476 MR. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT- 

 we have 



. . (60). 



126. Now, in discussing invariants and covariants in their canonical forms as solu- 

 tions of partial differential equations, we may, so far as they are functions of the 

 coefficients Q of the original differential equation, cease to consider them as explicit 

 functions of these quantities, and can consider them as functions of the priminvariants 

 and of differential coefficients of the priminvariants ; for each of the coefficients Q 

 can be expressed uniquely in this last form. Thus we have 



Qs = s> 



and so on. 



Form-Equation and Index-Equation of a Concomitant. 



127. We may therefore define the most general covariant possible when in its 

 canonical form as a function of (i) the dependent variables, original and associate, 

 and of their differential coefficients, and of (ii) the priminvariants and their diffe- 

 rential coefficients, which is such that, when the same function is formed for the trans- 

 formed differential equation in its canonical form, the relation 



u (m) 



t 



is satisfied, X being the index, and the bracketted numerical exponents denoting 

 differentiation of corresponding order with regard to the respective independent 

 variables. 



128. As an example, sufficiently indicative of the general case, consider identical 

 covariants which are functions of u and its derivatives alone, so that we may write 



* (u, ...) = 



?, V>V, ) 



