DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 425 



rank ; in the case when n is even there is one dependent variable of self-complementary 

 r;ink. Each pair has the index of the factor power of z different from that for any 

 other pair. The simplest case of this arrangement is that which combines in a pair 

 the original variable y and the variable t p _ l of the "adjoint " equation of LAORANOE ; 

 and the two dependent variables have the same functional transformation. Since t f _ } 

 is thus a covariant for the transformation, HALPHEJJ infers that the invariants of the 

 equation satisfied by t f _ l are invariants of the original equation, and he has used this 

 proposition to construct 6 3 (see G). 



Reference to Invariants. 



GO. Before proceeding further with the associate variables, there is one point which 

 may be considered conveniently here. The quantities y,, y,, . . ., y,, from which the 

 associate variables are constructed, are, all of them, covariants with the same index ; 

 and, in particular, the associate of the first rank is in each case the Jacobian of two 

 of these covariants. Now, when we were considering invariants we were led to new 

 invariants by forming the Jacobian of any two ; and thus there is suggested a new 

 means of forming invariants, if, e.g., for three, which by involution to suitable powers 

 can be made to have the same index, we form the same function as the associates of 

 the second rank are for the special values of the original variable. It may, however, 

 be easily proved that such invariants are composite. For let <I>, , X be three such 

 invariants with the same index 6 (e.g., they might be ej", 0\ 6J", and 8 = \fip) ; 

 then the new function 



n = *, , x 



4>' ' X' 



<&'', ", x" 



Let [*] denote the quadriderivative of I> ; then 



and similarly for the others, so that after substitution for *", ", X" we have 



But the determinant on the right-hand side is 



X 

 X' 



*' 



MDCCCLXXXVIII. A. 



3 I 



