SURFACES IN HYPERSPACE. 285 



covariant system of order m to get a contra variant system of order p. 

 The proofs of these facts are as above for the special case when the 

 orders are equah 



The process by which covariant and contravariant systems are 

 combined in (17), (18), (19) to obtain a system of order equal to the 

 difference of the two orders is called composition}^ (In the definition 

 we have placed the common indices at the end. We may generalize 

 the definition by distributing the indices in any way. Thus (15) 

 and (16) may be considered as cases of composition of a system of 

 order 2m with one of order m.) 



Composition is very simple in matrical notation. 



X-Y° = X-A-i-Y = XY:A-i 



is clearly invariant. If proof were needed, we could write 



XY:A-i = [(XY):M-iM-i]:[MM:(A-0] = (XY):(A-0. 



We have simply to take into account what elements the dots actually 

 unite in the multiplications. 



11. Mutually reciprocal^® n-tuples. For any covariant sys- 

 tem Xr, consisting of n functions of the variables xi, x^,. . .Xn, and the 

 dual system X^''^ we have found by (15), (16) the relations 



X(^) = S.a(")X,, X. = S,a„X(^). (20) 



Suppose that we have n systems iXr, oXr, . . . , nXr and the corresponding 

 dual systems iX*''^ -iK^'^\. . ., nX^""'. The n systems fXr will be called 

 independent if the determinant | jX^ | does not vanish. As | a^g 1 9^ 0, 

 it follows at once that the dual systems are also independent. We 

 may define contravariant systems i\'^''> in terms of jX^by the equations 



S,..-X'(«).,X.= e.,, ers= I il'ij; (21) 



15 Composition is a sort of inverse of multiplication in that the result of 

 composition is to subtract the orders of the factors, whereas multiplication 

 adds the orders. Composition itself may be regarded as a species of multipU- 

 cation in the general sense in which Gibbs used the term, and has close 

 analogies with regressive multiplication or with the inner product as defined 

 by G. N. Lewis, Proc. Amer. Acad. Arts Sci., 46, 165-181 (1910), also (with 

 Wilson) Ibid., 48, 389-507 (1912), especially § 29. 



16 See note 14. 



