SURFACES IN HYPERSPACE. 



295 



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



XVX = [ViViVo + V1V2V2 + V1VV2] 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 Vx = 

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

 therefore be written 



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]-V,Xc-(A-i) + VyVy: 



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



The elements of this triadic are (compare 33) 



dXrdXs 



+ 2 



'P9 



pqn 



dOrq dcirq _ do,, 



dx,. dx, dx„ 



d{('qn) d(apn) d{apg) 



= 2 Zpqypifi 



dijp dyq 



r s 



7rp7sg(a""0 



dy 



{qp) 



L 9 J 



— 2 2,,,„7rp7s5(a("')) 



P Q 

 n 



