SURFACES IN HYPERSPACE. 309 



from the symmetric system brs and call it the second fundamental 

 form of the surface, defined by the first form (p as one of a class of ap- 

 plicables. and thus we haxe the surface defined by (p and i/' as a rigid 

 surface. In higher dimensions we construct n — 2 forms ypi , yp2,- ■ ■, 

 ypn-2 (two, when n = 4) from the n — 2 symmetric systems ibrs and this 

 set of n — 2 forms are the second fundamental forms. The different 

 forms are not, however, entirely determined because with a different 

 choice of the unit vectors Zi , Zo , . . . , z„_2 in the normal (/( — 2)-space, 

 there is a change in the quantities ibrs ■ The set of forms tpi taken 

 with <p and the generalized Gauss-Codazzi relations (57), (57'), (58), 

 (58') will determine the surface as a rigid surface in ^/-dimensions.^'^ 

 We shall not, however, enter into a proof of this proposition which is 

 adequately treated by Ricci and not important for our work. 



It has been stated that the systems Jjrs are not entirely determined. 

 The relations between different systems may be illustrated in the case 

 n = 4. We had yrs = 6rsZ + r„w, that is, brs and €„ are respectively 

 the components of y^ along z and along w. If a new choice z', w' 

 were made, the quantities b'rs , c'rs would be the components of yrs 

 along z', w'. Hence the relations 6'„ , c'rs and brs , Crs are those which 

 express a rotation, namely, 



brs = b'rsCOSd — c'rsSmd, Crs = b' rsS'md + c'rsCOHd. 



In general if n > 4, the relations between ife„ and {b'n must be those 

 which determine an orthogonal transformation in the normal (n — 2)- 

 space, since ibrs and ib'rs are merely the components of Yrs along two 

 different systems of orthogonal lines in that space. This amount 

 and only this amount of indetermination is involved in our set of 

 second fundamental forms ypi . 



27 The generalization of the Gauss-Codazzi equations to hypersurfaces (for 

 which the element of arc is a quadratic form of class 1) has been obtained by a 

 number of authors, including Ricci, and do not contain the i/'s which by Ricci's 

 development {Lezioni, Introduction, Chap. 4) are necessary in case the class 

 of the surface is greater than one. Levi (loc. cit., note 2) develops the theory 

 of surfaces in a very different way. For him the element of arc is apparently 

 not a particularly fundamental form but merely one of a set of fundamental 

 forms. That is to say where we, following Ricci, have a first fundamental 

 form (which is scalar) and a second fundamental form (60) which is vectorial, 

 both quadratic, Levi has an infinite set of {n + j')-linear forms F^^ (n, v = 

 1, 2, . . . ) of which the first, Fn, is ds"-. He shows that the problem of finding 

 the absolute invariants reduces to that of finding the simultaneous invariants 

 of the forms F^^ and he finds five special invariants Ai (i = 1,. ., 5) which 

 form a complete system of independent invariants. Our analysis leads us 

 very naturally to five invariants which are equivalent to Levi's (see note 39). 



