DKHIVATIVKS ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 489 



It IB necessary, then, to show that U*U M is covariantive ; if it be so, it must have 

 its index equal to n |(n 1 ) = (3 n), and so the relation 



= 



must be satisfied. Now, by (58), we have 



u*{l + i( 



and therefore, by (57), 



u s u<"> = 



/rft\ (S-) 



="' (2) . 



showing the covariantive nature of the function. 



A similar conclusion as to limitation of number holds with regard to the identical 

 covariants in the associate variables. 



MDCCCI.XXXVIII. A. 3R 



