SURFACES IN HYPERSPACE. 



295 



where the symbol V applies to the variables X and (X). Next, 

 XVX = [V1V1V2 + V1V2V2 + ViV Vo] yi-(X)y2-(X) 



may be obtained by interchanging the first and second extensive 

 magnitudes. And 



XXV = [V1V2V1 + V1V2V2 + V1V2V] yi-(X)y2«(X) 



follows from another interchange. Now 



VXX + XVX - XXV = 2ViViV2yi-(X)y2-(X) 



+ [V V1V2 + ViV V2 - ViV2V]yi'(X)y2-(X), 



because V2ViV2yi*(X)y2'(X) and ViV2Viyi'(X)y2«(X) are the 

 same. Thus far V has denoted differentiation by x. But Vz = 

 V^yVj/ = Vy(V). The terms in the bracket on the right may 

 therefore be WTitten 



V V1V2 = Vy(V)ViV2yi-(X)y2-(X) = VyVyVyi (VXX), 



and so on; hence 



VXX+ XVX - XXV = 2VVy(XX)-Vyc + VyVy: [(VXX + XvX 



- XXV)] -Vyc 



or 



2vvy = [VXX + XVX - XXv]-Vj,xc-(A-i) + VyVy: 



[(VXX+ XVX - XXV)]-A-i). 



The elements of this triadic are (compare 33) 



dXrdXs 



'P5 



da 



rq 



+ 



da.rq da,. 



d.i'r dx 



dx„ 



c.M"'^) 



+ 2 



pqn 



d{aqn) d{apn) _ d{apg) 



= 2 SpgTpta'"^^ 



r s 



dyq dyr. 



- 2 Spg„TrpT«9(a^"'0 



7rpT.3(a<"'0 



P 9 

 n 



