42 PROCEEDINGS OF THE AMERICAN ACADEMY. 



so that we have 



\ dx l dyi J 



= c w I 



di+JCLpq 



d&dyi ' 



an equation which not only shows that I, if it be a polynomial, is 

 isobaric, but in other cases is commonly used to define what is meant 

 by isobaric with the given system of weights. We shall speak of w as 

 the weight of the invariant even when it is not a polynomial. 



The proposition holds also for covariants if, in the case of covariant 



Qa+ ••■11 



differential expressions, we attribute to r the weight, with re- 

 spect to Xi, — a, and if, in the case of covariant differential forms, we 

 attribute to dx^ the weight one, to dxj, j ^ i, the weight zero, with 

 respect to x%. 



Proposition 20. An invariant may or may not be homogeneous; 

 but if not, it is a mere sum of invariants which are homogeneous. 



This is the counterpart of proposition 5, page 19, and the proof is 

 similar in the two cases ; for, as noted above, page 40, the as and their 

 derivatives are linear in the a's and their derivatives. So that if we 

 represent by G n (a) the terms of 1(a) of degree n, the corresponding 

 part of 7(a), namely G n (a), will be of degree n in the a's and their 

 derivatives. 



This proposition may be extended to both kinds of covariants, for 

 the d%'s are linear in the dx's ; and again, as also noted above, the de- 

 rivatives of u with respect to the x's are linear in the derivatives of u 

 with respect to the £'s; and this statement may evidently be reversed. 



§ 13. Particular Invariants and Covariants. 

 For a differential expression of the second order, 



l{u) = 2 *a ~£- + 2 «* £ + au > 



dxidxj ^n dxi 



certain simple invariants and covariants may be deduced by the 

 following considerations. 



so that 



dxi " ~ ' dxi d£k dxj ~< dxj dik 



— "V _£*: __ ___ — "V 



" ^p dxi d$k dxj "" ** 



d 2 u _ -^ d~£k du ^ dik d£i 6 2 w 



dxidxj ~< dxidXj d$k ~\ dxi dxj d$kdii 



