488 MR. A. E. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



fractions in the priminvariants alone, the part played by the class (iv) is in making 

 equal to one another the individxial fractions in these three principal sets. 



We thus have, by means of the functions previously obtained, the full number of 

 subsidiary integrals necessary to construct the most general solution of the form- 

 equation ; and it follows that any concomitant can be algebraically expressed in terms 

 of the concomitants previously given. 



Hence the aggregate of the concomitants (or their simplified algebraical equivalents), 

 obtainable by the quadriderivative and Jacobian operations from the priminvariants 

 and the dependent variables, is functionally complete. 



Limitation in the Number of Identical Covariants. 



138. This for particular cases has already ( 76-78) been indicated; without 

 entering at present on the details of the general case, it will be sufficient to obtain 

 the general result, which, by means of the result of 1 32, can be simplified. In fact, 

 IT,, of grade n, can be replaced by a function the first term of which is either UU M or 

 u z u ( *\ according as the grade n is even or odd ; and our present purpose will be 

 effected by showing that U S U M , which will include both cases, is covariantive. For, 

 since the differential equation 



<">+ V - Q r t*'"- r) = 



r = r\ n rl ^ 



is permanently true, we shall have 



a r v" _? ! n ( - r > 



a covariant, if U~U M be a covariant. Now, as has been implicitly proved in the last 

 paragraph, this covariant is expressible in terms of invariants and covariants already 

 obtained, the identical covariant of highest grade in such an expression being U,_ 3 ; 

 and the expression is therefore an equivalent for U S U M . On the other hand, viewed 

 as an identical covariant, U Z U M differs from !! (or U B , in the case of n even) by an 

 aggregate of terms each of which can be resolved into factors of lower grade ; and 

 therefore, since the aggregate is covariantive, on the hypothesis of the covariantive 

 property of U*U M , it is expressible in terms of identical covariants of lower grade. A 

 comparison of the two expressions thus obtained for U?U M gives U in terms of 

 covariants of Jower grade, so that U is reducible ; and all succeeding identical 

 covariants are also reducible. 



