308 WILSON AND MOORE. 



If i 9^ j, we set 



Zt*Zy|4 = Vijis, Zj-Zns = — f0|3 = Vjiis, (56) 



and 



Zilr'Ys = — ibrs- (56') 



We can then obtain by the same process as before, 



ibrst ibrts — ^i=i {ibrs^ji\t — jbrtVjils), \p 



(pr,st) = S"r' {ibps-ibrt — ibpfibrs)- (57') 



Moreover we may obtain by a somewhat detailed analysis in the case 

 /I = 4, 5, ... a relation involving the second derivatives of v as 



Vrs — Vsr = -pg (^prCgs — b psC gr) O^^'^ H = 4, (58) 



V ii\Ts — Vji\sT-T ^1=1 \VljirVli\s — Vlj\sVlHr) 



= Zp^a'P'^^ (ibpr.jbgs - ibps-ibgr)]. (58') 



Tn the case of a binary (first) fundamental form tp = 'Zursdxrd.Vs , 

 the Riemann symbol {pr, si) reduces to a single one, namely (12, 12), 

 and we may write 



(12, 12) = aG, (59) 



where G is an invariant, (G) = G, called the Gaussian invariant or 

 Gaussian curvature. If w = 4 equation (53") may be written 



\b\ + \c\ = aG, (59') 



and in higher dimensions we have, from (57') , 



2i \ib\ = aG, (59") 



where \b\, \ c \, \ ib \ are the determinants formed of the terms. 6rs, 

 Crs , ibrs • In case n = 3 we have simply | 6 | = aG. 



25. The Vector Second Form. In three dimensions we 

 construct a form, 



;/' = ZbrsdXrdXs , 



