1888.] associated with Linear Differential Equations. 



313 



variants of the cubic and the quartic in forms which differ from the 

 canonical form herein adopted. 



In the third section derived invariants are obtained, all in their 

 canonical forms ; they are derived from the priminvariants by one or 

 other of two processes called the quadriderivative and the Jacobian. 

 The irreducible invariants are ranged in classes according to their 

 degrees. The quadrinvariants consist of n — 2 functions, 



and of n — 3 independent functions of the form 



and every class of invariants of degree higher than the second 

 contains n— 2 invariants, each in that class associated with one of the 

 priminvariants in successive derivation according to the law 



o*,r = ^G(7e' £r ,r-i-r(^+i)e; / e air _ 1 . 



Propositions relating to the dependence of the derived invariants are 

 proved in the section ; and simpler equivalent forms are obtained later 

 in the memoir. 



In the fourth section covariants are discussed. The transformation 

 of the dependent variable in the second section shows that, with the 

 adopted definition of in variance, viz., reproduction save as to a power 

 of z', the dependent variable is a covariant. A set of dependent 

 variables, associate with the original dependent variable, is obtained 

 by the application of a theorem due to Clebsch. Denoting these by v 2 , 

 fjjj, . . . ., Vn_i, for the untrans formed equation, and by £>, . . . ., 

 t n _ h for the transformed equation, we have 



so that these associate variables are covariants. The variable v p satisfies 

 a linear differential equation of order ^ , n _^ j ; and, in particular, 



Vn—i the variable of Lagrange's "adjoint" equation. The follow- 

 ing inferences relating to these variables and equations are made : — 



(<z) The dependent variables form a complete system, that is, 

 functional combinations of them, similar to those by which 

 they are obtained, are expressible in terms of members of 

 the system ; 



(/3) The associate linear equations in variables which have the 



same index are mutually adjoint ; 

 {7) The invariants of the associate linear equations are expressible 



in terms of the invariants of the original equation. 



