DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 431 



save as to a power of z, the number of the covariants of this type is (with the 

 reservation of 77) unlimited ; and in the case when the second form is the canonical 

 form, so that Qj and Q 3 vanish, all the covariants thus obtained for this form of 

 quantic are purely identical, that is, they do not involve the coefficients Q. 



Identical Covariants in the Associate Variables. 



68. Two kinds of identical covariants are possible. First, there are those which 

 involve only a single one of the set of dependent variables ; and the aggregate of 

 these, for each of the associate variables, is similar to that just given for the original 

 dependent variable. Second, there are those which involve more than a single one of 

 the set of dependent variables, and which, therefore, may be called simultaneous ; but 

 it will appear (see 72) that this class need not be retained, for they can all be 

 derived from proper invariants and covariants by purely algebraical processes of 

 multiplication and the like. Hence the former class alone requires to be retained. 



To find all the covariants depending on the associate variable of general rankp 1, 

 the process is the same as before ; we take the variable in connexion with the normal 

 form of the fundamental differential equation and, denoting it as in (xv.) by v p with 

 index %p (n p), we find a quadriderivative function 



which is covariantive, or say 



with index (np p 3 2). When the series of Jacobians of v p and the functions 

 V are formed in succession, they are found to be 



t;', > . . . . (xxi.) 



and the general term in the succession is 



v f . . . . (xxiii.) 



The index of the covariant V A , is s (np p 2 2). The number of covariants in 

 the succession is (with a reservation similar to that in 77, post) infinite when the 

 associate v p is regarded as the variable of an unretained associated differential quantic ; 



