DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 485 



Now, by (65), 



V^ = ty, 



so that 



J&- ( \ ty- v a *_ 



v " ae<r> "**' a) ae<r = ae<r" 



and, therefore, for all values of <r, 



Hence, 



................ (68). 



We now have 



and i/ therefore satisfies the index-equation. 



The fact that the invariant i|> satisfies the form-equation is the justification of the 

 statement made earlier ( 131), that the application of the Jacobian operation enables 

 us to obtain the successive integrals of the subsidiary equations necessary for the 

 construction of the general solution. 



Functional Completeness of the Set of Concomitants. 



137. A set of concomitants will be considered functionally complete when any 

 concomitant whatever can be expressed as an algebraical function of members of the 

 set ; and this we shall prove to hold of the aggregate of invariants and covariants 

 which have been obtained in Sections II., III., V. 



Let a concomitant $ have as elements entering into its expression u, u\ u u ..... 

 U M , where r is less than n ; v f) v 1 ,,. . . . , v "^, for values 2, 3, . . . , n 1 of p, where 



