478 MR. A. a. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



covariants in each of the associate variables there are pairs of equations exactly 

 similar to (62) ; and the equations which determine the invariants are 



T 2 



130. The index-equation involves operations the only effect of the application of 

 which is a change in the numerical coefficients of the various terms in the concomitant 

 to which they are applied. The form-equation involves operations which replace any 

 derivative of an element of the function by the derivative of that element of order 

 next lower ; and, if the aggregate of the orders of the various derivatives entering 

 into the composition of any term be called the grade of that term, the effect of the 

 operations in the form-equation is to replace such a term by a set of terms of grade 

 less by unity. 



From the facts that both the characteristic equations satisfied by a concomitant 

 are linear and that the algebraico-differential operations which occur in them leave a 

 term unaltered in order in the variables and degree in the invariants, coupled with 

 the preceding conclusion as to the modification of the grade of the term, we can 

 derive the inference that every concomitant, if not irreducible, can be resolved into 

 sums and products of irreducible concomitants each of which has the property of 

 being an aggregate of terms such that, for the aggregate, the orders of the different 

 terms in the dependent variables are separately the same throughout, the degree in 

 any invariant is the same throughout, the dimension-number for every term is the 

 same, and the grade of every term is the same. For instance, 



is a concomitant of index 2p. + 2 (n 1) p (n p) ; the different terms are 

 resoluble into products of concomitants each of which has the preceding properties. 

 Hence for every irreducible concomitant there are three kinds of numbers which are 

 characteristic, viz., the separate orders in the different dependent variables, the 

 separate degrees in the different invariants, and the grade of the concomitant ; and a 

 knowledge of these numbers gives the dimension-number, and thence the index, of 

 the concomitant. 



Applications of the Differential Equations. 



131. Example I. The identical covariants which are functions of u. 



In order to obtain all such identical covariants, it is necessary to obtain the most 



