310 WILSON AND MOORE. 



Instead of carrying n — 2 second fundamental forms ypi we shall 

 combine them into a single vector second fundamental form 



^ = Zil/'i + Z2\l/2 + + Zn-2'/'n-2 = I^YrsdXrdXs (60) 



in the normal {n — 2)-space. If the vector form is regarded as given, 

 the surface may be regarded as not fixed relative to arbitrary axes in 

 space; only the shape of the surface is determined. 



26. Canonical orthogonal curve systems. ^^ We have defined 

 a set of curves on a surface by the differential equations obtained by 

 equating the ratios dxr'. X^'')(§13; here r = 1, 2). The quantities X^*") 

 are the contravariant system defining the curves; the dual system X^ 

 is a covariant system which may also be regarded as defining the curves. 

 We have defined perpendicularity and hence orthogonal systems of 

 curves. If we give the definition 



X(^) = ^ (61) 



ds 



we have a special system X'''^ which satisfies the relation 



2,X,X(''^ = 1, (61') 



and we shall here assume this system. The orthogonal curves defined 

 by X^*"^ or Xr will satisfy the relation 



2A^'->Xr = 2,X,X('> = 0. (62) 



If we impose the further condition 



SXXM = 1, (62') 



we have a set of relations which will determine X^*"^ or Xr except for 

 sign (the arbitrariness of sign corresponds to the two opposite direc- 

 tions along the curve). For from (62) X^'') = (-l)''+VV+i » it being 

 understood that all even values of the index are equivalent and all 

 odd values also equivalent. Then from (62'), 



2„X(^^Xf^)a„ = 1 = S„p-(-l)'-+^Xr+iXs+ia„. 



28 Ricci, Lezioni, p. 106, and Atli. R. 1st. Veneto, (7) 4, 1-29 (1893). 



