280 WILSON AND MOORE. 



of the coefficients of (3). Let M~^ and Mc be the reciprocal and 

 conjugate. Then, 



X = M«y, y = M-^-x, u = vM-\ v = U'M, 



provided the products are defined as usual and the dot is used in the 

 sense of Gibbs. The transformations of X and XY into (X) and (XY) 

 are 



X-x= X-M-y = (X)-y, (X) = X-M, X = iX)'M-\ 



XY:x| = XY:[M-y M-il] = (XY):yil, 



(XY) = XY:MM, XY = (XY):M-iM-i. 



The double products (containing two dots) are to be interpreted as 

 indicating the union of corresponding elements, that is, 



XY:MM = [X-M] [Y-M] and XY:x| = [X-x] [Y-|]. 



The expression XY:MM may be written also as Mc*XY«M. The 

 use of a formal product of the dyad type XY for any system of the 

 second order is legitimate because the systems are linear. 



8. An adjoined quadratic form. Now if we have a given funda- 

 mental quadratic form 



ZiijdijXiXj, dij = dji, \fl) 



the transformation of the coefficients an will be the same as that of the 

 elements Xij found above. It is for this reason that we say that the 

 system Xij transforms covariantly with an or with the given form 

 (8) ; and we shall further say that a system of X's with any number 

 of (lower) indices which transforms according to (7) or (7') is a covariant 

 system. The simplest case, given by (5) or (5'), shows that the covari- 

 ant system of order 1 is transformed like the contragredient variables 

 in (4). 



If now we have a system, which we may denote by X^ *' , instead of 

 by Xi, which transforms like the cogredient variables, or in an analo- 

 gous manner, we call the system contravariant; ^^ that is if, 



13 Grossmann employs Greek letters mth lower indices to designate a contra- 

 variant system instead of Ricci's letters with upper indices. A trial of this 

 notation convinces us that however awkward Ricci's notation appears it is 

 more convenient than that of Grossmann; especially in \aew of the principal 

 of duality (§ 9), the lack of correspondence between Greek and Roman letters, 

 and the undesirability of immobilizing alphabets in a definite sense. 



