308 WILSON AND MOORE. 



If i 9^ j, we set 



Zt'ZjIs = I'i/ls, ZfZi\s = — V^s = Vii\s, (56) 



and 



Zilr'ys = — ibrs- (56') 



We can then obtain by the same process as before, 



ihrst — ibrts = ^j=i Kprs^ ii\t — ibrt^ji\s), (O'j 



(pr,st) = S"^! (ibps-ibrt — ibpfibrs)- (57') 



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

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



Vrs — Vsr = 2pg (6prCgs — bpgCqr) A^^'^ U = 4, (58) 



Vji\Ts — Vji\sr + 2;=i" {vij\TVli\s — Vij^sVli\r) 



= Sp^afp^) {ibpr.jbgs - ibps.jbgr)]. (58') 



Tn the case of a binary (first) fundamental form (p = '21arsd.Vrd,Vs , 

 the Riemann symbol (pr, st) 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 n = 4 equation (53") may be written 



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



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



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



where \b\, \ c \, \ ib \ are the determinants formed of the terms 6r», 

 Cts , ibrs ■ In case n = 3 we have simply | ^ | = aG. 



25. The Vector Second Form. In three dimensions we 

 construct a form, 



