") 



With these values, yn, yi2, y^, etc. may be calculated. 

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



1 1 + ^pc 

 4>s = Ga = \ 'EpiQi 



^PiTi 



1 + ^Pf 



^p,q 



^Piti 



1 + ^qr ^Qiti 



^qn-: 



^piqi 



^1 ill 



^PiSi 



= z^nti - Si') 



ZpiQi 

 'ZpiSi 

 1 + 2».-2 



1 + 2^^2 29.-5i 

 ZqiSi 

 Zpiqi 



Vq.2 



^^O I 



- pi'{l + Xqi') 



Zpiqi l+2gi2 



- qi'il + i:pi') + 2piqiLpiqi 



+ Sij(/irj - SiSj) [- pipj{l + S^i^) + piqjEpiqi 

 + qipjlpiqi — qiqj{l + Zpi^)]. 



(108) 



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

 and 2, the formula becomes, 



Ga = {nh - 51-) (1 + P-^ + 92^) + {r.h - S2') (l+pi" + qi') 



— {hr2 + r2h — 2^152) (pijh + 9192). (108') 



The case 71 > 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 n = 4 



being 



(^i'-2 + r2^i — 2^152) {pip2 + 9152) = 0. 



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

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



