DERIVAT1VKS ASSOCIATED WITH I.INKAU DIFFKKKNTIAL EQUATIONS. 435 



composite ; the same holds of those formed with 0, (u\ and v r For, denoting the 

 first of such Jacobians by T, we have 



whence by means of the expression for 0,(u). 2 , in (iii) of 70, which is a composite 

 covariant, it follows that 



= {cr + 1 - *( - 1) W)i[( - l)ut/, -p(n -J) vO 



But the simultaneous identical covariant on the right hand side is composite ; hence 

 T is composite. So for the others in succession. 



Lastly, as in (iv.), the Jacobian of any two mixed covariants of the first order in 

 any variables is composite. For taking 



it is easy to show that 



( _ i) MW + {p+ 1 - \p(n -p)}0,(v f ) l 0.(u) 3 

 = 2<r- n 



It has just been proved that the second factor on the right hand side is composite ; 

 and therefore W is composite. 



It follows, from all these results and the propositions proved in 48, that all the 

 simultaneous identical covariants are composite. 



74. The general conclusion as to the aggregate of covariantive concomitants is 

 thus : 



The aggregate of proper concomitants associated with a differential quantic or a 

 differential equation is composed of three classes 



(A.) INVARIANTS, being functions of the coefficients of the quantic or equation ; 



(B.) IDENTICAL COVARIANTS, being (i.) functions of the dependent variable and 

 its derivatives (which when of sufficiently high order change into mixed 

 covariants if associated with a differential equation) ; and (ii.) functions of 

 the associate dependent variables and their derivatives ; but any function 

 involving more than one dependent variable is composite ; 



(C.) MIXED COVARIANTS, being functions of the dependent variables, original 

 and associate (but not involving more than one dependent variable), and 

 of the invariants and their derivatives. 



3 K 2 



