272 WILSON AND MOORE. 



perfectly flexible inextensible membrane. There is no restriction to a 

 rigid surface in space and none upon the number of dimensions in 

 which the surface may lie; we work entirely on the surface itself. 

 Hence all the results which Ricci obtained in Part I of his Lezioni 

 are true without any modification of the proofs or the interpretation 

 in any number of dimensions. 



3. Quadratic differential forms. When we wish to interpret a 

 manifold defined by a binary quadratic differential form as a surface 

 in space we have to introduce a set of variables such that 



« 



ds- = ^ij-aadxidxi = dyi^ + dy-i^ + dy^"^, i,j = 1, 2; 



and it is the determination of this set of variables which leads to the 

 second fundamental form. It is a fundamental proposition in the 

 theory of binary quadratic forms that such a form may be written as 

 the sum of three squares. Hence for the interpretation of a binary 

 differential form as a surface, three dimensions are sufficient. When 

 the theory of the ternary differential quadratic form is studied with 

 reference to its reduction to a sum of squares, it is found that in general 

 six variables are needed. Hence to interpret the theory of the ternary 

 form we must in general go to a spread V^ of three dimensions in Sq. 

 It is clear from this that the theory of the F3 in S4 does not correspond 

 with the theory of any but a very special class of ternary forms. 

 Hence from the point of view of the quadratic form the theory of 

 surfaces does not generalize very simply. In general for a quadratic 

 differential form in k variables the reduction to the simi of squares 

 may require k{k — l)/2 variables, and the minimum niunber of addi- 

 tional variables required, above k, is called the class of the form.^ 



When in our special case of a binary quadratic form, we wish to 

 interpret the form as a surface in Sn, we have to determine n variables 

 y so that ds'^ = 2i dyi^, i = 1, • • -,71. The determination is made 

 by the properties of systems of partial differential equations, in par- 

 ticular complete systems. This has been accomplished by Ricci in 

 the general case and his result is stated in a theorem.^ 



5 Ricci, Lezioni, Introduction, Chap. 4. 



6 Ricci, Lezioni, pp. 90-91. Ricci has also treated the more general question 

 of a variety of n dimensions immersed in a variety (not a Euclidean space) of 

 n + VI dimensions and the transformation of ^arsdxrdxs, r, s = 1,. . ., n, into 

 ^Cuvdyudyv, u, v= 1,..., n -f m. Rend. R. Ace. Lincei, (o) 11, 355-362 

 (1902J. 



