434 MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



(iv) Similarly the Jacobian of <r (u) l and any invariant H p of index p is composite; 

 for, denoting it by V, it is easy to show that 



2 p p ^(n) 2 -f (n - 1) teV = {2cr + 2 - n - 1} 0,(u) l p (u\, 



whence it follows that V is composite. 



In the same way it may be proved that the Jacobian of 6, (u) l and f (u) t is com- 

 posite. (See also 43.) 



71. The general result, therefore, is that all the covariants, which can be obtained 

 by the methods used, are expressible in terms of the identical covariants u, U 2 , U 3 , . . . 

 and of the mixed covariants of the first order 6, (u) l ; all of these are proper, i.e., they 

 cannot be expressed in terms of invariants and covariants of earlier rank, but all the 

 mixed covariants can be expressed in terms of any one of them and of invariants. 



Mixed Covariants in the Associate Variables. 



. ,' f __ > 



72. The aggregate of mixed covariants, which involve in their expressions only a 

 single associate variable, is for each associate composed similarly to the corresponding 

 aggregate in the original variable ; and all the covariants, which can be obtained by 

 the methods employed, can be expressed in terms of the identical covariants 

 "Up, Vj, i8 , V pi3 , . . . , and of mixed covariants 



6. (v p ) } = 2er@y p + p(n-p) v f #. 



of the first order. These mixed covariants are proper, but they can all be expressed 

 in terms of any one of them, and of invariants. 



By retaining as proper covariants one at least of these mixed covariants of the first 

 order in each of the associate dependent variables, we are enabled to dispense with 

 the simultaneous identical covariants ( 68) as being composite. For the simplest 

 simultaneous identical covariant is the Jacobian of two of the dependent variables, 

 6ay u and v p ; and it is easily proved that 



(n - 1) u0, (v p \ -p(n- p) v p e, (u), = 2<r> (r {(n -l)w' f -p (n - p) v f v'}, 



so that this Jacobian is composite. 



The application of the analysis of 60 (which shows that the invariant function 

 obtained by constructing a function for invariants, similar to them in the same way as 

 v p is to u) to covariantive combinations of more than two of the associate variables 

 taken simultaneously shows that such combinations can be expressed in terms of the 

 covariants already obtained, and are therefore composite. 



73. It has been shown, in (iii) of 70, that successive Jucobians of d.(u), and u are 



