312 WILSON AND MOORE. 



The solution for the 6's gives at once, 



brs = aXrXs+ IJ.(\rK + K^s) + /SX^Xs. (65') 



The determinant of the b's is then 



\b\ = aZAsb^''^ = a(a/3 - m')- (65") 



If we are working with several systems ibrs we have for each a set of 

 invariants ai, I3i, /Xi formed from (65). The second fundamental 

 forms are therefore 



\pi = "^rsWiW + f^iO^r^s + XrXs) + j3i\r\s\dXrdXa. (66) 



The vector fundamental form is 



^ = S,,[aXA. + H^(X.X, + X,X.) + PXrX.] dx4xs (67) 



where a = :Sa.Zi, jx = XinZi, p = S/^iZ,, (67') 



{ running from 1 to w— 2. The vectors a, P, |X are invariant vectors in 

 the normal (/i— 2)-space. From (65'), (67') we have immediately, 



y,, = aX.Xa + |A(X.X, + X,X,) + pX A^. (68) 



Then from (65") and (59") we have, 



G = Xiia^i-^i") = a.p - \y.\ (69) 



Hence the result: The Gaussian invariant G is the scalar product of 

 the vector invariants a and P diminished by the square of the vector 

 invariant |i. 



28. Moving rectangular axes. The elements y^ or yk\T, h = 

 1, 2, . . ., n, are tangent to the surface. If we form • 



I = s.x^'-Vr = ^Xy^'K -n = 2,X(^Vr = ^.x.yc-), (70) 



we have two vectors tangent to the surface. Moreover these are: 

 1°, unit vectors; 2°, mutually perpendicular, 3°, tangent respectively 



