40 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



IV. Change of Independent Variables; Invariants and 



COVARIANTS. 



§ 12. General Properties. 



We come now to change of independent variables and the invariants 

 and covariants of this transformation. A differential expression in the 

 independent variables X\, . 

 variables 



x m goes over, under the change of 



fe{ — Ci(#l> . . . X m ) } 



i = 1, 2, 



m, 



into another of the same order. With regard to the coefficients of the 

 latter, which we may call a, let us note, in the general case, certain facts, 

 sufficient for our purposes. 



Any derivative of order k of u with respect to the x's is a polynomial 

 in the derivatives, of order k and less, of u with respect to the £'s, and 

 in the derivatives of the |'s with respect to the x's, and is linear in the 

 former set of arguments. These facts follow at once, directly for the 

 first derivatives, by mathematical induction for the higher derivatives, 

 from the formula 



d -^-v d£j d 



" ^7~ 



i 



dxi 



dxi d£/ 



Hence the a's are polynomials in the a's and in the derivatives of the 

 £'s with respect to the x's, linear in the a's. The derivatives of the a's, 

 on the other hand, with respect to the |'s, are linear polynomials in the 

 a's and their derivatives with respect to the #'s, with coefficients poly- 

 nomials in the derivatives of the |'s with respect to the #'s, the whole 

 divided by a power of the functional determinant of the £'s with re- 

 spect to the x's, 



J = 



dxi 



<0l 



dx m 



dim 

 dxi 



dL 



dx 



m 



This follows from the formula 





2dXj d -y^ J{j d 



