SURFACES IN HYPERSPACE. 303 



CHAPTER 11. THE GENERAL THEORY OF SURFACES. 



21. Normalization of element of arc. In ordinary surface 

 theory the second fundamental form may be derived ^^ by consider- 

 ing a change of variable from the given or first fundamental form, 



ip — ZarsdXrdXg to ((f) = '^dl/k', 



where r, s have the range 1, 2, and k the range 1, 2, 3. We shall 

 refer to Ricci ^^ for this development and proceed to the case in which 

 we are interested, namely, in which the surface lies not in 3 but in 

 n > 3 dimensions. The proof for this case is similar to Ricci's. We 

 shall treat first the simplest assumption, namely, that n = 4, and 

 shall mention the generalization to n > 4 for the most part without 

 proof. 



To simplify notations we shall use a small amount of vector analysis. 

 A set of values of the variables yi may be written simply as y. A sum 

 of the form l^kVk'Zk is then the scalar product yz. The use of vector 

 analysis is possible and entirely appropriate when operating as now 

 in a Euclidean space of n dimensions. If any question as to the 

 legitimacy of the application of Ricci's rules for the absolute calculus 

 arises we may revert at once to the ordinary form of analysis without 

 vectors by taking components (supposed to be along fixed orthogonal 

 directions) of vector equations and by replacing scalar products by 

 sums. 



We have, then, 



'^kdyi? = dydy = Zarsdxrdxs , k = 1,. . ., 4; r, s = 1, 2, 

 by virtue of some transformation 



Vk = yk{xi,X2) or y = y(.ri,.r2). 

 Now if Yr denote the partial derivative of y by Xr, 



c/y • (/y = 2 rsYr ' Ysdxrdxs , (44; 



22 It is not ordinarily derived in this way. 



23 Lezioni, Part II, Chap. 1. 



