312 WILSON AND MOORE. 



The solution for the 6's gives at once, 



brs = a\r\s+ m(>'7-Xs + X,X,) + /3XAs. (65') 



The determinant of the b's is then 



\b\ = aXAsb^^'^ = a(a/3 - /x'). (65") 



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

 invariants af, jSf, Mi formed from (65). The second fundamental 

 forms are therefore 



l/'i = I,rs[ai\r\ + f^iiK^^s + XAs) + (3i\r><s]d^rdXs. (66) 



The vector fundamental form is 



^ = 2„[aXA« + |A(X,X, + X,X,) + PX,X,] dxrdx, (67) 



where a = Sa.-Zi, |i = S/xiZ», p = SitJiZ,-, (67') 



z running from 1 to n— 2. The vectors a, P, |x are invariant vectors in 

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



y,3 = aX.Xs + |A(X,X, + X,Xs) + PX,X,. (68) 



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



G = Si(ai/3i - Mr) = a.p - jjl^. (69) 



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

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

 invariant p.. 



28. Moving rectangular axes. The elements y^ or yh\r, h = 

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



I = SA<^Vr = 2,Xry«, -n = ^.XO-Vr = ^r\rY^''\ (70) 



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

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



