SURFACES IN HYPERSPACE. 275 



we find, on working out the details, 



yd) = x(i) ^ + Z<2) ^ , F(2) = A'C) ^ + Z^^) ^ • (2) 

 <9a-i ' dx-i dxi dx^ 



We can write the transformation from X to Y in the two cases in the 

 forms 



It = 2s Xs - — ' (1 ) 



yW^v^^YW^. (2') 



dXs 



The system Xi, X2 is said to be transformed covariantly . The 

 system Y^'^\ Y^"^^ is said to be transformed contravariantly. Further- 

 more if we have any system of X's which is so defined that the 

 transformed system follows the rule (1'), it is called covariant and 

 the members of the system are denoted by lower indices; whereas if 

 the system follows the rule of transformation (2'), it is called contra- 

 variant and the members of the system are denoted by upper indices. 

 The system of differentials dx\, dx-i, by the formula for the total 

 differential, is seen to follow rule (2') and therefore the system of 

 differentials of the independent variables are the members of a contra- 

 variant system; but we observe that the indices are lower, in con- 

 formity with ordinary practice, and not upper as the rule here would 

 require. 



If we were dealing with more than two variables Xi, xi, we should 

 still find that the transformation of the differential 



, Xidxi + .Yof/.co + + XndXn 



led to the rule (!'), where s runs from 1 to n, for changing X's into 

 y's ; and that the transformation of the system of equations 



dxi dx-2 dxn 



j^ii) ~ x(2) ^ ^ x^^) 



led to the set of equations (2'), where s runs from 1 to n. 



