310 WILSON AND MOORE. 



Instead of carrying n — 2 second fundamental forms \pi we shaH 

 combine them into a single vector second fundamental form 



^ = ZnAl + Z2tA2 + . . . . + Z„_2l/'n-2 = '^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 dxri X^''K§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 



2,X^^^Xr = 2,XrX(^) = 0. (62) 



If we impose the further condition 



ZrXrV'-^ = 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)''+VXr+i ,. it being 

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

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



S,,X(^^X'^)a.3 = 1 = S„p2(-l)^+^Xr+iX3+ia„ . 



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



