DERIVATIVES ASSOCIATED WITH LINEAR 1)1 Fi I III NTI AL KQ NATIONS. 439 



First for the case of the derived invariants, we may take (xi.) as n representative, 

 viz.: 



To transform this we write 



e 

 



BO that 



Hence, 



O 

 If then we write 



this equation comes to be 



'. //^^r . 



and therefore 



or by re-substituting we have 



*"' 



'" 



_ 



the symbolical form for the derived invariants. 



Similarly, for the identical covariants, it may be proved that 



and that 



Corresponding expressions may easily be found for the mixed covarianta. 



